1 Setup
1.1 Load packages
knitr::opts_chunk$set(error = TRUE)library(INLA)## Loading required package: Matrix## Loading required package: foreach## Loading required package: parallel## Loading required package: sp## This is INLA_22.05.07 built 2022-05-07 09:58:26 UTC.
## - See www.r-inla.org/contact-us for how to get help.
## - To enable PARDISO sparse library; see inla.pardiso()library(DClusterm)## Loading required package: spacetime## Loading required package: DCluster## Loading required package: boot## Loading required package: spdep## Loading required package: spData## To access larger datasets in this package, install the spDataLarge
## package with: `install.packages('spDataLarge',
## repos='https://nowosad.github.io/drat/', type='source')`## Loading required package: sf## Linking to GEOS 3.10.2, GDAL 3.4.2, PROJ 8.2.1; sf_use_s2() is TRUE## Loading required package: MASSlibrary(tidyverse)## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.1 ──## ✔ ggplot2 3.3.6 ✔ purrr 0.3.4
## ✔ tibble 3.1.7 ✔ dplyr 1.0.9
## ✔ tidyr 1.2.0 ✔ stringr 1.4.0
## ✔ readr 2.1.2 ✔ forcats 0.5.1## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ purrr::accumulate() masks foreach::accumulate()
## ✖ tidyr::expand() masks Matrix::expand()
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ✖ tidyr::pack() masks Matrix::pack()
## ✖ dplyr::select() masks MASS::select()
## ✖ tidyr::unpack() masks Matrix::unpack()
## ✖ purrr::when() masks foreach::when()data(brainNM)1.2 Utility functions
#' @param code model code as a character string
#' @return name of the loaded DLL
tmb_compile_and_load <- function(code) {
f <- tempfile(fileext = ".cpp")
writeLines(mod, f)
TMB::compile(f)
dyn.load(TMB::dynlib(tools::file_path_sans_ext(f)))
basename(tools::file_path_sans_ext(f))
}
nm.adj <- poly2nb(brainst@sp)
adj.mat <- as(nb2mat(nm.adj, style = "B"), "Matrix")
data <- brainst@dataNote that area ID = 11 does not have any observed events and very low expected cases.
This creates some issues later for the improper models.
filter(data, ID == 11)Code factor versions of ID variables
data$ID <- as.integer(data$ID)
data$IDf <- factor(sprintf("%02d", data$ID))
data$Yearf <- factor(data$Year, unique(data$Year))
data$ID.Year <- data$Year - 1973 + 1
data$ID2 <- data$ID
data$area.year <- interaction(data$IDf, data$Year)
data$id.area.year <- as.integer(data$area.year)prec.prior <- list(prec = list(param = c(0.001, 0.001)))
rho.prior <- list(rho = list(param = c(0, 0.15)))
diagval <- INLA:::inla.set.f.default()$diagonal
brain.st <- inla(Observed ~ 1 + f(Year, model = "rw1",
hyper = prec.prior) +
f(as.numeric(ID), model = "besag", graph = adj.mat,
hyper = prec.prior),
data = data, E = Expected, family = "poisson",
control.predictor = list(compute = TRUE, link = 1))## as(<dgCMatrix>, "dgTMatrix") is deprecated since Matrix 1.5-0; do as(., "TsparseMatrix") insteadsummary(brain.st)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.82, Running = 0.528, Post = 0.0229, Total = 4.37
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.048 0.038 -0.126 -0.047 0.025 NA 0
##
## Random effects:
## Name Model
## Year RW1 model
## as.numeric(ID) Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for Year 500.80 554.54 56.97 335.62 1958.54 NA
## Precision for as.numeric(ID) 59.22 74.05 6.94 37.17 248.24 NA
##
## Marginal log-Likelihood: -824.58
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')names(inla.models()$group)## [1] "exchangeable" "exchangeablepos" "ar1" "ar"
## [5] "rw1" "rw2" "besag" "iid"Use options control.inla = list(strategy = "gaussian", int.strategy = "eb") so that we are doing the same thing as in TMB.
INLA defaults: * Intercept: flat prior * Fixed effects: N(0, prec = 0.001)
brain.st2 <- inla(Observed ~ 1 +
f(as.numeric(ID2), model = "besag", graph = adj.mat,
group = ID.Year, control.group = list(model = "rw1"),
hyper = prec.prior),
data = data, E = Expected, family = "poisson",
control.compute = list(config = TRUE),
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.predictor = list(compute = TRUE, link = 1))2 Model 1: intercept only
R-INLA default is a improper flat prior on the intercept
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
DATA_VECTOR(y);
DATA_VECTOR(E);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
vector<Type> mu(beta0 + log(E));
val -= dpois(y, exp(mu), true).sum();
return val;
}
'dll <- tmb_compile_and_load(mod)## Warning in checkMatrixPackageVersion(): Package version inconsistency detected.
## TMB was built with Matrix version 1.4.1
## Current Matrix version is 1.5.1
## Please re-install 'TMB' from source using install.packages('TMB', type = 'source') or ask CRAN for a binary version of 'TMB' matching CRAN's 'Matrix' packagetmbdata <- list(y = data$Observed,
E = data$Expected)
tmbpar <- list(beta0 = 0)
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c(),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr1 <- TMB::sdreport(obj)
inlafit <- inla(Observed ~ 1,
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"))
mgcvfit1 <- mgcv::gam(Observed ~ 1, data = data, offset = log(Expected), family = "poisson")
summary(sdr1)## Estimate Std. Error
## beta0 0 0.029173summary(inlafit)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.71, Running = 0.192, Post = 0.0056, Total = 2.91
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) 0 0.029 -0.057 0 0.057 NA 0
##
## Marginal log-Likelihood: -794.43
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')inlafit$summary.fixed3 Model 2: ICAR model
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_space);
DATA_SPARSE_MATRIX(Q); // Structure matrix for ICAR area model
DATA_SCALAR(Qrank);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space);
// Note: dgamma() is parameterised as (shape, scale); INLA parameterised as (shape, rate)
val -= dlgamma(log_prec_space, Type(0.001), Type(1.0 / 0.001), true);
PARAMETER_VECTOR(u_space);
val -= Qrank * 0.5 * log_prec_space -
0.5 * exp(log_prec_space) * (u_space * (Q * u_space)).sum();
val -= dnorm(u_space.sum(), Type(0.0), Type(0.001) * u_space.size(), true); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_space * u_space +
log(E));
val -= dpois(y, exp(mu), true).sum();
return val;
}
'dll <- tmb_compile_and_load(mod)
Q <- diag(rowSums(adj.mat)) - adj.mat
Qadj <- Q + Matrix::Diagonal(ncol(Q), rep(1e-6, ncol(Q)))
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_space = Matrix::sparse.model.matrix(~0 + IDf, data),
Q = Qadj,
Qrank = as.integer(rankMatrix(Q)))
tmbpar <- list(beta0 = 0,
log_prec_space = 0,
u_space = numeric(ncol(tmbdata$Z_space)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_space"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr,
control = list(iter.max = 1000,
eval.max = 1000))
tmbfit <- optim(obj$par, obj$fn, obj$gr, method = "BFGS")
sdr2 <- TMB::sdreport(obj)
summary(sdr2, "all")## Estimate Std. Error
## log_prec_space 3.454993755 0.90168419
## beta0 -0.036929365 0.03758052
## u_space 0.116341634 0.05679451
## u_space -0.005569479 0.09728607
## u_space -0.003047975 0.07857896
## u_space -0.012811869 0.10193937
## u_space 0.019197311 0.10817920
## u_space -0.012068927 0.08628529
## u_space -0.047357624 0.08086184
## u_space -0.119175956 0.09817941
## u_space -0.104867950 0.10879945
## u_space 0.006599596 0.08429197
## u_space 0.001514533 0.09252039
## u_space -0.106317999 0.14601458
## u_space -0.136769001 0.10502048
## u_space -0.028572076 0.07155156
## u_space 0.099932294 0.10772916
## u_space -0.113386078 0.11062324
## u_space 0.095412258 0.10147128
## u_space 0.015925590 0.08228574
## u_space -0.097450140 0.08612561
## u_space 0.010179504 0.07498311
## u_space 0.068422967 0.07927720
## u_space -0.017357517 0.08534305
## u_space 0.084443415 0.07571724
## u_space 0.010845101 0.09890691
## u_space 0.027969043 0.06806072
## u_space 0.087735790 0.06953996
## u_space -0.032312032 0.07470064
## u_space 0.038042081 0.08827612
## u_space 0.002791407 0.10506727
## u_space 0.044646923 0.07100095
## u_space 0.006475032 0.11286903
## u_space 0.100590145 0.08345504prec.prior <- list(prec = list(param = c(0.001, 0.001)))
inlafit <- inla(Observed ~ f(as.integer(IDf), model = "besag",
hyper = prec.prior, graph = adj.mat, constr = TRUE,
diagonal = diagval),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
summary(inlafit)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.94, Running = 0.238, Post = 0.0101, Total = 3.19
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.037 0.036 -0.108 -0.037 0.034 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for as.integer(IDf) 65.81 90.27 7.68 39.27 289.83 NA
##
## Marginal log-Likelihood: -818.47
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')inlafit$internal.summary.hyperparinlafit$misc$configs$config[[1]]$theta## Log precision for as.integer(IDf)
## 3.451179summary(sdr2, "fixed")## Estimate Std. Error
## log_prec_space 3.454994 0.9016842inlafit$summary.fixedinlafit$summary.random[[1]]plot(inlafit$summary.random[[1]][ , 2], summary(sdr2, "random")[-1, 1])
abline(0, 1, col = "red")
plot(inlafit$summary.random[[1]][ , 3], summary(sdr2, "random")[-1, 2])
abline(0, 1, col = "red")
3.1 INLA with custom Cmatrix
diagval <- INLA:::inla.set.f.default()$diagonal
inlafitC <- inla(Observed ~ f(as.integer(IDf), model = "generic0", Cmatrix = Q,
hyper = prec.prior, diagonal = diagval, constr = TRUE),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))These match:
summary(inlafit)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.94, Running = 0.238, Post = 0.0101, Total = 3.19
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.037 0.036 -0.108 -0.037 0.034 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for as.integer(IDf) 65.81 90.27 7.68 39.27 289.83 NA
##
## Marginal log-Likelihood: -818.47
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafitC)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.79, Running = 0.227, Post = 0.01, Total = 3.03
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.037 0.036 -0.108 -0.037 0.034 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for as.integer(IDf) 65.81 90.27 7.68 39.27 289.83 NA
##
## Marginal log-Likelihood: -818.47
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')By setting constr = TRUE above, INLA infers the rank deficiency of one for the Cmatrix.
grep("rank", inlafitC$logfile, value = TRUE)## [1] " computed/guessed rank-deficiency = [1]"3.2 With constr = FALSE
If I change constr = FALSE, for the ICAR model R-INLA still calculates the rank deficiency of 1 based on the number of connected components of the graph.
But for the "generic0" version, R-INLA no longer detects rank deficiency of the Cmatrix.
inlafit_unconstr <- inla(Observed ~ f(as.integer(IDf), model = "besag",
hyper = prec.prior, graph = adj.mat, constr = FALSE,
diagonal = diagval),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
inlafitC_unconstr_bad <- inla(Observed ~ f(as.integer(IDf), model = "generic0", Cmatrix = Q,
hyper = prec.prior, diagonal = diagval, constr = FALSE),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep("rank", inlafit_unconstr$logfile, value = TRUE)## [1] " rank-deficiency is *defined* [1]"grep("rank", inlafitC_unconstr_bad$logfile, value = TRUE)## [1] " computed/guessed rank-deficiency = [0]"Consequently, the precision estimate is different:
summary(inlafit_unconstr)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.81, Running = 0.226, Post = 0.00992, Total = 3.05
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.037 17.676 -34.681 -0.037 34.608 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for as.integer(IDf) 67.52 93.54 7.81 40.08 299.39 NA
##
## Marginal log-Likelihood: -812.89
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafitC_unconstr_bad)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.83, Running = 0.217, Post = 0.00977, Total = 3.06
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.031 17.678 -34.68 -0.031 34.618 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for as.integer(IDf) 167.00 361.37 10.99 74.06 909.84 NA
##
## Marginal log-Likelihood: -811.87
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')We need to specify f(..., rankdef = 1) for the results to align
inlafitC_unconstr <- inla(Observed ~ f(as.integer(IDf), model = "generic0", Cmatrix = Q,
hyper = prec.prior, diagonal = diagval, constr = FALSE, rankdef = 1),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep("rank", inlafitC_unconstr$logfile, value = TRUE)## [1] " rank-deficiency is *defined* [1]"summary(inlafit_unconstr)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.81, Running = 0.226, Post = 0.00992, Total = 3.05
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.037 17.676 -34.681 -0.037 34.608 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for as.integer(IDf) 67.52 93.54 7.81 40.08 299.39 NA
##
## Marginal log-Likelihood: -812.89
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafitC_unconstr)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.18, Running = 0.263, Post = 0.0101, Total = 3.46
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.037 17.676 -34.681 -0.037 34.608 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for as.integer(IDf) 67.52 93.54 7.81 40.08 299.39 NA
##
## Marginal log-Likelihood: -812.89
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')3.3 MRF in mgcv
rownames(Q) <- colnames(Q) <- levels(data$IDf)
mgcvfit2 <- mgcv::gam(Observed ~ 0 + s(IDf, bs = "mrf", xt = list(penalty = as.matrix(Q))),
data = data, offset = log(Expected), family = "poisson")
sm <- smoothCon(s(IDf, bs = "mrf", xt = list(penalty = as.matrix(Q))), data,
scale.penalty = FALSE)## Error in smoothCon(s(IDf, bs = "mrf", xt = list(penalty = as.matrix(Q))), : could not find function "smoothCon"sm_abs <- smoothCon(s(IDf, bs = "mrf", xt = list(penalty = as.matrix(Q))), data,
absorb.cons = TRUE, scale.penalty = FALSE)## Error in smoothCon(s(IDf, bs = "mrf", xt = list(penalty = as.matrix(Q))), : could not find function "smoothCon"vcov(mgcvfit2)## s(IDf).1 s(IDf).2 s(IDf).3 s(IDf).4 s(IDf).5
## s(IDf).1 5.607235e-03 -3.256655e-04 -1.051554e-03 -7.569044e-04 -5.258000e-04
## s(IDf).2 -3.256655e-04 3.444031e-03 -5.997532e-04 3.121777e-04 1.006941e-03
## s(IDf).3 -1.051554e-03 -5.997532e-04 6.083383e-03 -2.106369e-04 -4.179609e-04
## s(IDf).4 -7.569044e-04 3.121777e-04 -2.106369e-04 7.778673e-03 6.389922e-04
## s(IDf).5 -5.258000e-04 1.006941e-03 -4.179609e-04 6.389922e-04 4.401090e-03
## s(IDf).6 4.525914e-04 5.039069e-05 -7.878835e-04 -4.671260e-04 -2.743895e-04
## s(IDf).7 -3.260043e-04 1.425065e-03 -6.869935e-04 1.116531e-05 3.087268e-04
## s(IDf).8 1.939604e-03 -3.706481e-04 -1.203930e-03 -8.410539e-04 -6.560299e-04
## s(IDf).9 -4.545703e-04 1.890098e-04 -2.400342e-04 2.253405e-04 1.105704e-03
## s(IDf).10 -9.881689e-04 -4.299312e-04 2.134061e-03 1.420446e-04 -9.481295e-05
## s(IDf).11 1.128192e-03 -5.531212e-04 -1.456696e-03 -1.042883e-03 -9.014017e-04
## s(IDf).12 -4.902648e-04 1.461389e-03 -6.016915e-04 4.627439e-04 5.451214e-04
## s(IDf).13 1.517781e-04 6.679869e-04 -6.782604e-04 -1.478795e-04 7.819024e-04
## s(IDf).14 -5.983942e-04 -5.952229e-04 -8.657145e-05 -6.004300e-04 -6.373098e-04
## s(IDf).15 1.008692e-03 -3.090899e-04 -1.181305e-03 -8.059352e-04 -6.240693e-04
## s(IDf).16 -1.364779e-04 -5.288121e-04 -4.693805e-04 -6.169442e-04 -6.138397e-04
## s(IDf).17 -8.613567e-04 -5.559257e-04 1.796812e-03 -3.172783e-04 -4.247828e-04
## s(IDf).18 1.405826e-04 9.163465e-04 -7.778675e-04 -2.471332e-04 1.758007e-04
## s(IDf).19 -7.819577e-04 1.468720e-04 4.544417e-04 1.455805e-03 9.428606e-04
## s(IDf).20 -6.452559e-04 -5.799400e-04 4.686612e-04 -5.310249e-04 -5.895501e-04
## s(IDf).21 -6.564181e-04 1.120144e-03 -3.842821e-04 1.920690e-03 1.316545e-03
## s(IDf).22 -3.058367e-04 -5.195339e-04 -2.369731e-04 -5.623619e-04 -5.666640e-04
## s(IDf).23 -4.557600e-04 -5.712754e-04 -2.389629e-04 -6.088253e-04 -6.456822e-04
## s(IDf).24 -5.776670e-04 -2.306804e-04 4.674559e-04 4.961606e-05 8.980749e-05
## s(IDf).25 -4.485864e-04 -4.052522e-04 1.304746e-04 -3.726176e-04 -3.595971e-04
## s(IDf).26 1.402200e-03 5.632989e-06 -9.901771e-04 -6.003228e-04 -2.642594e-04
## s(IDf).27 1.427211e-03 -1.206689e-04 -8.308729e-04 -5.533908e-04 -1.934904e-04
## s(IDf).28 -9.403595e-04 -6.588717e-04 2.309782e-03 -4.726711e-04 -6.063121e-04
## s(IDf).29 1.393944e-05 -1.530872e-04 -3.822178e-04 -3.167127e-04 6.317366e-06
## s(IDf).30 -1.100185e-03 -4.386702e-04 2.584665e-03 2.753275e-04 -5.640103e-05
## s(IDf).31 8.405774e-04 -3.535710e-04 -5.701826e-04 -5.338547e-04 -4.079925e-04
## s(IDf).6 s(IDf).7 s(IDf).8 s(IDf).9 s(IDf).10
## s(IDf).1 4.525914e-04 -3.260043e-04 1.939604e-03 -4.545703e-04 -9.881689e-04
## s(IDf).2 5.039069e-05 1.425065e-03 -3.706481e-04 1.890098e-04 -4.299312e-04
## s(IDf).3 -7.878835e-04 -6.869935e-04 -1.203930e-03 -2.400342e-04 2.134061e-03
## s(IDf).4 -4.671260e-04 1.116531e-05 -8.410539e-04 2.253405e-04 1.420446e-04
## s(IDf).5 -2.743895e-04 3.087268e-04 -6.560299e-04 1.105704e-03 -9.481295e-05
## s(IDf).6 4.808030e-03 1.700080e-04 9.603200e-04 -3.468231e-04 -7.339732e-04
## s(IDf).7 1.700080e-04 5.815206e-03 -3.192860e-04 -1.326758e-04 -5.745456e-04
## s(IDf).8 9.603200e-04 -3.192860e-04 6.267345e-03 -6.518866e-04 -1.146901e-03
## s(IDf).9 -3.468231e-04 -1.326758e-04 -6.518866e-04 4.137889e-03 1.119833e-04
## s(IDf).10 -7.339732e-04 -5.745456e-04 -1.146901e-03 1.119833e-04 4.986939e-03
## s(IDf).11 1.124017e-03 -4.718479e-04 4.084654e-03 -9.055285e-04 -1.404273e-03
## s(IDf).12 -1.249642e-04 1.957628e-03 -5.218611e-04 -4.667433e-05 -4.475896e-04
## s(IDf).13 1.825555e-04 2.780384e-04 -2.713447e-05 6.933836e-04 -5.003975e-04
## s(IDf).14 -5.730544e-04 -6.017197e-04 -8.047694e-04 -4.640486e-04 -2.305652e-04
## s(IDf).15 1.800308e-03 -2.340759e-04 2.880838e-03 -6.465604e-04 -1.124829e-03
## s(IDf).16 -4.290781e-04 -5.245773e-04 -5.217467e-04 -4.902129e-04 -5.286854e-04
## s(IDf).17 -6.863245e-04 -6.258199e-04 -1.031158e-03 -1.620281e-04 1.405874e-03
## s(IDf).18 1.083740e-03 1.302868e-03 2.416926e-04 -8.216771e-05 -6.722370e-04
## s(IDf).19 -5.299522e-04 -1.651493e-04 -9.188036e-04 8.376321e-04 1.083357e-03
## s(IDf).20 -5.872203e-04 -5.972777e-04 -8.396835e-04 -4.135359e-04 1.272400e-04
## s(IDf).21 -3.308145e-04 5.909969e-04 -7.397244e-04 3.270168e-04 -7.613319e-05
## s(IDf).22 -4.667393e-04 -5.266383e-04 -6.009010e-04 -4.180250e-04 -3.307355e-04
## s(IDf).23 -5.128521e-04 -5.644556e-04 -6.978755e-04 -5.217546e-04 -3.765300e-04
## s(IDf).24 -4.752857e-04 -3.700706e-04 -7.490507e-04 6.891149e-04 9.728454e-04
## s(IDf).25 -4.447254e-04 -4.464410e-04 -6.308391e-04 -1.175778e-04 9.355135e-05
## s(IDf).26 1.256387e-03 3.226492e-05 1.682947e-03 -2.948881e-04 -9.063461e-04
## s(IDf).27 2.035960e-04 -2.141762e-04 5.087043e-04 -5.642087e-05 -7.346607e-04
## s(IDf).28 -7.364272e-04 -6.979327e-04 -1.098258e-03 -4.196565e-04 9.046130e-04
## s(IDf).29 -2.541050e-04 -2.909147e-04 -3.547384e-04 5.926875e-04 -2.258318e-04
## s(IDf).30 -7.940026e-04 -6.000732e-04 -1.249756e-03 3.687437e-05 2.432570e-03
## s(IDf).31 -1.749375e-04 -3.829546e-04 4.732293e-06 -2.404393e-04 -5.466790e-04
## s(IDf).11 s(IDf).12 s(IDf).13 s(IDf).14 s(IDf).15
## s(IDf).1 1.128192e-03 -4.902648e-04 1.517781e-04 -5.983942e-04 1.008692e-03
## s(IDf).2 -5.531212e-04 1.461389e-03 6.679869e-04 -5.952229e-04 -3.090899e-04
## s(IDf).3 -1.456696e-03 -6.016915e-04 -6.782604e-04 -8.657145e-05 -1.181305e-03
## s(IDf).4 -1.042883e-03 4.627439e-04 -1.478795e-04 -6.004300e-04 -8.059352e-04
## s(IDf).5 -9.014017e-04 5.451214e-04 7.819024e-04 -6.373098e-04 -6.240693e-04
## s(IDf).6 1.124017e-03 -1.249642e-04 1.825555e-04 -5.730544e-04 1.800308e-03
## s(IDf).7 -4.718479e-04 1.957628e-03 2.780384e-04 -6.017197e-04 -2.340759e-04
## s(IDf).8 4.084654e-03 -5.218611e-04 -2.713447e-05 -8.047694e-04 2.880838e-03
## s(IDf).9 -9.055285e-04 -4.667433e-05 6.933836e-04 -4.640486e-04 -6.465604e-04
## s(IDf).10 -1.404273e-03 -4.475896e-04 -5.003975e-04 -2.305652e-04 -1.124829e-03
## s(IDf).11 1.207389e-02 -6.864468e-04 -2.888450e-04 -9.989356e-04 4.169407e-03
## s(IDf).12 -6.864468e-04 5.844722e-03 1.566026e-04 -6.040137e-04 -4.656424e-04
## s(IDf).13 -2.888450e-04 1.566026e-04 3.011263e-03 -5.652326e-04 -2.301055e-05
## s(IDf).14 -9.989356e-04 -6.040137e-04 -5.652326e-04 6.375944e-03 -8.174781e-04
## s(IDf).15 4.169407e-03 -4.656424e-04 -2.301055e-05 -8.174781e-04 6.424489e-03
## s(IDf).16 -7.310624e-04 -5.503810e-04 -4.440674e-04 4.484367e-04 -5.882915e-04
## s(IDf).17 -1.261928e-03 -5.711616e-04 -5.635204e-04 3.466040e-04 -1.021147e-03
## s(IDf).18 9.047285e-05 4.982407e-04 7.455521e-04 -6.182247e-04 4.232394e-04
## s(IDf).19 -1.159494e-03 1.177792e-04 -3.262249e-05 -4.857154e-04 -8.923261e-04
## s(IDf).20 -1.037731e-03 -5.874115e-04 -5.670069e-04 1.507025e-03 -8.472029e-04
## s(IDf).21 -9.522885e-04 1.516527e-03 1.671365e-04 -6.425001e-04 -6.956844e-04
## s(IDf).22 -7.907610e-04 -5.394860e-04 -4.478068e-04 1.294800e-03 -6.437304e-04
## s(IDf).23 -8.849353e-04 -5.762693e-04 -5.477417e-04 7.696665e-04 -7.239724e-04
## s(IDf).24 -9.577969e-04 -2.953722e-04 -1.124163e-04 -5.467367e-05 -7.466141e-04
## s(IDf).25 -7.996024e-04 -4.365862e-04 -3.110175e-04 1.050074e-03 -6.435124e-04
## s(IDf).26 1.347020e-03 -2.383317e-04 5.773656e-04 -6.791522e-04 1.690155e-03
## s(IDf).27 1.191528e-04 -3.381053e-04 6.832198e-04 -4.715406e-04 2.882550e-04
## s(IDf).28 -1.328121e-03 -6.573265e-04 -7.048416e-04 3.907549e-04 -1.084393e-03
## s(IDf).29 -5.979572e-04 -3.119514e-04 4.281922e-04 -1.465057e-04 -4.127683e-04
## s(IDf).30 -1.520473e-03 -4.450509e-04 -5.694659e-04 -4.097733e-04 -1.220285e-03
## s(IDf).31 -2.849779e-04 -4.364234e-04 -6.250523e-05 -6.392395e-07 -1.862954e-04
## s(IDf).16 s(IDf).17 s(IDf).18 s(IDf).19 s(IDf).20
## s(IDf).1 -0.0001364779 -8.613567e-04 1.405826e-04 -7.819577e-04 -6.452559e-04
## s(IDf).2 -0.0005288121 -5.559257e-04 9.163465e-04 1.468720e-04 -5.799400e-04
## s(IDf).3 -0.0004693805 1.796812e-03 -7.778675e-04 4.544417e-04 4.686612e-04
## s(IDf).4 -0.0006169442 -3.172783e-04 -2.471332e-04 1.455805e-03 -5.310249e-04
## s(IDf).5 -0.0006138397 -4.247828e-04 1.758007e-04 9.428606e-04 -5.895501e-04
## s(IDf).6 -0.0004290781 -6.863245e-04 1.083740e-03 -5.299522e-04 -5.872203e-04
## s(IDf).7 -0.0005245773 -6.258199e-04 1.302868e-03 -1.651493e-04 -5.972777e-04
## s(IDf).8 -0.0005217467 -1.031158e-03 2.416926e-04 -9.188036e-04 -8.396835e-04
## s(IDf).9 -0.0004902129 -1.620281e-04 -8.216771e-05 8.376321e-04 -4.135359e-04
## s(IDf).10 -0.0005286854 1.405874e-03 -6.722370e-04 1.083357e-03 1.272400e-04
## s(IDf).11 -0.0007310624 -1.261928e-03 9.047285e-05 -1.159494e-03 -1.037731e-03
## s(IDf).12 -0.0005503810 -5.711616e-04 4.982407e-04 1.177792e-04 -5.874115e-04
## s(IDf).13 -0.0004440674 -5.635204e-04 7.455521e-04 -3.262249e-05 -5.670069e-04
## s(IDf).14 0.0004484367 3.466040e-04 -6.182247e-04 -4.857154e-04 1.507025e-03
## s(IDf).15 -0.0005882915 -1.021147e-03 4.232394e-04 -8.923261e-04 -8.472029e-04
## s(IDf).16 0.0060276368 -2.151711e-04 -4.936220e-04 -5.900614e-04 4.768977e-04
## s(IDf).17 -0.0002151711 4.040963e-03 -6.824920e-04 1.583048e-04 1.003464e-03
## s(IDf).18 -0.0004936220 -6.824920e-04 3.944378e-03 -3.193838e-04 -6.241002e-04
## s(IDf).19 -0.0005900614 1.583048e-04 -3.193838e-04 3.394611e-03 -3.438806e-04
## s(IDf).20 0.0004768977 1.003464e-03 -6.241002e-04 -3.438806e-04 3.730178e-03
## s(IDf).21 -0.0006291297 -4.343799e-04 1.061542e-04 1.034686e-03 -5.900983e-04
## s(IDf).22 0.0012902347 8.835535e-05 -5.123133e-04 -4.842690e-04 9.562483e-04
## s(IDf).23 0.0020363743 8.624352e-05 -5.715180e-04 -5.512063e-04 1.294408e-03
## s(IDf).24 -0.0003137429 7.617351e-04 -3.930426e-04 6.601728e-04 7.105308e-05
## s(IDf).25 0.0001321844 6.321045e-04 -4.456409e-04 -1.773919e-04 7.749948e-04
## s(IDf).26 -0.0004348642 -8.400542e-04 8.687227e-04 -6.339882e-04 -7.050418e-04
## s(IDf).27 -0.0001157466 -6.512443e-04 1.730356e-04 -5.072515e-04 -5.105622e-04
## s(IDf).28 -0.0001871125 1.878646e-03 -7.631120e-04 -9.887988e-05 1.416754e-03
## s(IDf).29 -0.0001087377 -1.668035e-04 -1.497428e-04 -9.574169e-05 -1.868584e-04
## s(IDf).30 -0.0006476287 8.950556e-04 -7.242646e-04 1.393450e-03 -7.471344e-05
## s(IDf).31 0.0009391544 -3.775161e-04 -2.356954e-04 -4.905515e-04 -7.369045e-05
## s(IDf).21 s(IDf).22 s(IDf).23 s(IDf).24 s(IDf).25
## s(IDf).1 -6.564181e-04 -3.058367e-04 -4.557600e-04 -5.776670e-04 -4.485864e-04
## s(IDf).2 1.120144e-03 -5.195339e-04 -5.712754e-04 -2.306804e-04 -4.052522e-04
## s(IDf).3 -3.842821e-04 -2.369731e-04 -2.389629e-04 4.674559e-04 1.304746e-04
## s(IDf).4 1.920690e-03 -5.623619e-04 -6.088253e-04 4.961606e-05 -3.726176e-04
## s(IDf).5 1.316545e-03 -5.666640e-04 -6.456822e-04 8.980749e-05 -3.595971e-04
## s(IDf).6 -3.308145e-04 -4.667393e-04 -5.128521e-04 -4.752857e-04 -4.447254e-04
## s(IDf).7 5.909969e-04 -5.266383e-04 -5.644556e-04 -3.700706e-04 -4.464410e-04
## s(IDf).8 -7.397244e-04 -6.009010e-04 -6.978755e-04 -7.490507e-04 -6.308391e-04
## s(IDf).9 3.270168e-04 -4.180250e-04 -5.217546e-04 6.891149e-04 -1.175778e-04
## s(IDf).10 -7.613319e-05 -3.307355e-04 -3.765300e-04 9.728454e-04 9.355135e-05
## s(IDf).11 -9.522885e-04 -7.907610e-04 -8.849353e-04 -9.577969e-04 -7.996024e-04
## s(IDf).12 1.516527e-03 -5.394860e-04 -5.762693e-04 -2.953722e-04 -4.365862e-04
## s(IDf).13 1.671365e-04 -4.478068e-04 -5.477417e-04 -1.124163e-04 -3.110175e-04
## s(IDf).14 -6.425001e-04 1.294800e-03 7.696665e-04 -5.467367e-05 1.050074e-03
## s(IDf).15 -6.956844e-04 -6.437304e-04 -7.239724e-04 -7.466141e-04 -6.435124e-04
## s(IDf).16 -6.291297e-04 1.290235e-03 2.036374e-03 -3.137429e-04 1.321844e-04
## s(IDf).17 -4.343799e-04 8.835535e-05 8.624352e-05 7.617351e-04 6.321045e-04
## s(IDf).18 1.061542e-04 -5.123133e-04 -5.715180e-04 -3.930426e-04 -4.456409e-04
## s(IDf).19 1.034686e-03 -4.842690e-04 -5.512063e-04 6.601728e-04 -1.773919e-04
## s(IDf).20 -5.900983e-04 9.562483e-04 1.294408e-03 7.105308e-05 7.749948e-04
## s(IDf).21 4.563056e-03 -5.850386e-04 -6.408573e-04 -4.306882e-05 -4.078448e-04
## s(IDf).22 -5.850386e-04 3.148911e-03 1.373112e-03 -1.486960e-04 5.887282e-04
## s(IDf).23 -6.408573e-04 1.373112e-03 6.063116e-03 -2.422189e-04 3.022425e-04
## s(IDf).24 -4.306882e-05 -1.486960e-04 -2.422189e-04 2.994345e-03 3.961520e-04
## s(IDf).25 -4.078448e-04 5.887282e-04 3.022425e-04 3.961520e-04 2.507230e-03
## s(IDf).26 -4.316602e-04 -5.009007e-04 -5.970304e-04 -5.337499e-04 -4.969765e-04
## s(IDf).27 -4.275959e-04 -2.352749e-04 -3.837671e-04 -3.058177e-04 -2.728248e-04
## s(IDf).28 -5.842127e-04 1.182664e-04 2.151006e-04 2.405330e-04 3.413992e-04
## s(IDf).29 -2.569947e-04 -6.645153e-05 -2.336454e-04 3.763212e-04 2.528336e-04
## s(IDf).30 5.901561e-06 -4.715675e-04 -5.148755e-04 5.139891e-04 -1.134425e-04
## s(IDf).31 -5.005318e-04 5.070078e-04 2.625426e-04 -2.419080e-04 -1.839603e-05
## s(IDf).26 s(IDf).27 s(IDf).28 s(IDf).29 s(IDf).30
## s(IDf).1 1.402200e-03 1.427211e-03 -9.403595e-04 1.393944e-05 -1.100185e-03
## s(IDf).2 5.632989e-06 -1.206689e-04 -6.588717e-04 -1.530872e-04 -4.386702e-04
## s(IDf).3 -9.901771e-04 -8.308729e-04 2.309782e-03 -3.822178e-04 2.584665e-03
## s(IDf).4 -6.003228e-04 -5.533908e-04 -4.726711e-04 -3.167127e-04 2.753275e-04
## s(IDf).5 -2.642594e-04 -1.934904e-04 -6.063121e-04 6.317366e-06 -5.640103e-05
## s(IDf).6 1.256387e-03 2.035960e-04 -7.364272e-04 -2.541050e-04 -7.940026e-04
## s(IDf).7 3.226492e-05 -2.141762e-04 -6.979327e-04 -2.909147e-04 -6.000732e-04
## s(IDf).8 1.682947e-03 5.087043e-04 -1.098258e-03 -3.547384e-04 -1.249756e-03
## s(IDf).9 -2.948881e-04 -5.642087e-05 -4.196565e-04 5.926875e-04 3.687437e-05
## s(IDf).10 -9.063461e-04 -7.346607e-04 9.046130e-04 -2.258318e-04 2.432570e-03
## s(IDf).11 1.347020e-03 1.191528e-04 -1.328121e-03 -5.979572e-04 -1.520473e-03
## s(IDf).12 -2.383317e-04 -3.381053e-04 -6.573265e-04 -3.119514e-04 -4.450509e-04
## s(IDf).13 5.773656e-04 6.832198e-04 -7.048416e-04 4.281922e-04 -5.694659e-04
## s(IDf).14 -6.791522e-04 -4.715406e-04 3.907549e-04 -1.465057e-04 -4.097733e-04
## s(IDf).15 1.690155e-03 2.882550e-04 -1.084393e-03 -4.127683e-04 -1.220285e-03
## s(IDf).16 -4.348642e-04 -1.157466e-04 -1.871125e-04 -1.087377e-04 -6.476287e-04
## s(IDf).17 -8.400542e-04 -6.512443e-04 1.878646e-03 -1.668035e-04 8.950556e-04
## s(IDf).18 8.687227e-04 1.730356e-04 -7.631120e-04 -1.497428e-04 -7.242646e-04
## s(IDf).19 -6.339882e-04 -5.072515e-04 -9.887988e-05 -9.574169e-05 1.393450e-03
## s(IDf).20 -7.050418e-04 -5.105622e-04 1.416754e-03 -1.868584e-04 -7.471344e-05
## s(IDf).21 -4.316602e-04 -4.275959e-04 -5.842127e-04 -2.569947e-04 5.901561e-06
## s(IDf).22 -5.009007e-04 -2.352749e-04 1.182664e-04 -6.645153e-05 -4.715675e-04
## s(IDf).23 -5.970304e-04 -3.837671e-04 2.151006e-04 -2.336454e-04 -5.148755e-04
## s(IDf).24 -5.337499e-04 -3.058177e-04 2.405330e-04 3.763212e-04 5.139891e-04
## s(IDf).25 -4.969765e-04 -2.728248e-04 3.413992e-04 2.528336e-04 -1.134425e-04
## s(IDf).26 3.534611e-03 9.502890e-04 -9.225002e-04 -8.679086e-05 -9.955776e-04
## s(IDf).27 9.502890e-04 4.134678e-03 -7.593514e-04 5.849454e-04 -8.454406e-04
## s(IDf).28 -9.225002e-04 -7.593514e-04 6.812154e-03 -3.658278e-04 8.132525e-04
## s(IDf).29 -8.679086e-05 5.849454e-04 -3.658278e-04 2.778322e-03 -3.830148e-04
## s(IDf).30 -9.955776e-04 -8.454406e-04 8.132525e-04 -3.830148e-04 7.548388e-03
## s(IDf).31 6.996892e-05 7.365971e-04 -4.314612e-04 3.961727e-04 -6.524212e-04
## s(IDf).31
## s(IDf).1 8.405774e-04
## s(IDf).2 -3.535710e-04
## s(IDf).3 -5.701826e-04
## s(IDf).4 -5.338547e-04
## s(IDf).5 -4.079925e-04
## s(IDf).6 -1.749375e-04
## s(IDf).7 -3.829546e-04
## s(IDf).8 4.732293e-06
## s(IDf).9 -2.404393e-04
## s(IDf).10 -5.466790e-04
## s(IDf).11 -2.849779e-04
## s(IDf).12 -4.364234e-04
## s(IDf).13 -6.250523e-05
## s(IDf).14 -6.392395e-07
## s(IDf).15 -1.862954e-04
## s(IDf).16 9.391544e-04
## s(IDf).17 -3.775161e-04
## s(IDf).18 -2.356954e-04
## s(IDf).19 -4.905515e-04
## s(IDf).20 -7.369045e-05
## s(IDf).21 -5.005318e-04
## s(IDf).22 5.070078e-04
## s(IDf).23 2.625426e-04
## s(IDf).24 -2.419080e-04
## s(IDf).25 -1.839603e-05
## s(IDf).26 6.996892e-05
## s(IDf).27 7.365971e-04
## s(IDf).28 -4.314612e-04
## s(IDf).29 3.961727e-04
## s(IDf).30 -6.524212e-04
## s(IDf).31 3.007647e-033.4 Model 2b: express as TMB GMRF
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_space);
DATA_SPARSE_MATRIX(Q); // Structure matrix for ICAR area model
DATA_SCALAR(Qrank);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space);
// Note: dgamma() is parameterised as (shape, scale); INLA parameterised as (shape, rate)
val -= dlgamma(log_prec_space, Type(0.001), Type(1.0 / 0.001), true);
PARAMETER_VECTOR(u_space);
val += SCALE(GMRF(Q), exp(-0.5 * log_prec_space))(u_space);
val -= -(Q.cols() - Qrank) * 0.5 * (log_prec_space - log(2 * PI)); // adjust GMRF for rank deficiency
val -= dnorm(sum(u_space), Type(0.0), Type(0.001) * u_space.size(), true); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_space * u_space +
log(E));
val -= dpois(y, exp(mu), true).sum();
return val;
}
'dll <- tmb_compile_and_load(mod)
Q <- diag(rowSums(adj.mat)) - adj.mat
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_space = Matrix::sparse.model.matrix(~0 + IDf, data),
Q = Q,
Qrank = as.integer(rankMatrix(Q)))
tmbpar <- list(beta0 = 0,
log_prec_space = 0,
u_space = numeric(ncol(tmbdata$Z_space)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_space"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr,
control = list(iter.max = 1000,
eval.max = 1000))
tmbfit <- optim(obj$par, obj$fn, obj$gr, method = "BFGS")
sdr2b <- TMB::sdreport(obj)
summary(sdr2b, "all")## Estimate Std. Error
## log_prec_space 3.454994987 0.90168437
## beta0 -0.036929359 0.03758052
## u_space 0.116341626 0.05679451
## u_space -0.005569504 0.09728603
## u_space -0.003048042 0.07857893
## u_space -0.012811814 0.10193935
## u_space 0.019197287 0.10817919
## u_space -0.012068942 0.08628526
## u_space -0.047357667 0.08086183
## u_space -0.119175959 0.09817940
## u_space -0.104867964 0.10879944
## u_space 0.006599599 0.08429194
## u_space 0.001514570 0.09252036
## u_space -0.106318067 0.14601458
## u_space -0.136768980 0.10502046
## u_space -0.028572076 0.07155153
## u_space 0.099932313 0.10772913
## u_space -0.113386096 0.11062322
## u_space 0.095412270 0.10147126
## u_space 0.015925639 0.08228571
## u_space -0.097450131 0.08612558
## u_space 0.010179499 0.07498308
## u_space 0.068422990 0.07927717
## u_space -0.017357545 0.08534302
## u_space 0.084443436 0.07571722
## u_space 0.010845181 0.09890689
## u_space 0.027969057 0.06806070
## u_space 0.087735791 0.06953994
## u_space -0.032312076 0.07470062
## u_space 0.038042045 0.08827609
## u_space 0.002791464 0.10506725
## u_space 0.044646927 0.07100092
## u_space 0.006475053 0.11286901
## u_space 0.100590115 0.08345502Hyper parameters
c(inlafit$misc$theta.mode,
sqrt(diag(inlafit$misc$cov.intern)))## [1] 3.4511795 0.9189193summary(sdr2, "fixed")## Estimate Std. Error
## log_prec_space 3.454994 0.9016842summary(sdr2b, "fixed")## Estimate Std. Error
## log_prec_space 3.454995 0.9016844Intercept
inlafit$summary.fixed[ , 1:2]summary(sdr2, "random")[1, ]## Estimate Std. Error
## -0.03692936 0.03758052summary(sdr2b, "random")[1, ]## Estimate Std. Error
## -0.03692936 0.03758052Random effects
inlafit$summary.random[[1]]plot(summary(sdr2, "random")[-1, 1], summary(sdr2b, "random")[-1, 1])
abline(0, 1, col = "red")
plot(summary(sdr2, "random")[-1, 2], summary(sdr2b, "random")[-1, 2])
abline(0, 1, col = "red")
3.5 Model 2c: non-centered parameterisation
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_space);
DATA_SPARSE_MATRIX(Q); // Structure matrix for ICAR area model
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space);
// Note: TMB dlgamma() is parameterised as (shape, scale); INLA parameterised as (shape, rate)
val -= dlgamma(log_prec_space, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space(exp(-0.5 * log_prec_space));
PARAMETER_VECTOR(u_raw_space);
vector<Type> u_space(u_raw_space * sigma_space);
val += GMRF(Q)(u_raw_space);
val -= dnorm(sum(u_raw_space), Type(0.0), Type(0.001) * u_raw_space.size(), true); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_space * u_space +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space);
return val;
}
'dll <- tmb_compile_and_load(mod)
Q <- diag(rowSums(adj.mat)) - adj.mat
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_space = Matrix::sparse.model.matrix(~0 + IDf, data),
Q = Q)
tmbpar <- list(beta0 = 0,
log_prec_space = 0,
u_raw_space = numeric(ncol(tmbdata$Z_space)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
tmbfit <- optim(obj$par, obj$fn, obj$gr, method = "BFGS")
sdr2c <- TMB::sdreport(obj)
summary(sdr2c, "all")## Estimate Std. Error
## log_prec_space 3.454994987 0.90168437
## beta0 -0.036929359 0.03756763
## u_raw_space 0.654602649 0.33726113
## u_raw_space -0.031337125 0.54787505
## u_raw_space -0.017149977 0.44543008
## u_raw_space -0.072086386 0.56673687
## u_raw_space 0.108014605 0.59986703
## u_raw_space -0.067906575 0.48803808
## u_raw_space -0.266460556 0.47922819
## u_raw_space -0.670550182 0.52848137
## u_raw_space -0.590045449 0.54931439
## u_raw_space 0.037133014 0.47381670
## u_raw_space 0.008521815 0.52100250
## u_raw_space -0.598204534 0.78715827
## u_raw_space -0.769538295 0.52103017
## u_raw_space -0.160762378 0.39855590
## u_raw_space 0.562274733 0.56902599
## u_raw_space -0.637973196 0.55679270
## u_raw_space 0.536842461 0.54035944
## u_raw_space 0.089606496 0.46989550
## u_raw_space -0.548308598 0.43937933
## u_raw_space 0.057275521 0.41905655
## u_raw_space 0.384985769 0.43348488
## u_raw_space -0.097663196 0.48763002
## u_raw_space 0.475125702 0.41226262
## u_raw_space 0.061021017 0.56424288
## u_raw_space 0.157369459 0.38196837
## u_raw_space 0.493650321 0.35728735
## u_raw_space -0.181805700 0.43069509
## u_raw_space 0.214045686 0.47321300
## u_raw_space 0.015706330 0.59240296
## u_raw_space 0.251208428 0.39374926
## u_raw_space 0.036432249 0.63544269
## u_raw_space 0.565975895 0.39142775
## u_space 0.116341626 0.05678598
## u_space -0.005569504 0.09728106
## u_space -0.003048042 0.07857277
## u_space -0.012811814 0.10193460
## u_space 0.019197287 0.10817471
## u_space -0.012068942 0.08627964
## u_space -0.047357667 0.08085584
## u_space -0.119175959 0.09817446
## u_space -0.104867964 0.10879499
## u_space 0.006599599 0.08428619
## u_space 0.001514570 0.09251513
## u_space -0.106318067 0.14601126
## u_space -0.136768980 0.10501585
## u_space -0.028572076 0.07154476
## u_space 0.099932313 0.10772463
## u_space -0.113386096 0.11061885
## u_space 0.095412270 0.10146649
## u_space 0.015925639 0.08227983
## u_space -0.097450131 0.08611995
## u_space 0.010179499 0.07497663
## u_space 0.068422990 0.07927106
## u_space -0.017357545 0.08533735
## u_space 0.084443436 0.07571082
## u_space 0.010845181 0.09890199
## u_space 0.027969057 0.06805358
## u_space 0.087735791 0.06953298
## u_space -0.032312076 0.07469413
## u_space 0.038042045 0.08827060
## u_space 0.002791464 0.10506264
## u_space 0.044646927 0.07099410
## u_space 0.006475053 0.11286472
## u_space 0.100590115 0.08344921Hyper parameters
summary(sdr2, "fixed")## Estimate Std. Error
## log_prec_space 3.454994 0.9016842summary(sdr2b, "fixed")## Estimate Std. Error
## log_prec_space 3.454995 0.9016844summary(sdr2c, "fixed")## Estimate Std. Error
## log_prec_space 3.454995 0.9016844Intercept
summary(sdr2, "random")[1, ]## Estimate Std. Error
## -0.03692936 0.03758052summary(sdr2b, "random")[1, ]## Estimate Std. Error
## -0.03692936 0.03758052summary(sdr2c, "random")[1, ]## Estimate Std. Error
## -0.03692936 0.03756763Random effects
inlafit$summary.random[[1]]plot(summary(sdr2, "random")[-1, 1], summary(sdr2c, "report")[ , 1])
abline(0, 1, col = "red")
plot(summary(sdr2, "random")[-1 , 2], summary(sdr2c, "report")[ , 2])
abline(0, 1, col = "red")
3.6 Model 2d: explicit sum-to-zero constraint
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(L_space);
DATA_SPARSE_MATRIX(Z_space);
DATA_SPARSE_MATRIX(Q); // Structure matrix for ICAR area model
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space);
// Note: TMB dlgamma() is parameterised as (shape, scale); INLA parameterised as (shape, rate)
val -= dlgamma(log_prec_space, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space(exp(-0.5 * log_prec_space));
PARAMETER_VECTOR(u_raw_space);
vector<Type> u_space(L_space * u_raw_space * sigma_space);
val += GMRF(Q)(u_raw_space);
vector<Type> mu(beta0 +
Z_space * u_space +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space);
return val;
}
'dll <- tmb_compile_and_load(mod)
Q <- diag(rowSums(adj.mat)) - adj.mat
Aconstr <- matrix(1, ncol = ncol(Q))
qrc <- qr(t(Aconstr))
L_space <- qr.Q(qrc,complete=TRUE)[ , (nrow(Aconstr)+1):ncol(Aconstr)]
tmbdata <- list(y = data$Observed,
E = data$Expected,
L_space = L_space,
Z_space = Matrix::sparse.model.matrix(~0 + IDf, data),
Q = as(t(L_space) %*% Q %*% L_space, "dgCMatrix"))
tmbpar <- list(beta0 = 0,
log_prec_space = 0,
u_raw_space = numeric(ncol(tmbdata$L_space)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
tmbfit <- optim(obj$par, obj$fn, obj$gr, method = "BFGS")
sdr2d <- TMB::sdreport(obj)
summary(sdr2d, "all")## Estimate Std. Error
## log_prec_space 3.454994987 0.90168437
## beta0 -0.036929359 0.03756721
## u_raw_space -0.129672258 0.55359812
## u_raw_space -0.115485110 0.45926020
## u_raw_space -0.170421519 0.56898733
## u_raw_space 0.009679472 0.60688641
## u_raw_space -0.166241708 0.49738198
## u_raw_space -0.364795689 0.48822660
## u_raw_space -0.768885316 0.53531388
## u_raw_space -0.688380582 0.55490584
## u_raw_space -0.061202119 0.47949625
## u_raw_space -0.089813318 0.52550349
## u_raw_space -0.696539668 0.79430338
## u_raw_space -0.867873428 0.52611323
## u_raw_space -0.259097512 0.40466413
## u_raw_space 0.463939600 0.56743101
## u_raw_space -0.736308330 0.56267201
## u_raw_space 0.438507327 0.53706820
## u_raw_space -0.008728638 0.46974419
## u_raw_space -0.646643731 0.44469116
## u_raw_space -0.041059613 0.42860983
## u_raw_space 0.286650636 0.43162132
## u_raw_space -0.195998330 0.49815087
## u_raw_space 0.376790569 0.40343930
## u_raw_space -0.037314116 0.55630210
## u_raw_space 0.059034326 0.38294564
## u_raw_space 0.395315187 0.34950243
## u_raw_space -0.280140833 0.44160091
## u_raw_space 0.115710553 0.48020219
## u_raw_space -0.082628803 0.59232582
## u_raw_space 0.152873294 0.39027695
## u_raw_space -0.061902884 0.64200395
## u_raw_space 0.467640762 0.38800606
## u_space 0.116341626 0.05678571
## u_space -0.005569504 0.09728089
## u_space -0.003048042 0.07857257
## u_space -0.012811814 0.10193444
## u_space 0.019197287 0.10817456
## u_space -0.012068942 0.08627946
## u_space -0.047357667 0.08085564
## u_space -0.119175959 0.09817430
## u_space -0.104867964 0.10879484
## u_space 0.006599599 0.08428601
## u_space 0.001514570 0.09251496
## u_space -0.106318067 0.14601115
## u_space -0.136768980 0.10501570
## u_space -0.028572076 0.07154454
## u_space 0.099932313 0.10772449
## u_space -0.113386096 0.11061870
## u_space 0.095412270 0.10146633
## u_space 0.015925639 0.08227964
## u_space -0.097450131 0.08611977
## u_space 0.010179499 0.07497642
## u_space 0.068422990 0.07927086
## u_space -0.017357545 0.08533716
## u_space 0.084443436 0.07571062
## u_space 0.010845181 0.09890183
## u_space 0.027969057 0.06805335
## u_space 0.087735791 0.06953275
## u_space -0.032312076 0.07469392
## u_space 0.038042045 0.08827042
## u_space 0.002791464 0.10506249
## u_space 0.044646927 0.07099388
## u_space 0.006475053 0.11286458
## u_space 0.100590115 0.08344902Hyper parameters
c(inlafit$misc$theta.mode,
sqrt(diag(inlafit$misc$cov.intern)))## [1] 3.4511795 0.9189193summary(sdr2, "fixed")## Estimate Std. Error
## log_prec_space 3.454994 0.9016842summary(sdr2d, "fixed")## Estimate Std. Error
## log_prec_space 3.454995 0.9016844Intercept
inlafit$summary.fixed[ , 1:2]summary(sdr2, "random")[1, ]## Estimate Std. Error
## -0.03692936 0.03758052summary(sdr2d, "random")[1, ]## Estimate Std. Error
## -0.03692936 0.03756721Random effects
inlafit$summary.random[[1]]plot(summary(sdr2, "random")[-1, 1], summary(sdr2d, "report")[ , 1])
abline(0, 1, col = "red")
plot(summary(sdr2, "random")[-1 , 2], summary(sdr2d, "report")[ , 2])
abline(0, 1, col = "red")
4 Model 3: RW1 time trend
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_time);
DATA_SPARSE_MATRIX(R_time); // Structure matrix for RW1
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_time);
val -= dlgamma(log_prec_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_time(exp(-0.5 * log_prec_time));
PARAMETER_VECTOR(u_raw_time);
vector<Type> u_time(u_raw_time * sigma_time);
val += GMRF(R_time)(u_raw_time);
val -= dnorm(sum(u_raw_time), Type(0.0), Type(0.001) * u_raw_time.size(), true); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_time * u_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
Q <- diag(rowSums(adj.mat)) - adj.mat
D_time <- diff(diag(length(levels(data$Yearf))), differences = 1)
R_time <- Matrix::Matrix(t(D_time) %*% D_time)
R_time_adj <- R_time + Matrix::Diagonal(ncol(R_time), 1e-6)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_time = Matrix::sparse.model.matrix(~0 + Yearf, data),
R_time = R_time_adj)
tmbpar <- list(beta0 = 0,
log_prec_time = 0,
u_raw_time = numeric(ncol(tmbdata$Z_time)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_time"),
DLL = dll,
silent = TRUE)## Warning in getParameterOrder(data, parameters, new.env(), DLL = DLL): Expected
## sparse matrix of class 'dgTMatrix'.## Error in getParameterOrder(data, parameters, new.env(), DLL = DLL): Error when reading the variable: 'R_time'. Please check data and parameters.tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr3 <- TMB::sdreport(obj)
summary(sdr3, "all")## Estimate Std. Error
## log_prec_space 3.454994987 0.90168437
## beta0 -0.036929359 0.03756721
## u_raw_space -0.129672258 0.55359812
## u_raw_space -0.115485110 0.45926020
## u_raw_space -0.170421519 0.56898733
## u_raw_space 0.009679472 0.60688641
## u_raw_space -0.166241708 0.49738198
## u_raw_space -0.364795689 0.48822660
## u_raw_space -0.768885316 0.53531388
## u_raw_space -0.688380582 0.55490584
## u_raw_space -0.061202119 0.47949625
## u_raw_space -0.089813318 0.52550349
## u_raw_space -0.696539668 0.79430338
## u_raw_space -0.867873428 0.52611323
## u_raw_space -0.259097512 0.40466413
## u_raw_space 0.463939600 0.56743101
## u_raw_space -0.736308330 0.56267201
## u_raw_space 0.438507327 0.53706820
## u_raw_space -0.008728638 0.46974419
## u_raw_space -0.646643731 0.44469116
## u_raw_space -0.041059613 0.42860983
## u_raw_space 0.286650636 0.43162132
## u_raw_space -0.195998330 0.49815087
## u_raw_space 0.376790569 0.40343930
## u_raw_space -0.037314116 0.55630210
## u_raw_space 0.059034326 0.38294564
## u_raw_space 0.395315187 0.34950243
## u_raw_space -0.280140833 0.44160091
## u_raw_space 0.115710553 0.48020219
## u_raw_space -0.082628803 0.59232582
## u_raw_space 0.152873294 0.39027695
## u_raw_space -0.061902884 0.64200395
## u_raw_space 0.467640762 0.38800606
## u_space 0.116341626 0.05678571
## u_space -0.005569504 0.09728089
## u_space -0.003048042 0.07857257
## u_space -0.012811814 0.10193444
## u_space 0.019197287 0.10817456
## u_space -0.012068942 0.08627946
## u_space -0.047357667 0.08085564
## u_space -0.119175959 0.09817430
## u_space -0.104867964 0.10879484
## u_space 0.006599599 0.08428601
## u_space 0.001514570 0.09251496
## u_space -0.106318067 0.14601115
## u_space -0.136768980 0.10501570
## u_space -0.028572076 0.07154454
## u_space 0.099932313 0.10772449
## u_space -0.113386096 0.11061870
## u_space 0.095412270 0.10146633
## u_space 0.015925639 0.08227964
## u_space -0.097450131 0.08611977
## u_space 0.010179499 0.07497642
## u_space 0.068422990 0.07927086
## u_space -0.017357545 0.08533716
## u_space 0.084443436 0.07571062
## u_space 0.010845181 0.09890183
## u_space 0.027969057 0.06805335
## u_space 0.087735791 0.06953275
## u_space -0.032312076 0.07469392
## u_space 0.038042045 0.08827042
## u_space 0.002791464 0.10506249
## u_space 0.044646927 0.07099388
## u_space 0.006475053 0.11286458
## u_space 0.100590115 0.08344902inlafit <- inla(Observed ~ f(Year, model = "rw1", hyper = prec.prior),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
summary(inlafit)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.19, Running = 0.258, Post = 0.00996, Total = 3.46
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.01 0.03 -0.068 -0.01 0.048 NA 0
##
## Random effects:
## Name Model
## Year RW1 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for Year 482.69 522.63 54.92 327.37 1862.03 NA
##
## Marginal log-Likelihood: -800.46
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Hyper parameters
c(inlafit$misc$theta.mode,
sqrt(diag(inlafit$misc$cov.intern)))## [1] 5.8189878 0.9323871summary(sdr3, "fixed")## Estimate Std. Error
## log_prec_space 3.454995 0.9016844Intercept
inlafit$summary.fixed[ , 1:2]summary(sdr3, "random")[1, ]## Estimate Std. Error
## -0.03692936 0.03756721Random effects
inlafit$summary.random[[1]]plot(inlafit$summary.random[[1]][ , 2], summary(sdr3, "report")[ , 1])## Error in xy.coords(x, y, xlabel, ylabel, log): 'x' and 'y' lengths differabline(0, 1, col = "red")## Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yetplot(inlafit$summary.random[[1]][ , 3], summary(sdr3, "report")[ , 2])## Error in xy.coords(x, y, xlabel, ylabel, log): 'x' and 'y' lengths differabline(0, 1, col = "red")## Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet4.1 INLA with custom Cmatrix
inlafitC <- inla(Observed ~ f(ID.Year, model = "generic0", Cmatrix = R_time,
hyper = prec.prior, diagonal = diagval, constr = TRUE),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))These match:
summary(inlafit)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.19, Running = 0.258, Post = 0.00996, Total = 3.46
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.01 0.03 -0.068 -0.01 0.048 NA 0
##
## Random effects:
## Name Model
## Year RW1 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for Year 482.69 522.63 54.92 327.37 1862.03 NA
##
## Marginal log-Likelihood: -800.46
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafitC)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.96, Running = 0.222, Post = 0.0097, Total = 3.2
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.01 0.03 -0.068 -0.01 0.048 NA 0
##
## Random effects:
## Name Model
## ID.Year Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for ID.Year 482.69 522.63 54.92 327.37 1862.03 NA
##
## Marginal log-Likelihood: -800.46
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')4.2 Model 3b: Explicit sum-to-zero constraint
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(L_time);
DATA_SPARSE_MATRIX(Z_time);
DATA_SPARSE_MATRIX(LRL_time); // Structure matrix for RW1
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_time);
val -= dlgamma(log_prec_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_time(exp(-0.5 * log_prec_time));
PARAMETER_VECTOR(u_raw_time);
vector<Type> u_time(L_time * u_raw_time * sigma_time);
val += GMRF(LRL_time)(u_raw_time);
vector<Type> mu(beta0 +
Z_time * u_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
Aconstr <- matrix(1, ncol = ncol(R_time))
qrc <- qr(t(Aconstr))
L_time <- qr.Q(qrc, complete=TRUE)[ , (nrow(Aconstr)+1):ncol(Aconstr)]
tmbdata <- list(y = data$Observed,
E = data$Expected,
L_time = L_time,
Z_time = Matrix::sparse.model.matrix(~0 + Yearf, data),
LRL_time = as(t(L_time) %*% R_time %*% L_time, "dgCMatrix"))
tmbpar <- list(beta0 = 0,
log_prec_time = 0,
u_raw_time = numeric(ncol(tmbdata$L_time)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr3b <- TMB::sdreport(obj)
summary(sdr3b, "all")## Estimate Std. Error
## log_prec_time 5.819926913 0.93427166
## beta0 -0.010241374 0.02970886
## u_raw_time -0.340303008 1.08871281
## u_raw_time -0.863917041 1.01430121
## u_raw_time -1.389911868 0.98495013
## u_raw_time -1.419793635 1.01129192
## u_raw_time -1.417025875 1.02889253
## u_raw_time -1.517564716 1.02165287
## u_raw_time -1.326533189 1.02575823
## u_raw_time -1.331942219 1.02043588
## u_raw_time -0.702666799 1.02105504
## u_raw_time -0.169232317 1.02507324
## u_raw_time 0.186108464 1.05960919
## u_raw_time 1.158601663 1.03925667
## u_raw_time 1.550224579 1.07184424
## u_raw_time 1.529308478 1.16580106
## u_raw_time 1.759865065 1.20395636
## u_raw_time 1.973874589 1.23106156
## u_raw_time 1.638993208 1.39117791
## u_raw_time 1.823317259 1.49064774
## u_time -0.014265303 0.07695862
## u_time -0.021200916 0.06595329
## u_time -0.049726215 0.05652280
## u_time -0.078381214 0.05690575
## u_time -0.080009105 0.05498376
## u_time -0.079858324 0.05412832
## u_time -0.085335451 0.05656950
## u_time -0.074928488 0.05504278
## u_time -0.075223160 0.05956872
## u_time -0.040941670 0.05268751
## u_time -0.011881375 0.05082891
## u_time 0.007476781 0.05085719
## u_time 0.060455994 0.05487941
## u_time 0.081790718 0.05624031
## u_time 0.080651256 0.05204878
## u_time 0.093211453 0.05330813
## u_time 0.104870204 0.05632900
## u_time 0.086626630 0.05658407
## u_time 0.096668184 0.06623955Hyper parameters
c(inlafit$misc$theta.mode,
sqrt(diag(inlafit$misc$cov.intern)))## [1] 5.8189878 0.9323871summary(sdr3, "fixed")## Estimate Std. Error
## log_prec_space 3.454995 0.9016844summary(sdr3b, "fixed")## Estimate Std. Error
## log_prec_time 5.819927 0.9342717Intercept
inlafit$summary.fixed[ , 1:2]summary(sdr3, "random")[1, ]## Estimate Std. Error
## -0.03692936 0.03756721summary(sdr3b, "random")[1, ]## Estimate Std. Error
## -0.01024137 0.02970886Random effects
inlafit$summary.random[[1]]plot(summary(sdr3, "report")[ , 1], summary(sdr3b, "report")[ , 1])## Error in xy.coords(x, y, xlabel, ylabel, log): 'x' and 'y' lengths differabline(0, 1, col = "red")## Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yetplot(summary(sdr3, "report")[ , 2], summary(sdr3b, "report")[ , 2])## Error in xy.coords(x, y, xlabel, ylabel, log): 'x' and 'y' lengths differabline(0, 1, col = "red")## Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet5 Model 4: RW1 time + ICAR space
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_time);
DATA_SPARSE_MATRIX(R_time);
DATA_SPARSE_MATRIX(Z_space);
DATA_SPARSE_MATRIX(R_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_time);
val -= dlgamma(log_prec_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_time(exp(-0.5 * log_prec_time));
PARAMETER_VECTOR(u_raw_time);
vector<Type> u_time(u_raw_time * sigma_time);
val += GMRF(R_time)(u_raw_time);
val -= dnorm(sum(u_raw_time), Type(0.0), Type(0.001) * u_raw_time.size(), true); // soft sum-to-zero constraint
PARAMETER(log_prec_space);
val -= dlgamma(log_prec_space, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space(exp(-0.5 * log_prec_space));
PARAMETER_VECTOR(u_raw_space);
vector<Type> u_space(u_raw_space * sigma_space);
val += GMRF(R_space)(u_raw_space);
val -= dnorm(sum(u_raw_space), Type(0.0), Type(0.001) * u_raw_space.size(), true); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_time * u_time +
Z_space * u_space +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_time);
ADREPORT(u_space);
return val;
}
'dll <- tmb_compile_and_load(mod)
R_space <- diag(rowSums(adj.mat)) - adj.mat
R_space_adj <- R_space + Matrix::Diagonal(ncol(R_space), diagval)
D_time <- diff(diag(length(levels(data$Yearf))), differences = 1)
R_time <- Matrix::Matrix(t(D_time) %*% D_time)
R_time_adj <- R_time + Matrix::Diagonal(ncol(R_time), diagval)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_time = Matrix::sparse.model.matrix(~0 + Yearf, data),
R_time = R_time_adj,
Z_space = Matrix::sparse.model.matrix(~0 + IDf, data),
R_space = R_space_adj)
tmbpar <- list(beta0 = 0,
log_prec_time = 0,
u_raw_time = numeric(ncol(tmbdata$Z_time)),
log_prec_space = 0,
u_raw_space = numeric(ncol(tmbdata$Z_space)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_time", "u_raw_space"),
DLL = dll,
silent = TRUE)## Warning in getParameterOrder(data, parameters, new.env(), DLL = DLL): Expected
## sparse matrix of class 'dgTMatrix'.## Error in getParameterOrder(data, parameters, new.env(), DLL = DLL): Error when reading the variable: 'R_time'. Please check data and parameters.tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr4 <- TMB::sdreport(obj)
summary(sdr4, "all")## Estimate Std. Error
## log_prec_time 5.819926913 0.93427166
## beta0 -0.010241374 0.02970886
## u_raw_time -0.340303008 1.08871281
## u_raw_time -0.863917041 1.01430121
## u_raw_time -1.389911868 0.98495013
## u_raw_time -1.419793635 1.01129192
## u_raw_time -1.417025875 1.02889253
## u_raw_time -1.517564716 1.02165287
## u_raw_time -1.326533189 1.02575823
## u_raw_time -1.331942219 1.02043588
## u_raw_time -0.702666799 1.02105504
## u_raw_time -0.169232317 1.02507324
## u_raw_time 0.186108464 1.05960919
## u_raw_time 1.158601663 1.03925667
## u_raw_time 1.550224579 1.07184424
## u_raw_time 1.529308478 1.16580106
## u_raw_time 1.759865065 1.20395636
## u_raw_time 1.973874589 1.23106156
## u_raw_time 1.638993208 1.39117791
## u_raw_time 1.823317259 1.49064774
## u_time -0.014265303 0.07695862
## u_time -0.021200916 0.06595329
## u_time -0.049726215 0.05652280
## u_time -0.078381214 0.05690575
## u_time -0.080009105 0.05498376
## u_time -0.079858324 0.05412832
## u_time -0.085335451 0.05656950
## u_time -0.074928488 0.05504278
## u_time -0.075223160 0.05956872
## u_time -0.040941670 0.05268751
## u_time -0.011881375 0.05082891
## u_time 0.007476781 0.05085719
## u_time 0.060455994 0.05487941
## u_time 0.081790718 0.05624031
## u_time 0.080651256 0.05204878
## u_time 0.093211453 0.05330813
## u_time 0.104870204 0.05632900
## u_time 0.086626630 0.05658407
## u_time 0.096668184 0.06623955inlafit <- inla(Observed ~
f(Year, model = "rw1", hyper = prec.prior) +
f(as.integer(IDf), model = "besag", hyper = prec.prior, graph = adj.mat),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
summary(inlafit)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.19, Running = 0.276, Post = 0.0117, Total = 3.48
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.046 0.036 -0.118 -0.046 0.025 NA 0
##
## Random effects:
## Name Model
## Year RW1 model
## as.integer(IDf) Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for Year 500.80 554.54 56.97 335.62 1958.54 NA
## Precision for as.integer(IDf) 59.22 74.05 6.94 37.17 248.24 NA
##
## Marginal log-Likelihood: -824.58
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Hyper parameters
cbind("mean" = inlafit$misc$theta.mode,
"se" = sqrt(diag(inlafit$misc$cov.intern)))## mean se
## [1,] 5.821663 0.9330135
## [2,] 3.485399 0.8865806summary(sdr4, "fixed")## Estimate Std. Error
## log_prec_time 5.819927 0.9342717Intercept
inlafit$summary.fixed[ , 1:2]summary(sdr4, "random")[1, ]## Estimate Std. Error
## -0.01024137 0.02970886Random effects
sdr4sum <- summary(sdr4, "all")
plot(inlafit$summary.random[[1]][ , 2], sdr4sum[rownames(sdr4sum) == "u_time", 1],
main = "f(Year): mean")
abline(0, 1, col = "red")
plot(inlafit$summary.random[[1]][ , 3], sdr4sum[rownames(sdr4sum) == "u_time", 2],
main = "f(Year): sd")
abline(0, 1, col = "red")
plot(inlafit$summary.random[[2]][ , 2], sdr4sum[rownames(sdr4sum) == "u_space", 1],
main = "f(area): mean")## Error in xy.coords(x, y, xlabel, ylabel, log): 'x' and 'y' lengths differabline(0, 1, col = "red")
plot(inlafit$summary.random[[2]][ , 3], sdr4sum[rownames(sdr4sum) == "u_space", 2],
main = "f(area): sd")## Error in xy.coords(x, y, xlabel, ylabel, log): 'x' and 'y' lengths differabline(0, 1, col = "red")
5.1 R-INLA with custom Cmatrix
inlafitC <- inla(Observed ~
f(ID.Year, model = "generic0", Cmatrix = R_time,
hyper = prec.prior, diagonal = diagval, constr = TRUE) +
f(as.integer(IDf), model = "generic0", Cmatrix = R_space,
hyper = prec.prior, diagonal = diagval, constr = TRUE),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))These match:
summary(inlafit)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.19, Running = 0.276, Post = 0.0117, Total = 3.48
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.046 0.036 -0.118 -0.046 0.025 NA 0
##
## Random effects:
## Name Model
## Year RW1 model
## as.integer(IDf) Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for Year 500.80 554.54 56.97 335.62 1958.54 NA
## Precision for as.integer(IDf) 59.22 74.05 6.94 37.17 248.24 NA
##
## Marginal log-Likelihood: -824.58
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafitC)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.02, Running = 0.283, Post = 0.0121, Total = 3.32
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.046 0.036 -0.118 -0.046 0.025 NA 0
##
## Random effects:
## Name Model
## ID.Year Generic0 model
## as.integer(IDf) Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for ID.Year 500.80 554.54 56.97 335.62 1958.54 NA
## Precision for as.integer(IDf) 59.22 74.05 6.94 37.17 248.24 NA
##
## Marginal log-Likelihood: -824.58
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')5.2 Model 4b: Explicity sum-to-zero constraint
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(L_time);
DATA_SPARSE_MATRIX(Z_time);
DATA_SPARSE_MATRIX(LRL_time);
DATA_MATRIX(L_space);
DATA_SPARSE_MATRIX(Z_space);
DATA_SPARSE_MATRIX(LRL_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_time);
val -= dlgamma(log_prec_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_time(exp(-0.5 * log_prec_time));
PARAMETER_VECTOR(u_raw_time);
vector<Type> u_time(L_time * u_raw_time * sigma_time);
val += GMRF(LRL_time)(u_raw_time);
PARAMETER(log_prec_space);
val -= dlgamma(log_prec_space, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space(exp(-0.5 * log_prec_space));
PARAMETER_VECTOR(u_raw_space);
vector<Type> u_space(L_space * u_raw_space * sigma_space);
val += GMRF(LRL_space)(u_raw_space);
vector<Type> mu(beta0 +
Z_time * u_time +
Z_space * u_space +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_time);
ADREPORT(u_space);
return val;
using namespace density;
}
'dll <- tmb_compile_and_load(mod)
Aconstr <- matrix(1, ncol = ncol(R_time))
qrc <- qr(t(Aconstr))
L_time <- qr.Q(qrc, complete=TRUE)[ , (nrow(Aconstr)+1):ncol(Aconstr)]
Aconstr <- matrix(1, ncol = ncol(R_space))
qrc <- qr(t(Aconstr))
L_space <- qr.Q(qrc, complete=TRUE)[ , (nrow(Aconstr)+1):ncol(Aconstr)]
tmbdata <- list(y = data$Observed,
E = data$Expected,
L_time = L_time,
Z_time = Matrix::sparse.model.matrix(~0 + Yearf, data),
LRL_time = as(t(L_time) %*% R_time %*% L_time, "dgCMatrix"),
L_space = L_space,
Z_space = Matrix::sparse.model.matrix(~0 + IDf, data),
LRL_space = as(t(L_space) %*% R_space %*% L_space, "dgCMatrix"))
tmbpar <- list(beta0 = 0,
log_prec_time = 0,
u_raw_time = numeric(ncol(tmbdata$L_time)),
log_prec_space = 0,
u_raw_space = numeric(ncol(tmbdata$L_space)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_time", "u_raw_space"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr4b <- TMB::sdreport(obj)
summary(sdr4b, "all")## Estimate Std. Error
## log_prec_time 5.822729548 0.93680967
## log_prec_space 3.457439129 0.90709267
## beta0 -0.046644559 0.03791256
## u_raw_time -0.334372321 1.08902386
## u_raw_time -0.857805175 1.01451716
## u_raw_time -1.384457744 0.98512123
## u_raw_time -1.416341022 1.01121223
## u_raw_time -1.415143350 1.02871232
## u_raw_time -1.516860877 1.02158704
## u_raw_time -1.325856123 1.02578862
## u_raw_time -1.328882041 1.02104164
## u_raw_time -0.696509402 1.02171736
## u_raw_time -0.161414991 1.02578080
## u_raw_time 0.193453668 1.06013743
## u_raw_time 1.162676120 1.03964265
## u_raw_time 1.549263863 1.07187015
## u_raw_time 1.521775558 1.16543231
## u_raw_time 1.746051661 1.20331021
## u_raw_time 1.954769970 1.23023795
## u_raw_time 1.617165540 1.38996419
## u_raw_time 1.799825745 1.48957066
## u_raw_space -0.131964194 0.55383894
## u_raw_space -0.106128123 0.45926960
## u_raw_space -0.164341403 0.56927074
## u_raw_space 0.027605551 0.60732287
## u_raw_space -0.157950348 0.49748500
## u_raw_space -0.383200014 0.48854847
## u_raw_space -0.757333719 0.53534114
## u_raw_space -0.689390261 0.55533996
## u_raw_space -0.056795930 0.47960162
## u_raw_space -0.084298639 0.52565804
## u_raw_space -0.697549540 0.79491179
## u_raw_space -0.853806886 0.52610019
## u_raw_space -0.257735909 0.40464243
## u_raw_space 0.458399682 0.56761940
## u_raw_space -0.741697122 0.56301058
## u_raw_space 0.434508184 0.53742877
## u_raw_space -0.008262779 0.46988521
## u_raw_space -0.647602524 0.44465076
## u_raw_space -0.031691966 0.42881823
## u_raw_space 0.281221455 0.43176348
## u_raw_space -0.182748259 0.49826161
## u_raw_space 0.371657060 0.40347004
## u_raw_space -0.060006841 0.55669920
## u_raw_space 0.058795140 0.38306049
## u_raw_space 0.387272113 0.34963676
## u_raw_space -0.283659639 0.44204205
## u_raw_space 0.113267479 0.48039909
## u_raw_space -0.083409365 0.59254135
## u_raw_space 0.150476361 0.39030552
## u_raw_space -0.052446281 0.64237422
## u_raw_space 0.462597901 0.38815587
## u_time -0.013820195 0.07695264
## u_time -0.020769258 0.06595043
## u_time -0.049244756 0.05649817
## u_time -0.077895410 0.05687708
## u_time -0.079629906 0.05497251
## u_time -0.079564751 0.05412916
## u_time -0.085098331 0.05660677
## u_time -0.074707398 0.05506524
## u_time -0.074872012 0.05957044
## u_time -0.040470034 0.05262085
## u_time -0.011360131 0.05076732
## u_time 0.007945233 0.05078479
## u_time 0.060672323 0.05495932
## u_time 0.081703251 0.05630191
## u_time 0.080207847 0.05202984
## u_time 0.092408789 0.05326340
## u_time 0.103763364 0.05625256
## u_time 0.085397199 0.05656253
## u_time 0.095334176 0.06618416
## u_space 0.115673201 0.05676854
## u_space -0.006048613 0.09718461
## u_space -0.001462412 0.07878018
## u_space -0.011795942 0.10175211
## u_space 0.022276860 0.10843133
## u_space -0.010661456 0.08622668
## u_space -0.050645875 0.08066725
## u_space -0.117058931 0.09802953
## u_space -0.104998182 0.10878723
## u_space 0.007294622 0.08421497
## u_space 0.002412574 0.09237963
## u_space -0.106446549 0.14591777
## u_space -0.134184033 0.10482635
## u_space -0.028374546 0.07148126
## u_space 0.098747797 0.10761159
## u_space -0.114283255 0.11078226
## u_space 0.094506780 0.10154262
## u_space 0.015909817 0.08219893
## u_space -0.097580376 0.08627444
## u_space 0.011750866 0.07502375
## u_space 0.067296614 0.07916307
## u_space -0.015063372 0.08539172
## u_space 0.083349979 0.07565134
## u_space 0.006724648 0.09935686
## u_space 0.027813373 0.06796984
## u_space 0.086121832 0.06919868
## u_space -0.032976308 0.07462168
## u_space 0.037482843 0.08816465
## u_space 0.002570430 0.10499146
## u_space 0.044087849 0.07086826
## u_space 0.008066735 0.11278183
## u_space 0.099493030 0.08331491Hyper parameters
cbind(inlafit$misc$theta.mode,
sqrt(diag(inlafit$misc$cov.intern)))## [,1] [,2]
## [1,] 5.821663 0.9330135
## [2,] 3.485399 0.8865806summary(sdr4, "fixed")## Estimate Std. Error
## log_prec_time 5.819927 0.9342717summary(sdr4b, "fixed")## Estimate Std. Error
## log_prec_time 5.822730 0.9368097
## log_prec_space 3.457439 0.9070927Intercept
inlafit$summary.fixed[ , 1:2]summary(sdr4, "random")[1, ]## Estimate Std. Error
## -0.01024137 0.02970886summary(sdr4b, "random")[1, ]## Estimate Std. Error
## -0.04664456 0.03791256Random effects
sdr4bsum <- summary(sdr4, "all")
plot(sdr4sum[rownames(sdr4sum) == "u_time", 1],
sdr4bsum[rownames(sdr4bsum) == "u_time", 1],
main = "f(Year): mean")
abline(0, 1, col = "red")
plot(sdr4sum[rownames(sdr4sum) == "u_time", 2],
sdr4bsum[rownames(sdr4bsum) == "u_time", 2],
main = "f(Year): sd")
abline(0, 1, col = "red")
plot(sdr4sum[rownames(sdr4sum) == "u_space", 1],
sdr4bsum[rownames(sdr4bsum) == "u_space", 1],
main = "f(area): mean")## Warning in min(x): no non-missing arguments to min; returning Inf## Warning in max(x): no non-missing arguments to max; returning -Inf## Warning in min(x): no non-missing arguments to min; returning Inf## Warning in max(x): no non-missing arguments to max; returning -Inf## Error in plot.window(...): need finite 'xlim' values
abline(0, 1, col = "red")
plot(sdr4sum[rownames(sdr4sum) == "u_space", 2],
sdr4bsum[rownames(sdr4bsum) == "u_space", 2],
main = "f(area): sd")## Warning in min(x): no non-missing arguments to min; returning Inf
## Warning in min(x): no non-missing arguments to max; returning -Inf## Warning in min(x): no non-missing arguments to min; returning Inf## Warning in max(x): no non-missing arguments to max; returning -Inf## Error in plot.window(...): need finite 'xlim' valuesabline(0, 1, col = "red")6 Model 5: ICAR space x IID time
R-INLA automatically applies a sum-to-zero constraint for the spatial field at each time.
This is too many constraints for an intercept plus interaction only model because it imposes that the average is the intercept at each time, and hence there is no time trend in the model.
The correct specification for this model should probably be a single sum-to-zero constraint to account for the lost degree of freedom from the intercept term
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_time);
DATA_SPARSE_MATRIX(R_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time * sigma_space_time);
val += SEPARABLE(GMRF(R_time), GMRF(R_space))(u_raw_space_time);
for (int i = 0; i < u_raw_space_time.cols(); i++) {
val -= dnorm(u_raw_space_time.col(i).sum(), Type(0), Type(0.001) * u_raw_space_time.rows(), true);
}
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
R_space <- diag(rowSums(adj.mat)) - adj.mat
R_space_scaled <- inla.scale.model(R_space, constr = list(A = matrix(1, ncol = ncol(R_space)), e = 0))
R_space_adj <- R_space + Matrix::Diagonal(ncol(R_space), 1e-6)
R_space_scaled_adj <- R_space_scaled + Matrix::Diagonal(ncol(R_space_scaled), 1e-6)
R_time <- Matrix::sparseMatrix(1:19, 1:19, x = rep(1L, 19))
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_time = R_time,
R_space = R_space_adj)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = array(0, c(nrow(tmbdata$R_space), nrow(tmbdata$R_time))))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr5 <- TMB::sdreport(obj)
summary(sdr5, "all")## Estimate Std. Error
## log_prec_space_time 3.875642e+00 1.69376067
## beta0 -5.221556e-03 0.03087570
## u_raw_space_time 1.308886e-01 0.55247732
## u_raw_space_time -4.285736e-02 0.59498647
## u_raw_space_time -8.313194e-02 0.53112358
## u_raw_space_time 2.183067e-02 0.64685309
## u_raw_space_time -2.420405e-02 0.79633707
## u_raw_space_time -1.755869e-02 0.52215069
## u_raw_space_time -2.887748e-01 0.73554616
## u_raw_space_time -1.032285e-01 0.71275944
## u_raw_space_time -1.193131e-01 0.70252654
## u_raw_space_time 1.006872e-02 0.49533710
## u_raw_space_time 3.315063e-02 0.56396085
## u_raw_space_time -1.004277e-01 0.92801559
## u_raw_space_time -9.336400e-02 0.71626007
## u_raw_space_time -5.157709e-02 0.43276821
## u_raw_space_time 9.007125e-02 0.68328891
## u_raw_space_time -1.865883e-01 0.72777583
## u_raw_space_time 1.698843e-01 0.70649390
## u_raw_space_time 7.492797e-02 0.52536011
## u_raw_space_time -1.196505e-01 0.55607880
## u_raw_space_time -9.305697e-03 0.47152689
## u_raw_space_time 1.313731e-01 0.54522772
## u_raw_space_time -6.911261e-02 0.57214033
## u_raw_space_time 1.129330e-01 0.51394219
## u_raw_space_time 1.858706e-01 0.72184983
## u_raw_space_time 9.579130e-02 0.47127808
## u_raw_space_time 1.254013e-01 0.47368989
## u_raw_space_time -1.312330e-01 0.51615232
## u_raw_space_time -2.017725e-02 0.50658819
## u_raw_space_time 8.297281e-02 0.70014488
## u_raw_space_time 7.243029e-02 0.42369564
## u_raw_space_time -8.054346e-04 0.69395080
## u_raw_space_time 1.237210e-01 0.49052084
## u_raw_space_time 1.144052e-01 0.54995256
## u_raw_space_time -2.483533e-02 0.59437092
## u_raw_space_time -7.904423e-02 0.53060391
## u_raw_space_time 3.176839e-02 0.64697515
## u_raw_space_time -2.398830e-02 0.79611944
## u_raw_space_time -4.653276e-02 0.52307455
## u_raw_space_time -1.060127e-01 0.70576684
## u_raw_space_time -1.643581e-01 0.71926656
## u_raw_space_time -1.227527e-01 0.70321209
## u_raw_space_time -2.863258e-03 0.49525041
## u_raw_space_time 2.376624e-02 0.56366747
## u_raw_space_time -1.280301e-01 0.93018005
## u_raw_space_time -1.152796e-01 0.71777854
## u_raw_space_time -4.761776e-02 0.43251652
## u_raw_space_time 1.383954e-01 0.68819294
## u_raw_space_time -7.157237e-02 0.71566891
## u_raw_space_time 6.777669e-02 0.69645917
## u_raw_space_time 6.258305e-02 0.52448453
## u_raw_space_time -1.182550e-01 0.55613437
## u_raw_space_time -1.291015e-03 0.47138452
## u_raw_space_time 1.283248e-01 0.54507516
## u_raw_space_time -5.260036e-02 0.57093717
## u_raw_space_time 1.055591e-01 0.51324216
## u_raw_space_time 4.229727e-02 0.70843666
## u_raw_space_time 6.759793e-02 0.46817643
## u_raw_space_time 1.587165e-01 0.48005595
## u_raw_space_time -8.310016e-02 0.51033658
## u_raw_space_time 2.002198e-02 0.50660461
## u_raw_space_time 2.655899e-02 0.69779182
## u_raw_space_time 7.331439e-02 0.42383141
## u_raw_space_time -4.854805e-03 0.69388482
## u_raw_space_time 1.319291e-01 0.49201324
## u_raw_space_time 1.100895e-02 0.54439880
## u_raw_space_time -4.533652e-02 0.59506596
## u_raw_space_time -1.277151e-02 0.52757692
## u_raw_space_time 8.496770e-02 0.64980753
## u_raw_space_time 9.597296e-02 0.79925765
## u_raw_space_time -6.127313e-04 0.52192467
## u_raw_space_time -1.234287e-01 0.70745119
## u_raw_space_time -8.582377e-02 0.71163242
## u_raw_space_time -1.001411e-01 0.70100259
## u_raw_space_time -5.159546e-03 0.49524603
## u_raw_space_time 4.995088e-02 0.56458159
## u_raw_space_time -4.466161e-02 0.92526327
## u_raw_space_time -4.531635e-02 0.71343692
## u_raw_space_time 1.146774e-02 0.43119519
## u_raw_space_time -4.510800e-03 0.68031418
## u_raw_space_time -1.059000e-01 0.71826194
## u_raw_space_time 6.609647e-02 0.69660285
## u_raw_space_time 3.425496e-02 0.52303392
## u_raw_space_time -4.021830e-02 0.54988830
## u_raw_space_time 2.301604e-02 0.47156719
## u_raw_space_time 3.318582e-02 0.53690919
## u_raw_space_time -8.434402e-03 0.56970523
## u_raw_space_time 1.089203e-02 0.50754323
## u_raw_space_time 8.286120e-02 0.71054099
## u_raw_space_time -3.173361e-02 0.46611332
## u_raw_space_time 2.976208e-02 0.46497165
## u_raw_space_time -5.213421e-02 0.50790719
## u_raw_space_time -2.867734e-02 0.50676764
## u_raw_space_time 6.255649e-02 0.69899741
## u_raw_space_time -6.047746e-03 0.42015053
## u_raw_space_time 1.382432e-01 0.70259886
## u_raw_space_time 6.659588e-03 0.48117902
## u_raw_space_time 6.412325e-03 0.54349504
## u_raw_space_time 1.671102e-03 0.59391090
## u_raw_space_time -1.161308e-01 0.53472310
## u_raw_space_time 1.021664e-01 0.65092670
## u_raw_space_time 1.596426e-01 0.80602685
## u_raw_space_time 1.879601e-03 0.52186922
## u_raw_space_time -1.558878e-01 0.71044386
## u_raw_space_time -1.888817e-01 0.72260897
## u_raw_space_time -1.164674e-03 0.69631044
## u_raw_space_time 3.423759e-02 0.49590593
## u_raw_space_time 9.624408e-02 0.56803694
## u_raw_space_time -3.574285e-03 0.92433022
## u_raw_space_time -1.903285e-01 0.72739261
## u_raw_space_time -3.489461e-02 0.43177497
## u_raw_space_time 5.674398e-03 0.68005102
## u_raw_space_time -2.361397e-02 0.71324307
## u_raw_space_time 3.611641e-02 0.69472076
## u_raw_space_time 6.805396e-02 0.52471297
## u_raw_space_time -1.381145e-01 0.55906118
## u_raw_space_time 8.425244e-02 0.47594247
## u_raw_space_time 2.691577e-02 0.53642531
## u_raw_space_time 5.573308e-03 0.56961178
## u_raw_space_time -2.089748e-03 0.50727343
## u_raw_space_time -1.761748e-02 0.70742635
## u_raw_space_time 7.899645e-02 0.46893902
## u_raw_space_time 3.486576e-02 0.46477816
## u_raw_space_time -1.546868e-02 0.50623385
## u_raw_space_time -1.032427e-02 0.50625635
## u_raw_space_time 4.508852e-02 0.69793002
## u_raw_space_time 1.615407e-02 0.42019742
## u_raw_space_time 9.961705e-02 0.69779692
## u_raw_space_time -5.531461e-03 0.48095211
## u_raw_space_time -1.336789e-01 0.54875807
## u_raw_space_time -3.010039e-03 0.59381192
## u_raw_space_time -5.290486e-03 0.52713775
## u_raw_space_time 3.553717e-03 0.64626616
## u_raw_space_time 2.739708e-01 0.82300013
## u_raw_space_time 4.721282e-02 0.52274016
## u_raw_space_time 2.850487e-02 0.69978344
## u_raw_space_time 3.034519e-02 0.70777205
## u_raw_space_time 2.232039e-02 0.69605882
## u_raw_space_time 2.646828e-02 0.49551244
## u_raw_space_time 4.480132e-02 0.56422484
## u_raw_space_time 7.750424e-02 0.92569688
## u_raw_space_time -1.036706e-02 0.71220068
## u_raw_space_time -5.766787e-03 0.43101092
## u_raw_space_time -1.298251e-01 0.68664984
## u_raw_space_time 9.768532e-03 0.71272108
## u_raw_space_time -1.341004e-01 0.70098596
## u_raw_space_time -2.249770e-02 0.52244309
## u_raw_space_time -4.635611e-03 0.54874926
## u_raw_space_time 1.025381e-01 0.47762324
## u_raw_space_time -7.987312e-02 0.53896310
## u_raw_space_time 1.199546e-01 0.57627775
## u_raw_space_time -1.172486e-01 0.51374185
## u_raw_space_time -6.977335e-02 0.70805934
## u_raw_space_time 3.909590e-02 0.46602296
## u_raw_space_time -1.103769e-01 0.47070056
## u_raw_space_time 1.338559e-02 0.50595683
## u_raw_space_time -2.246766e-02 0.50646397
## u_raw_space_time -1.861021e-02 0.69728052
## u_raw_space_time -3.739341e-02 0.42068121
## u_raw_space_time 9.095611e-02 0.69722065
## u_raw_space_time -2.548661e-02 0.48077991
## u_raw_space_time -7.041154e-03 0.54149423
## u_raw_space_time -3.345969e-03 0.59371843
## u_raw_space_time -3.837130e-02 0.52782304
## u_raw_space_time -1.008739e-02 0.64622997
## u_raw_space_time 9.571626e-02 0.79852145
## u_raw_space_time -1.427004e-02 0.52187531
## u_raw_space_time 6.858278e-02 0.70066349
## u_raw_space_time -5.217278e-02 0.70868789
## u_raw_space_time -8.036810e-03 0.69573849
## u_raw_space_time -2.025133e-02 0.49543557
## u_raw_space_time -4.528160e-03 0.56307044
## u_raw_space_time 7.510197e-02 0.92556334
## u_raw_space_time 9.892625e-02 0.71582280
## u_raw_space_time -2.471932e-02 0.43141557
## u_raw_space_time -5.600173e-03 0.67951897
## u_raw_space_time 4.477202e-02 0.71302121
## u_raw_space_time -5.065126e-02 0.69473555
## u_raw_space_time 1.035683e-02 0.52217716
## u_raw_space_time -2.631333e-02 0.54884741
## u_raw_space_time 1.363966e-02 0.47111648
## u_raw_space_time -1.616105e-02 0.53577173
## u_raw_space_time 1.038580e-02 0.56921058
## u_raw_space_time 7.275886e-03 0.50676701
## u_raw_space_time -8.272235e-02 0.70900744
## u_raw_space_time -2.737752e-02 0.46510827
## u_raw_space_time -5.431422e-02 0.46536821
## u_raw_space_time 1.636330e-02 0.50579624
## u_raw_space_time 4.376757e-03 0.50608118
## u_raw_space_time 5.350496e-02 0.69855323
## u_raw_space_time -2.652193e-02 0.42034125
## u_raw_space_time -1.060265e-02 0.69357258
## u_raw_space_time -1.592581e-02 0.48048728
## u_raw_space_time 1.487273e-02 0.54056389
## u_raw_space_time 2.816299e-02 0.59401655
## u_raw_space_time 8.274891e-02 0.53067626
## u_raw_space_time -2.461366e-02 0.64634235
## u_raw_space_time -2.958155e-02 0.79541896
## u_raw_space_time 2.747356e-02 0.52207552
## u_raw_space_time 5.735322e-02 0.69943252
## u_raw_space_time 2.236568e-02 0.70698088
## u_raw_space_time 3.340354e-02 0.69581076
## u_raw_space_time -1.301039e-02 0.49520279
## u_raw_space_time -3.601390e-02 0.56369337
## u_raw_space_time 2.934474e-02 0.92376843
## u_raw_space_time -5.355533e-02 0.71257996
## u_raw_space_time 2.301655e-02 0.43113437
## u_raw_space_time -6.548983e-02 0.68115949
## u_raw_space_time 1.964178e-02 0.71199176
## u_raw_space_time 1.256267e-01 0.70126484
## u_raw_space_time -5.284800e-02 0.52362330
## u_raw_space_time 6.027437e-02 0.55001792
## u_raw_space_time -2.332032e-02 0.47134746
## u_raw_space_time -6.405316e-02 0.53783037
## u_raw_space_time 4.463402e-02 0.57028301
## u_raw_space_time -2.001729e-02 0.50650772
## u_raw_space_time 5.564781e-03 0.70558250
## u_raw_space_time -8.936728e-02 0.46997028
## u_raw_space_time -4.375987e-02 0.46437515
## u_raw_space_time 4.617672e-02 0.50669596
## u_raw_space_time 9.085536e-03 0.50605867
## u_raw_space_time -7.195174e-02 0.69933317
## u_raw_space_time -1.546408e-02 0.41992723
## u_raw_space_time -2.623128e-02 0.69376161
## u_raw_space_time -5.158641e-04 0.48011602
## u_raw_space_time 2.036293e-01 0.55351157
## u_raw_space_time -2.925762e-03 0.59358046
## u_raw_space_time -1.430064e-01 0.53504858
## u_raw_space_time 4.703541e-02 0.64664400
## u_raw_space_time -1.647410e-01 0.80445366
## u_raw_space_time -2.627221e-02 0.52163623
## u_raw_space_time -1.172558e-01 0.70165742
## u_raw_space_time -2.750168e-01 0.73394145
## u_raw_space_time -4.720736e-02 0.69561215
## u_raw_space_time 5.734539e-03 0.49510668
## u_raw_space_time 4.825733e-02 0.56398212
## u_raw_space_time -9.936026e-02 0.92580120
## u_raw_space_time -1.843081e-01 0.72167204
## u_raw_space_time -4.960853e-02 0.43195213
## u_raw_space_time 1.531602e-01 0.68709535
## u_raw_space_time -1.172092e-01 0.71617658
## u_raw_space_time 4.355566e-02 0.69327591
## u_raw_space_time 7.833852e-02 0.52449471
## u_raw_space_time -1.537951e-01 0.55816225
## u_raw_space_time -2.880388e-03 0.47093143
## u_raw_space_time 1.369505e-01 0.54351771
## u_raw_space_time -6.823916e-02 0.57032080
## u_raw_space_time 1.776390e-01 0.52107871
## u_raw_space_time 9.437219e-02 0.70744205
## u_raw_space_time 5.613924e-02 0.46588240
## u_raw_space_time 1.156824e-01 0.46898368
## u_raw_space_time -7.609357e-02 0.50786464
## u_raw_space_time 4.822309e-02 0.50758002
## u_raw_space_time 9.418357e-02 0.69968141
## u_raw_space_time 7.874278e-02 0.42321353
## u_raw_space_time 6.697353e-02 0.69541424
## u_raw_space_time 7.929642e-02 0.48228850
## u_raw_space_time 1.041176e-01 0.54260804
## u_raw_space_time 3.294131e-02 0.59404861
## u_raw_space_time -2.695617e-02 0.52611780
## u_raw_space_time -3.257266e-02 0.64635705
## u_raw_space_time -1.665125e-01 0.80525763
## u_raw_space_time -3.158015e-02 0.52189272
## u_raw_space_time 5.258956e-02 0.69835406
## u_raw_space_time -4.582035e-02 0.70649245
## u_raw_space_time -1.282043e-02 0.69483391
## u_raw_space_time -2.732196e-02 0.49547927
## u_raw_space_time -3.170895e-02 0.56335948
## u_raw_space_time 1.113578e-02 0.92309673
## u_raw_space_time -5.859979e-02 0.71100367
## u_raw_space_time -1.267377e-02 0.43072601
## u_raw_space_time 7.653691e-02 0.68114213
## u_raw_space_time 2.240880e-03 0.71122520
## u_raw_space_time 2.041516e-01 0.71109432
## u_raw_space_time -1.294660e-02 0.52187062
## u_raw_space_time -4.048379e-02 0.54833036
## u_raw_space_time -6.554062e-02 0.47349669
## u_raw_space_time 3.782379e-02 0.53567563
## u_raw_space_time -7.748869e-02 0.57155129
## u_raw_space_time 8.496206e-02 0.50949914
## u_raw_space_time 5.717772e-02 0.70596197
## u_raw_space_time -8.001184e-02 0.46875089
## u_raw_space_time 9.462682e-03 0.46269053
## u_raw_space_time 2.219395e-02 0.50546440
## u_raw_space_time 1.868432e-02 0.50611504
## u_raw_space_time -2.142552e-02 0.69684030
## u_raw_space_time 1.149849e-02 0.41963209
## u_raw_space_time -3.224617e-02 0.69372566
## u_raw_space_time 5.111696e-02 0.48089187
## u_raw_space_time 1.903872e-03 0.53805278
## u_raw_space_time -3.669596e-02 0.59405429
## u_raw_space_time 3.814427e-02 0.52610858
## u_raw_space_time 1.330141e-02 0.64561684
## u_raw_space_time 3.995397e-01 0.85475982
## u_raw_space_time 4.783267e-02 0.52250060
## u_raw_space_time -5.180985e-02 0.69733555
## u_raw_space_time 9.931582e-02 0.70916377
## u_raw_space_time -6.211288e-02 0.69613129
## u_raw_space_time 1.285576e-02 0.49508676
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## u_space_time -3.159688e-02 0.08951582
## u_space_time -6.425322e-03 0.09358569
## u_space_time -2.760672e-02 0.12181134
## u_space_time -1.973316e-02 0.08088701
## u_space_time 2.023603e-02 0.10595487
## u_space_time -4.422977e-02 0.12186929
## u_space_time -9.887568e-03 0.10101535
## u_space_time -1.932055e-04 0.07127522
## u_space_time -5.145380e-03 0.08144616
## u_space_time -1.307461e-02 0.13423166
## u_space_time -5.215146e-02 0.12987403
## u_space_time -6.358606e-03 0.06260862
## u_space_time 2.886265e-02 0.10707694
## u_space_time -2.295745e-03 0.10216128
## u_space_time 1.843645e-02 0.10295627
## u_space_time 3.571092e-03 0.07514558
## u_space_time -6.003454e-03 0.07887719
## u_space_time -1.528581e-02 0.07178899
## u_space_time 1.199864e-02 0.07858970
## u_space_time -3.425188e-02 0.09742329
## u_space_time 3.484647e-02 0.09012589
## u_space_time 2.410836e-02 0.10720479
## u_space_time 1.504711e-02 0.07102575
## u_space_time 2.600118e-02 0.07715020
## u_space_time 2.746315e-03 0.07282407
## u_space_time 1.665531e-02 0.07784120
## u_space_time -7.074699e-03 0.10106732
## u_space_time 1.710858e-02 0.06584761
## u_space_time -1.355527e-02 0.10219713
## u_space_time 1.968648e-02 0.07481543
## u_space_time -7.640046e-03 0.07793085
## u_space_time -9.409427e-03 0.08665435
## u_space_time -1.825453e-02 0.08078871
## u_space_time 2.126194e-02 0.09852892
## u_space_time 8.422921e-03 0.11516741
## u_space_time -6.442249e-03 0.07576560
## u_space_time -4.413318e-02 0.12102760
## u_space_time -3.445404e-02 0.11517657
## u_space_time -2.574042e-02 0.10777102
## u_space_time 3.298152e-03 0.07145774
## u_space_time 1.436563e-02 0.08389237
## u_space_time -3.195563e-02 0.14209762
## u_space_time -1.950832e-02 0.10646020
## u_space_time -9.772061e-03 0.06380830
## u_space_time 4.660751e-02 0.12327871
## u_space_time -3.132269e-02 0.11328681
## u_space_time 7.886248e-03 0.10013285
## u_space_time 2.031633e-02 0.08132667
## u_space_time -3.079981e-02 0.09233627
## u_space_time 6.035267e-03 0.06836628
## u_space_time 2.630925e-02 0.08713899
## u_space_time -9.566186e-03 0.08325135
## u_space_time 2.221774e-02 0.08066193
## u_space_time 2.211630e-02 0.10652584
## u_space_time 7.673777e-03 0.06765217
## u_space_time 2.980614e-02 0.08178365
## u_space_time -2.141078e-02 0.07985085
## u_space_time -2.895319e-03 0.07294347
## u_space_time 2.836830e-02 0.10949239
## u_space_time 3.652906e-03 0.06063771
## u_space_time 1.859496e-02 0.10392080
## u_space_time 1.637208e-02 0.07403157
## u_space_time 2.324341e-02 0.08431494
## u_space_time 9.043665e-03 0.08661048
## u_space_time 3.422585e-03 0.07592691
## u_space_time -9.200639e-03 0.09405543
## u_space_time -3.572364e-02 0.12799049
## u_space_time -5.585083e-03 0.07550036
## u_space_time 2.612874e-02 0.10804757
## u_space_time -8.912607e-03 0.10237942
## u_space_time 2.159107e-02 0.10590385
## u_space_time -2.289834e-03 0.07135012
## u_space_time -8.721535e-03 0.08216308
## u_space_time 1.060357e-02 0.13374419
## u_space_time -1.381075e-02 0.10426651
## u_space_time 6.392973e-03 0.06291791
## u_space_time 1.300388e-02 0.09977478
## u_space_time 9.945831e-03 0.10321701
## u_space_time -1.708149e-02 0.10373700
## u_space_time -3.174227e-03 0.07518148
## u_space_time 8.725393e-04 0.07857796
## u_space_time -1.148803e-02 0.07011019
## u_space_time 1.131957e-03 0.07681336
## u_space_time -1.688392e-02 0.08601385
## u_space_time -2.099311e-04 0.07256776
## u_space_time -1.737039e-02 0.10506211
## u_space_time -1.482694e-03 0.06674965
## u_space_time 2.094804e-02 0.07457475
## u_space_time 8.827188e-03 0.07387255
## u_space_time 8.158700e-03 0.07398770
## u_space_time -1.269611e-02 0.10228510
## u_space_time 9.397817e-03 0.06218459
## u_space_time -1.299166e-02 0.10193415
## u_space_time 4.914656e-03 0.06927625
## u_space_time 7.342234e-02 0.13001521
## u_space_time -1.451149e-02 0.08814773
## u_space_time 1.319218e-02 0.08029091
## u_space_time -2.850155e-03 0.09316833
## u_space_time 5.494582e-03 0.11496925
## u_space_time 2.076212e-03 0.07527048
## u_space_time -6.979204e-02 0.14473084
## u_space_time -4.277683e-03 0.10149758
## u_space_time -4.745408e-02 0.12294449
## u_space_time 1.512039e-02 0.07567449
## u_space_time 4.193779e-03 0.08116565
## u_space_time -6.244358e-02 0.16333006
## u_space_time -1.333455e-02 0.10346579
## u_space_time -4.017794e-03 0.06208262
## u_space_time 2.867131e-02 0.10574864
## u_space_time -6.118283e-02 0.13805981
## u_space_time 2.510930e-02 0.10556672
## u_space_time 1.090830e-02 0.07639337
## u_space_time -3.097236e-02 0.09046985
## u_space_time 3.042454e-03 0.06798074
## u_space_time 1.477871e-02 0.07900475
## u_space_time -3.438956e-03 0.08181583
## u_space_time 2.503405e-02 0.07992942
## u_space_time 6.472713e-03 0.10089860
## u_space_time 2.540323e-02 0.07769494
## u_space_time 3.612087e-02 0.08485824
## u_space_time -3.231950e-02 0.08688473
## u_space_time 7.413111e-03 0.07397947
## u_space_time 3.231235e-03 0.10012387
## u_space_time 2.281411e-02 0.06924415
## u_space_time -3.787337e-03 0.10003685
## u_space_time 2.789408e-02 0.08014784
## u_space_time 3.782612e-02 0.09397646
## u_space_time -9.966103e-03 0.08683156
## u_space_time -2.047701e-02 0.08158493
## u_space_time 1.435316e-02 0.09554644
## u_space_time -2.478585e-02 0.12084608
## u_space_time -8.474224e-03 0.07603598
## u_space_time -2.292674e-02 0.10467613
## u_space_time -9.404543e-03 0.10209358
## u_space_time -2.292543e-02 0.10573525
## u_space_time 9.337412e-04 0.07128497
## u_space_time 1.211920e-02 0.08323771
## u_space_time -1.487056e-02 0.13425016
## u_space_time -2.629333e-02 0.10956177
## u_space_time -9.496211e-03 0.06358588
## u_space_time 1.947488e-02 0.10155770
## u_space_time -2.402167e-02 0.10815323
## u_space_time 6.121592e-03 0.09968540
## u_space_time 1.788031e-02 0.07989828
## u_space_time -1.200596e-02 0.08017086
## u_space_time 5.062310e-03 0.06836130
## u_space_time 2.075936e-02 0.08281303
## u_space_time -1.792951e-02 0.08622186
## u_space_time 1.632818e-02 0.07600076
## u_space_time 9.717804e-03 0.10149511
## u_space_time 1.241475e-02 0.06943600
## u_space_time 1.906591e-02 0.07185254
## u_space_time -1.450600e-02 0.07554637
## u_space_time -6.741988e-03 0.07366244
## u_space_time 3.103424e-02 0.11161438
## u_space_time 7.198686e-03 0.06114397
## u_space_time 9.615550e-03 0.10098701
## u_space_time 4.916587e-03 0.06906401
## u_space_time 3.820150e-03 0.07682920
## u_space_time 2.348066e-03 0.08544317
## u_space_time 1.878562e-02 0.08129810
## u_space_time -1.599662e-02 0.09638354
## u_space_time 1.421323e-02 0.11651659
## u_space_time 5.442892e-03 0.07548267
## u_space_time 1.143575e-02 0.10061117
## u_space_time 1.316719e-02 0.10338320
## u_space_time 1.827933e-03 0.09970371
## u_space_time -3.081688e-03 0.07145100
## u_space_time -1.282184e-02 0.08358528
## u_space_time 1.038566e-03 0.13257345
## u_space_time 1.500332e-02 0.10470948
## u_space_time 5.001395e-03 0.06241164
## u_space_time -2.104665e-02 0.10312166
## u_space_time 1.327769e-02 0.10399119
## u_space_time 2.218014e-02 0.10655703
## u_space_time -1.737570e-02 0.08000478
## u_space_time 8.466343e-03 0.07949983
## u_space_time 2.145371e-03 0.06779782
## u_space_time -1.516604e-02 0.08031475
## u_space_time 1.434195e-02 0.08493887
## u_space_time -4.073315e-03 0.07265032
## u_space_time -1.423751e-02 0.10309268
## u_space_time -1.191424e-02 0.06936055
## u_space_time -1.707515e-02 0.07153763
## u_space_time 8.957369e-03 0.07374860
## u_space_time 3.242363e-03 0.07298644
## u_space_time -1.962988e-02 0.10486259
## u_space_time -4.704072e-03 0.06079594
## u_space_time -1.288933e-02 0.10197621
## u_space_time 5.323558e-03 0.06948592inlafit <- inla(Observed ~
f(as.integer(IDf), model = "besag", hyper = prec.prior, scale.model = FALSE,
graph = adj.mat, diagonal = 1e-6,
group = ID.Year, control.group = list(model = "iid")),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
summary(inlafit)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.91, Running = 0.324, Post = 0.0251, Total = 3.26
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.005 0.03 -0.063 -0.005 0.053 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant
## Precision for as.integer(IDf) 359.96 1250.87 22.33 121.43 2294.40
## mode
## Precision for as.integer(IDf) NA
##
## Marginal log-Likelihood: -1197.30
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Hyper parameter comparison
cbind("mean" = inlafit$misc$theta.mode,
"se" = sqrt(diag(inlafit$misc$cov.intern)))## mean se
## [1,] 3.895976 1.933834summary(sdr5, "fixed")## Estimate Std. Error
## log_prec_space_time 3.875642 1.693761Fixed effects (Intercept)
inlafit$summary.fixed[ , 1:2]summary(sdr5, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.005221556 0.0308757Random effects mean and standard deviation
plot(inlafit$summary.random[[1]][,2], summary(sdr5, "report")[,1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(inlafit$summary.random[[1]][,3], summary(sdr5, "report")[,2],
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation")
abline(0, 1, col = "red")
6.1 R-INLA with custom Cmatrix
R_space_time <- kronecker(R_time, R_space)
Aconstr <- t(model.matrix(~0+factor(ID.Year), data[order(data$id.area.year), ]))
inlafitC <- inla(Observed ~
f(id.area.year, model = "generic0", Cmatrix = R_space_time,
hyper = prec.prior, diagonal = 1e-6,
extraconstr = list(A = Aconstr, e = numeric(nrow(Aconstr)))),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))These match:
summary(inlafit)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.91, Running = 0.324, Post = 0.0251, Total = 3.26
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.005 0.03 -0.063 -0.005 0.053 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant
## Precision for as.integer(IDf) 359.96 1250.87 22.33 121.43 2294.40
## mode
## Precision for as.integer(IDf) NA
##
## Marginal log-Likelihood: -1197.30
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafitC)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.14, Running = 0.301, Post = 0.0154, Total = 3.46
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.005 0.03 -0.063 -0.005 0.053 NA 0
##
## Random effects:
## Name Model
## id.area.year Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for id.area.year 359.96 1250.87 22.33 121.43 2294.40 NA
##
## Marginal log-Likelihood: -1197.30
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')6.2 Model 5b: Express GMRF as kronecker product
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(Aconstr)
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_space_time);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_raw_space_time_v(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time_v * sigma_space_time);
val += GMRF(R_space_time)(u_raw_space_time);
val -= dnorm(Aconstr * u_raw_space_time_v, Type(0), Type(0.001) * u_raw_space_time.rows(), true).sum(); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
R_space_time_adj <- R_space_time + Matrix::Diagonal(ncol(R_space_time), diagval)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Aconstr = Aconstr,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_space_time = R_space_time_adj)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = array(0, c(nrow(tmbdata$R_space), nrow(tmbdata$R_time))))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr5b <- TMB::sdreport(obj)
summary(sdr5b, "all")## Estimate Std. Error
## log_prec_space_time 3.8715748006 1.69019544
## beta0 -0.0052445905 0.03088873
## u_raw_space_time 0.1311370709 0.55260594
## u_raw_space_time -0.0428548132 0.59525451
## u_raw_space_time -0.0831857252 0.53138813
## u_raw_space_time 0.0218988990 0.64708667
## u_raw_space_time -0.0241687172 0.79648261
## u_raw_space_time -0.0175082821 0.52247525
## u_raw_space_time -0.2891936119 0.73562418
## u_raw_space_time -0.1033044691 0.71290466
## u_raw_space_time -0.1194218163 0.70268647
## u_raw_space_time 0.0101508183 0.49569318
## u_raw_space_time 0.0332521681 0.56425399
## u_raw_space_time -0.1004831612 0.92810146
## u_raw_space_time -0.0934290579 0.71640608
## u_raw_space_time -0.0515910873 0.43316219
## u_raw_space_time 0.0902472504 0.68347840
## u_raw_space_time -0.1868167318 0.72790119
## u_raw_space_time 0.1702189965 0.70665573
## u_raw_space_time 0.0750947798 0.52566009
## u_raw_space_time -0.1197617340 0.55632175
## u_raw_space_time -0.0092619062 0.47188749
## u_raw_space_time 0.1316353764 0.54548172
## u_raw_space_time -0.0691559484 0.57239868
## u_raw_space_time 0.1131554717 0.51420071
## u_raw_space_time 0.1862311497 0.72200410
## u_raw_space_time 0.0960180560 0.47163079
## u_raw_space_time 0.1256582998 0.47398914
## u_raw_space_time -0.1313770287 0.51642613
## u_raw_space_time -0.0201448080 0.50692338
## u_raw_space_time 0.0831490324 0.70034264
## u_raw_space_time 0.0726144086 0.42409358
## u_raw_space_time -0.0007664703 0.69417678
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## u_space_time 0.0055416803 0.07569874
## u_space_time 0.0115514633 0.10084912
## u_space_time 0.0132909238 0.10363668
## u_space_time 0.0019098017 0.09993332
## u_space_time -0.0030154698 0.07163822
## u_space_time -0.0127898178 0.08375906
## u_space_time 0.0011176642 0.13285621
## u_space_time 0.0151352292 0.10496931
## u_space_time 0.0050971049 0.06261251
## u_space_time -0.0210493163 0.10330789
## u_space_time 0.0134045521 0.10424760
## u_space_time 0.0223477651 0.10684051
## u_space_time -0.0173618847 0.08015505
## u_space_time 0.0085724739 0.07971731
## u_space_time 0.0022339185 0.06799653
## u_space_time -0.0151444121 0.08046931
## u_space_time 0.0144752840 0.08518390
## u_space_time -0.0040105941 0.07282404
## u_space_time -0.0142140695 0.10329088
## u_space_time -0.0118812334 0.06951069
## u_space_time -0.0170638044 0.07166879
## u_space_time 0.0090674751 0.07396427
## u_space_time 0.0033314384 0.07319217
## u_space_time -0.0196233255 0.10505561
## u_space_time -0.0046457760 0.06095962
## u_space_time -0.0128569990 0.10218470
## u_space_time 0.0054209155 0.06968715Hyper parmaters
summary(sdr5, "fixed")## Estimate Std. Error
## log_prec_space_time 3.875642 1.693761summary(sdr5b, "fixed")## Estimate Std. Error
## log_prec_space_time 3.871575 1.690195Intercept
summary(sdr5, "random")[1, ]## Estimate Std. Error
## -0.005221556 0.030875695summary(sdr5b, "random")[1, ]## Estimate Std. Error
## -0.00524459 0.03088873Random effects
plot(summary(sdr5, "report")[ , 1],
summary(sdr5b, "report")[ , 1])
abline(0, 1, col = "red")
plot(summary(sdr5, "report")[ , 2],
summary(sdr5b, "report")[ , 2])
abline(0, 1, col = "red")
6.3 Model 5c: explicit sum-to-zero constraint
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(L_space_time)
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(LRL_space_time);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_VECTOR(u_raw_space_time);
vector<Type> u_space_time(L_space_time * u_raw_space_time * sigma_space_time);
val += GMRF(LRL_space_time)(u_raw_space_time);
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
qrc <- qr(t(Aconstr))
L_space_time <- qr.Q(qrc, complete=TRUE)[ , (nrow(Aconstr)+1):ncol(Aconstr)]
R_space_time_adj <- R_space_time + Matrix::Diagonal(ncol(R_space_time), diagval)
tmbdata <- list(y = data$Observed,
E = data$Expected,
L_space_time = L_space_time,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
LRL_space_time = as(t(L_space_time) %*% R_space_time %*% L_space_time, "dgCMatrix"))
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = numeric(ncol(tmbdata$L_space_time)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr5c <- TMB::sdreport(obj)
summary(sdr5c, "all")## Estimate Std. Error
## log_prec_space_time 3.875661e+00 1.69378108
## beta0 -5.221459e-03 0.03087566
## u_raw_space_time -2.896786e-02 0.49002883
## u_raw_space_time 1.117100e-01 0.53582481
## u_raw_space_time -8.877444e-02 0.59474099
## u_raw_space_time 9.327017e-02 0.49013512
## u_raw_space_time 1.662071e-01 0.71131738
## u_raw_space_time 7.612835e-02 0.47365455
## u_raw_space_time 1.057382e-01 0.45291800
## u_raw_space_time -1.508944e-01 0.53754828
## u_raw_space_time -3.983940e-02 0.51205428
## u_raw_space_time 6.331016e-02 0.70261911
## u_raw_space_time 5.276750e-02 0.40952776
## u_raw_space_time -2.046752e-02 0.70584993
## u_raw_space_time 1.040578e-01 0.46992149
## u_raw_space_time 1.254558e-01 0.56131317
## u_raw_space_time -1.378408e-02 0.60364491
## u_raw_space_time -6.799266e-02 0.54046659
## u_raw_space_time 4.281944e-02 0.65596208
## u_raw_space_time -1.293708e-02 0.80311525
## u_raw_space_time -3.548134e-02 0.53336628
## u_raw_space_time -9.496102e-02 0.71299924
## u_raw_space_time -1.533059e-01 0.72586155
## u_raw_space_time -1.117008e-01 0.71024056
## u_raw_space_time 8.187905e-03 0.50659036
## u_raw_space_time 3.481731e-02 0.57389435
## u_raw_space_time -1.169782e-01 0.93547937
## u_raw_space_time -1.042277e-01 0.72481393
## u_raw_space_time -3.656632e-02 0.44488580
## u_raw_space_time 1.494457e-01 0.69758261
## u_raw_space_time -6.052097e-02 0.72307000
## u_raw_space_time 7.882757e-02 0.70509371
## u_raw_space_time 7.363387e-02 0.53588237
## u_raw_space_time -1.072031e-01 0.56510612
## u_raw_space_time 9.760119e-03 0.48332324
## u_raw_space_time 1.393751e-01 0.55678205
## u_raw_space_time -4.154895e-02 0.58035940
## u_raw_space_time 1.166096e-01 0.52535747
## u_raw_space_time 5.334841e-02 0.71668358
## u_raw_space_time 7.864860e-02 0.48103445
## u_raw_space_time 1.697666e-01 0.49371254
## u_raw_space_time -7.204852e-02 0.52046641
## u_raw_space_time 3.107290e-02 0.51799457
## u_raw_space_time 3.761017e-02 0.70605369
## u_raw_space_time 8.436500e-02 0.43807581
## u_raw_space_time 6.196488e-03 0.70196189
## u_raw_space_time 1.429793e-01 0.50505933
## u_raw_space_time 2.918076e-02 0.55342567
## u_raw_space_time -2.716430e-02 0.60238630
## u_raw_space_time 5.400429e-03 0.53647385
## u_raw_space_time 1.031391e-01 0.65840469
## u_raw_space_time 1.141441e-01 0.80643522
## u_raw_space_time 1.755919e-02 0.53106364
## u_raw_space_time -1.052559e-01 0.71263649
## u_raw_space_time -6.765124e-02 0.71725778
## u_raw_space_time -8.196845e-02 0.70648369
## u_raw_space_time 1.301246e-02 0.50474837
## u_raw_space_time 6.812252e-02 0.57385190
## u_raw_space_time -2.648955e-02 0.93004208
## u_raw_space_time -2.714413e-02 0.71959422
## u_raw_space_time 2.963948e-02 0.44255459
## u_raw_space_time 1.366129e-02 0.68722560
## u_raw_space_time -8.772737e-02 0.72355504
## u_raw_space_time 8.426789e-02 0.70439432
## u_raw_space_time 5.242673e-02 0.53272270
## u_raw_space_time -2.204619e-02 0.55797802
## u_raw_space_time 4.118785e-02 0.48209850
## u_raw_space_time 5.135759e-02 0.54633895
## u_raw_space_time 9.737600e-03 0.57795249
## u_raw_space_time 2.906395e-02 0.51710288
## u_raw_space_time 1.010325e-01 0.71840330
## u_raw_space_time -1.356130e-02 0.47558813
## u_raw_space_time 4.793382e-02 0.47580551
## u_raw_space_time -3.396200e-02 0.51639123
## u_raw_space_time -1.050523e-02 0.51565214
## u_raw_space_time 8.072804e-02 0.70666886
## u_raw_space_time 1.212424e-02 0.43131094
## u_raw_space_time 1.564140e-01 0.71134443
## u_raw_space_time 2.483145e-02 0.49122805
## u_raw_space_time 6.030684e-03 0.55574872
## u_raw_space_time 1.289575e-03 0.60510126
## u_raw_space_time -1.165114e-01 0.54702943
## u_raw_space_time 1.017843e-01 0.66121472
## u_raw_space_time 1.592598e-01 0.81439821
## u_raw_space_time 1.498130e-03 0.53456636
## u_raw_space_time -1.562681e-01 0.71972983
## u_raw_space_time -1.892618e-01 0.73172032
## u_raw_space_time -1.546229e-03 0.70587604
## u_raw_space_time 3.385589e-02 0.50928111
## u_raw_space_time 9.586200e-02 0.57979409
## u_raw_space_time -3.955809e-03 0.93155313
## u_raw_space_time -1.907085e-01 0.73644144
## u_raw_space_time -3.527582e-02 0.44700446
## u_raw_space_time 5.293025e-03 0.68985073
## u_raw_space_time -2.399538e-02 0.72257146
## u_raw_space_time 3.573469e-02 0.70434177
## u_raw_space_time 6.767211e-02 0.53739861
## u_raw_space_time -1.384949e-01 0.57082525
## u_raw_space_time 8.387036e-02 0.48990829
## u_raw_space_time 2.653423e-02 0.54881194
## u_raw_space_time 5.191773e-03 0.58127295
## u_raw_space_time -2.471113e-03 0.52033771
## u_raw_space_time -1.799865e-02 0.71683707
## u_raw_space_time 7.861442e-02 0.48310875
## u_raw_space_time 3.448407e-02 0.47903921
## u_raw_space_time -1.585014e-02 0.51930421
## u_raw_space_time -1.070567e-02 0.51932984
## u_raw_space_time 4.470692e-02 0.70749842
## u_raw_space_time 1.577249e-02 0.43589527
## u_raw_space_time 9.923498e-02 0.70740016
## u_raw_space_time -5.912874e-03 0.49471035
## u_raw_space_time -1.259230e-01 0.56658354
## u_raw_space_time 4.745194e-03 0.61089577
## u_raw_space_time 2.464954e-03 0.54622606
## u_raw_space_time 1.130891e-02 0.66203265
## u_raw_space_time 2.817239e-01 0.83664971
## u_raw_space_time 5.496777e-02 0.54237877
## u_raw_space_time 3.625995e-02 0.71447087
## u_raw_space_time 3.810031e-02 0.72229526
## u_raw_space_time 3.007552e-02 0.71078922
## u_raw_space_time 3.422332e-02 0.51607180
## u_raw_space_time 5.255617e-02 0.58249138
## u_raw_space_time 8.525893e-02 0.93704587
## u_raw_space_time -2.611536e-03 0.72640450
## u_raw_space_time 1.988553e-03 0.45420126
## u_raw_space_time -1.220690e-01 0.70084663
## u_raw_space_time 1.752381e-02 0.72703459
## u_raw_space_time -1.263443e-01 0.71488884
## u_raw_space_time -1.474238e-02 0.54167604
## u_raw_space_time 3.119781e-03 0.56712501
## u_raw_space_time 1.102925e-01 0.49949162
## u_raw_space_time -7.211745e-02 0.55727731
## u_raw_space_time 1.277089e-01 0.59461491
## u_raw_space_time -1.094927e-01 0.53269327
## u_raw_space_time -6.201786e-02 0.72217941
## u_raw_space_time 4.685072e-02 0.48797762
## u_raw_space_time -1.026210e-01 0.49134554
## u_raw_space_time 2.114077e-02 0.52598577
## u_raw_space_time -1.471229e-02 0.52626047
## u_raw_space_time -1.085494e-02 0.71182631
## u_raw_space_time -2.963799e-02 0.44425195
## u_raw_space_time 9.871061e-02 0.71232730
## u_raw_space_time -1.773143e-02 0.50168151
## u_raw_space_time -4.610952e-04 0.54995250
## u_raw_space_time 3.234263e-03 0.60135866
## u_raw_space_time -3.179073e-02 0.53618806
## u_raw_space_time -3.507134e-03 0.65322428
## u_raw_space_time 1.022957e-01 0.80459983
## u_raw_space_time -7.689696e-03 0.53047366
## u_raw_space_time 7.516242e-02 0.70746395
## u_raw_space_time -4.559210e-02 0.71488205
## u_raw_space_time -1.456461e-03 0.70223090
## u_raw_space_time -1.367097e-02 0.50445977
## u_raw_space_time 2.052053e-03 0.57111380
## u_raw_space_time 8.168161e-02 0.93073615
## u_raw_space_time 1.055056e-01 0.72262246
## u_raw_space_time -1.813890e-02 0.44171332
## u_raw_space_time 9.799461e-04 0.68623349
## u_raw_space_time 5.135191e-02 0.71959203
## u_raw_space_time -4.407075e-02 0.70110391
## u_raw_space_time 1.693687e-02 0.53095267
## u_raw_space_time -1.973287e-02 0.55696472
## u_raw_space_time 2.021974e-02 0.48081462
## u_raw_space_time -9.580816e-03 0.54419284
## u_raw_space_time 1.696595e-02 0.57723660
## u_raw_space_time 1.385587e-02 0.51582970
## u_raw_space_time -7.614156e-02 0.71510206
## u_raw_space_time -2.079713e-02 0.47468300
## u_raw_space_time -4.773364e-02 0.47479278
## u_raw_space_time 2.294338e-02 0.51485525
## u_raw_space_time 1.095689e-02 0.51508337
## u_raw_space_time 6.008461e-02 0.70533331
## u_raw_space_time -1.994157e-02 0.43093247
## u_raw_space_time -4.022360e-03 0.70008239
## u_raw_space_time -9.345570e-03 0.48986777
## u_raw_space_time 6.939797e-02 0.55814917
## u_raw_space_time 8.268845e-02 0.61029828
## u_raw_space_time 1.372738e-01 0.55226795
## u_raw_space_time 2.991210e-02 0.65892462
## u_raw_space_time 2.494433e-02 0.80540528
## u_raw_space_time 8.199898e-02 0.54062001
## u_raw_space_time 1.118785e-01 0.71453788
## u_raw_space_time 7.689117e-02 0.72046688
## u_raw_space_time 8.792903e-02 0.70988866
## u_raw_space_time 4.151535e-02 0.51212018
## u_raw_space_time 1.851199e-02 0.57736571
## u_raw_space_time 8.387032e-02 0.93422165
## u_raw_space_time 9.708175e-04 0.72249292
## u_raw_space_time 7.754202e-02 0.45300377
## u_raw_space_time -1.096374e-02 0.69116599
## u_raw_space_time 7.416741e-02 0.72507168
## u_raw_space_time 1.801511e-01 0.71993259
## u_raw_space_time 1.678007e-03 0.53732248
## u_raw_space_time 1.147995e-01 0.56942241
## u_raw_space_time 3.120550e-02 0.48841694
## u_raw_space_time -9.527073e-03 0.55052012
## u_raw_space_time 9.915926e-02 0.58834367
## u_raw_space_time 3.450840e-02 0.52279501
## u_raw_space_time 6.009026e-02 0.71854398
## u_raw_space_time -3.484092e-02 0.48263452
## u_raw_space_time 1.076601e-02 0.48052597
## u_raw_space_time 1.007020e-01 0.52676866
## u_raw_space_time 6.361112e-02 0.52392672
## u_raw_space_time -1.742557e-02 0.70874622
## u_raw_space_time 3.906163e-02 0.43965369
## u_raw_space_time 2.829454e-02 0.70535913
## u_raw_space_time 5.400971e-02 0.49845411
## u_raw_space_time 2.253524e-01 0.57176083
## u_raw_space_time 1.879848e-02 0.60758994
## u_raw_space_time -1.212813e-01 0.54802289
## u_raw_space_time 6.875960e-02 0.66017808
## u_raw_space_time -1.430155e-01 0.81269941
## u_raw_space_time -4.547891e-03 0.53724305
## u_raw_space_time -9.553095e-02 0.71206034
## u_raw_space_time -2.532906e-01 0.74149695
## u_raw_space_time -2.548296e-02 0.70707519
## u_raw_space_time 2.745878e-02 0.51198782
## u_raw_space_time 6.998141e-02 0.57956613
## u_raw_space_time -7.763547e-02 0.93381424
## u_raw_space_time -1.625827e-01 0.73080617
## u_raw_space_time -2.788396e-02 0.44996366
## u_raw_space_time 1.748836e-01 0.70139923
## u_raw_space_time -9.548427e-02 0.72623962
## u_raw_space_time 6.528005e-02 0.70560946
## u_raw_space_time 1.000625e-01 0.54168720
## u_raw_space_time -1.320698e-01 0.57036517
## u_raw_space_time 1.884389e-02 0.48853393
## u_raw_space_time 1.586740e-01 0.56120552
## u_raw_space_time -4.651460e-02 0.58392774
## u_raw_space_time 1.993622e-01 0.54031097
## u_raw_space_time 1.160962e-01 0.72035673
## u_raw_space_time 7.786324e-02 0.48476841
## u_raw_space_time 1.374061e-01 0.48881903
## u_raw_space_time -5.436886e-02 0.52277652
## u_raw_space_time 6.994693e-02 0.52499241
## u_raw_space_time 1.159074e-01 0.71289887
## u_raw_space_time 1.004666e-01 0.44442770
## u_raw_space_time 8.869740e-02 0.70848157
## u_raw_space_time 1.010203e-01 0.50086118
## u_raw_space_time 1.286865e-01 0.55961215
## u_raw_space_time 5.751098e-02 0.60829104
## u_raw_space_time -2.386206e-03 0.54087857
## u_raw_space_time -8.002434e-03 0.65806706
## u_raw_space_time -1.419413e-01 0.81254931
## u_raw_space_time -7.010069e-03 0.53653739
## u_raw_space_time 7.715893e-02 0.71102010
## u_raw_space_time -2.125026e-02 0.71722485
## u_raw_space_time 1.174965e-02 0.70616524
## u_raw_space_time -2.751855e-03 0.51088722
## u_raw_space_time -7.138769e-03 0.57681697
## u_raw_space_time 3.570567e-02 0.93203605
## u_raw_space_time -3.402961e-02 0.72143854
## u_raw_space_time 1.189620e-02 0.44886102
## u_raw_space_time 1.011062e-01 0.69438344
## u_raw_space_time 2.681080e-02 0.72262122
## u_raw_space_time 2.287198e-01 0.72627498
## u_raw_space_time 1.162345e-02 0.53677764
## u_raw_space_time -1.591363e-02 0.56206419
## u_raw_space_time -4.097022e-02 0.48855617
## u_raw_space_time 6.239345e-02 0.55142827
## u_raw_space_time -5.291827e-02 0.58391560
## u_raw_space_time 1.095313e-01 0.52724203
## u_raw_space_time 8.174728e-02 0.71829796
## u_raw_space_time -5.544124e-02 0.48337941
## u_raw_space_time 3.403254e-02 0.47999113
## u_raw_space_time 4.676366e-02 0.52193833
## u_raw_space_time 4.325409e-02 0.52236253
## u_raw_space_time 3.144643e-03 0.70788762
## u_raw_space_time 3.606829e-02 0.43878688
## u_raw_space_time -7.675990e-03 0.70470926
## u_raw_space_time 7.568648e-02 0.49871800
## u_raw_space_time 3.599749e-03 0.54549388
## u_raw_space_time -3.499965e-02 0.60071149
## u_raw_space_time 3.984015e-02 0.53370522
## u_raw_space_time 1.499742e-02 0.65179253
## u_raw_space_time 4.012326e-01 0.85974469
## u_raw_space_time 4.952847e-02 0.53015578
## u_raw_space_time -5.011341e-02 0.70300672
## u_raw_space_time 1.010111e-01 0.71488112
## u_raw_space_time -6.041636e-02 0.70179258
## u_raw_space_time 1.455181e-02 0.50311558
## u_raw_space_time 1.964120e-02 0.56986149
## u_raw_space_time -7.499287e-02 0.92905095
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## u_space_time -2.760656e-02 0.12181043
## u_space_time -1.973311e-02 0.08088638
## u_space_time 2.023538e-02 0.10595377
## u_space_time -4.422932e-02 0.12186863
## u_space_time -9.887661e-03 0.10101446
## u_space_time -1.934717e-04 0.07127441
## u_space_time -5.145540e-03 0.08144535
## u_space_time -1.307466e-02 0.13423052
## u_space_time -5.215088e-02 0.12987339
## u_space_time -6.358770e-03 0.06260795
## u_space_time 2.886196e-02 0.10707588
## u_space_time -2.295986e-03 0.10216033
## u_space_time 1.843596e-02 0.10295526
## u_space_time 3.570812e-03 0.07514476
## u_space_time -6.003661e-03 0.07887642
## u_space_time -1.528582e-02 0.07178839
## u_space_time 1.199825e-02 0.07858886
## u_space_time -3.425159e-02 0.09742273
## u_space_time 3.484568e-02 0.09012500
## u_space_time 2.410777e-02 0.10720375
## u_space_time 1.504660e-02 0.07102487
## u_space_time 2.600053e-02 0.07714936
## u_space_time 2.745989e-03 0.07282326
## u_space_time 1.665476e-02 0.07784025
## u_space_time -7.074780e-03 0.10106638
## u_space_time 1.710806e-02 0.06584677
## u_space_time -1.355529e-02 0.10219622
## u_space_time 1.968595e-02 0.07481459
## u_space_time -7.639904e-03 0.07793014
## u_space_time -9.409322e-03 0.08665348
## u_space_time -1.825428e-02 0.08078797
## u_space_time 2.126160e-02 0.09852801
## u_space_time 8.422746e-03 0.11516631
## u_space_time -6.442170e-03 0.07576478
## u_space_time -4.413251e-02 0.12102678
## u_space_time -3.445351e-02 0.11517567
## u_space_time -2.574006e-02 0.10777014
## u_space_time 3.298075e-03 0.07145693
## u_space_time 1.436540e-02 0.08389153
## u_space_time -3.195518e-02 0.14209648
## u_space_time -1.950805e-02 0.10645927
## u_space_time -9.771939e-03 0.06380759
## u_space_time 4.660671e-02 0.12327773
## u_space_time -3.132224e-02 0.11328594
## u_space_time 7.886144e-03 0.10013190
## u_space_time 2.031601e-02 0.08132590
## u_space_time -3.079935e-02 0.09233554
## u_space_time 6.035154e-03 0.06836551
## u_space_time 2.630883e-02 0.08713822
## u_space_time -9.566055e-03 0.08325052
## u_space_time 2.221737e-02 0.08066119
## u_space_time 2.211595e-02 0.10652489
## u_space_time 7.673657e-03 0.06765142
## u_space_time 2.980562e-02 0.08178293
## u_space_time -2.141048e-02 0.07985016
## u_space_time -2.895314e-03 0.07294266
## u_space_time 2.836784e-02 0.10949143
## u_space_time 3.652830e-03 0.06063699
## u_space_time 1.859465e-02 0.10391980
## u_space_time 1.637177e-02 0.07403079
## u_space_time 2.324291e-02 0.08431419
## u_space_time 9.043397e-03 0.08660957
## u_space_time 3.422370e-03 0.07592609
## u_space_time -9.200624e-03 0.09405453
## u_space_time -3.572317e-02 0.12798950
## u_space_time -5.585131e-03 0.07549957
## u_space_time 2.612817e-02 0.10804658
## u_space_time -8.912593e-03 0.10237848
## u_space_time 2.159058e-02 0.10590283
## u_space_time -2.289931e-03 0.07134932
## u_space_time -8.721528e-03 0.08216227
## u_space_time 1.060329e-02 0.13374296
## u_space_time -1.381066e-02 0.10426559
## u_space_time 6.392726e-03 0.06291714
## u_space_time 1.300352e-02 0.09977379
## u_space_time 9.945559e-03 0.10321603
## u_space_time -1.708129e-02 0.10373608
## u_space_time -3.174308e-03 0.07518069
## u_space_time 8.724006e-04 0.07857714
## u_space_time -1.148797e-02 0.07010950
## u_space_time 1.131810e-03 0.07681257
## u_space_time -1.688378e-02 0.08601309
## u_space_time -2.100362e-04 0.07256701
## u_space_time -1.737020e-02 0.10506119
## u_space_time -1.482804e-03 0.06674890
## u_space_time 2.094754e-02 0.07457394
## u_space_time 8.826922e-03 0.07387175
## u_space_time 8.158439e-03 0.07398686
## u_space_time -1.269602e-02 0.10228416
## u_space_time 9.397529e-03 0.06218383
## u_space_time -1.299158e-02 0.10193320
## u_space_time 4.914463e-03 0.06927548
## u_space_time 7.342093e-02 0.13001446
## u_space_time -1.451162e-02 0.08814700
## u_space_time 1.319152e-02 0.08028987
## u_space_time -2.850380e-03 0.09316743
## u_space_time 5.494118e-03 0.11496809
## u_space_time 2.075804e-03 0.07526961
## u_space_time -6.979129e-02 0.14473045
## u_space_time -4.278040e-03 0.10149661
## u_space_time -4.745372e-02 0.12294400
## u_space_time 1.511978e-02 0.07567350
## u_space_time 4.193411e-03 0.08116476
## u_space_time -6.244300e-02 0.16332934
## u_space_time -1.333472e-02 0.10346492
## u_space_time -4.018099e-03 0.06208192
## u_space_time 2.867059e-02 0.10574759
## u_space_time -6.118224e-02 0.13805938
## u_space_time 2.510864e-02 0.10556567
## u_space_time 1.090785e-02 0.07639250
## u_space_time -3.097224e-02 0.09046941
## u_space_time 3.042067e-03 0.06797993
## u_space_time 1.477823e-02 0.07900390
## u_space_time -3.439273e-03 0.08181500
## u_space_time 2.503341e-02 0.07992859
## u_space_time 6.472406e-03 0.10089764
## u_space_time 2.540249e-02 0.07769397
## u_space_time 3.612001e-02 0.08485736
## u_space_time -3.231937e-02 0.08688438
## u_space_time 7.412620e-03 0.07397856
## u_space_time 3.230928e-03 0.10012288
## u_space_time 2.281343e-02 0.06924325
## u_space_time -3.787574e-03 0.10003588
## u_space_time 2.789333e-02 0.08014692
## u_space_time 3.782561e-02 0.09397594
## u_space_time -9.965863e-03 0.08683065
## u_space_time -2.047661e-02 0.08158412
## u_space_time 1.435304e-02 0.09554556
## u_space_time -2.478534e-02 0.12084494
## u_space_time -8.474014e-03 0.07603514
## u_space_time -2.292633e-02 0.10467521
## u_space_time -9.404371e-03 0.10209262
## u_space_time -2.292500e-02 0.10573428
## u_space_time 9.338094e-04 0.07128417
## u_space_time 1.211910e-02 0.08323690
## u_space_time -1.487029e-02 0.13424895
## u_space_time -2.629284e-02 0.10956078
## u_space_time -9.495982e-03 0.06358512
## u_space_time 1.947469e-02 0.10155683
## u_space_time -2.402123e-02 0.10815227
## u_space_time 6.121633e-03 0.09968446
## u_space_time 1.788013e-02 0.07989757
## u_space_time -1.200573e-02 0.08017005
## u_space_time 5.062297e-03 0.06836055
## u_space_time 2.075914e-02 0.08281235
## u_space_time -1.792915e-02 0.08622098
## u_space_time 1.632804e-02 0.07600010
## u_space_time 9.717789e-03 0.10149419
## u_space_time 1.241464e-02 0.06943531
## u_space_time 1.906571e-02 0.07185194
## u_space_time -1.450571e-02 0.07554558
## u_space_time -6.741789e-03 0.07366161
## u_space_time 3.103382e-02 0.11161350
## u_space_time 7.198670e-03 0.06114329
## u_space_time 9.615493e-03 0.10098604
## u_space_time 4.916623e-03 0.06906328
## u_space_time 3.819871e-03 0.07682847
## u_space_time 2.347824e-03 0.08544227
## u_space_time 1.878509e-02 0.08129718
## u_space_time -1.599658e-02 0.09638272
## u_space_time 1.421276e-02 0.11651543
## u_space_time 5.442588e-03 0.07548181
## u_space_time 1.143537e-02 0.10061021
## u_space_time 1.316678e-02 0.10338218
## u_space_time 1.827712e-03 0.09970276
## u_space_time -3.081848e-03 0.07145022
## u_space_time -1.282185e-02 0.08358453
## u_space_time 1.038361e-03 0.13257224
## u_space_time 1.500287e-02 0.10470843
## u_space_time 5.001103e-03 0.06241086
## u_space_time -2.104651e-02 0.10312083
## u_space_time 1.327726e-02 0.10399017
## u_space_time 2.217949e-02 0.10655591
## u_space_time -1.737563e-02 0.08000413
## u_space_time 8.466008e-03 0.07949897
## u_space_time 2.145108e-03 0.06779703
## u_space_time -1.516601e-02 0.08031409
## u_space_time 1.434149e-02 0.08493793
## u_space_time -4.073476e-03 0.07264960
## u_space_time -1.423749e-02 0.10309181
## u_space_time -1.191425e-02 0.06935991
## u_space_time -1.707507e-02 0.07153706
## u_space_time 8.957015e-03 0.07374778
## u_space_time 3.242099e-03 0.07298560
## u_space_time -1.962978e-02 0.10486174
## u_space_time -4.704203e-03 0.06079527
## u_space_time -1.288934e-02 0.10197529
## u_space_time 5.323245e-03 0.06948513Hyper parameters
cbind(inlafit$misc$theta.mode,
sqrt(diag(inlafit$misc$cov.intern)))## [,1] [,2]
## [1,] 3.895976 1.933834summary(sdr5, "fixed")## Estimate Std. Error
## log_prec_space_time 3.875642 1.693761summary(sdr5c, "fixed")## Estimate Std. Error
## log_prec_space_time 3.875661 1.693781Intercept
inlafit$summary.fixed[ , 1:2]summary(sdr5, "random")[1, ]## Estimate Std. Error
## -0.005221556 0.030875695summary(sdr5c, "random")[1, ]## Estimate Std. Error
## -0.005221459 0.030875656Random effects
sdr5csum <- summary(sdr5, "all")
plot(summary(sdr5, "report")[ , 1],
summary(sdr5c, "report")[ , 1])
abline(0, 1, col = "red")
plot(summary(sdr5, "report")[ , 2],
summary(sdr5c, "report")[ , 2])
abline(0, 1, col = "red")
7 Model 6: IID x RW1
Use and IID spatial effect, but a RW1 time effect. In this case, R-INLA by default does not add any constraint because the main effect is proper.
To match the Cmatrix implementation with the IID x RW1 implementation with the group option, the argument rankdef needs to be specified as the rank deficiency of the Kronecker product precision matrix.
This is the number of areas.
inlafit6 <- inla(Observed ~
f(as.integer(IDf), model = "iid", hyper = prec.prior,
graph = adj.mat, diagonal = diagval,
group = ID.Year, control.group = list(model = "rw1", scale.model = TRUE)),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))Note that rank deficiency line in the log file pertains to the rank deficiency of the main IID effect, which has zero rank deficiency:
grep(".*rank.*", inlafit6$logfile, value = TRUE)## [1] " computed/guessed rank-deficiency = [0]"For the group model, the R-INLA log file does not make any comment about the rank
deficiency. However, we will see below that internally that the rank deficiency
which is imposed RW1 group model is accounted for.
7.1 R-INLA with custom Cmatrix
R_space <- diag(ncol(adj.mat))
D_time <- diff(diag(length(levels(data$Yearf))), differences = 1)
R_time <- Matrix::Matrix(t(D_time) %*% D_time)
R_time_adj <- R_time + Matrix::Diagonal(ncol(R_time), 1e-6)
R_time_scaled <- inla.scale.model(R_time, constr = list(A = matrix(1, ncol = ncol(R_time)), e = 0))
R_space_time <- kronecker(R_time_scaled, R_space)
inlafit6C <- inla(Observed ~
f(id.area.year, model = "generic0", Cmatrix = R_space_time, hyper = prec.prior,
diagonal = diagval, rankdef = 32),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))Here’s the rankdef comment in the logfile:
grep(".*rank.*", inlafit6C$logfile, value = TRUE)## [1] " rank-deficiency is *defined* [32]"These results match:
summary(inlafit6)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.81, Running = 0.283, Post = 0.0147, Total = 3.11
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.265 4.067 -8.236 -0.265 7.706 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) IID model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for as.integer(IDf) 463.12 695.03 42.18 261.51 2170.08 NA
##
## Marginal log-Likelihood: -1089.57
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafit6C)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.76, Running = 0.25, Post = 0.0143, Total = 3.02
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.265 4.067 -8.236 -0.265 7.706 NA 0
##
## Random effects:
## Name Model
## id.area.year Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for id.area.year 463.11 694.95 42.18 261.52 2169.93 NA
##
## Marginal log-Likelihood: -1089.57
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Marginal likelihood
inlafit6$mlik## [,1]
## log marginal-likelihood (integration) -1090.779
## log marginal-likelihood (Gaussian) -1089.572inlafit6C$mlik## [,1]
## log marginal-likelihood (integration) -1090.779
## log marginal-likelihood (Gaussian) -1089.572Same constraints (none)
all.equal(inlafit6$misc$configs$constr,
inlafit6C$misc$configs$constr)## [1] TRUEHyper parameters
inlafit6$internal.summary.hyperpar[, 1:2]inlafit6C$internal.summary.hyperpar[, 1:2]Fixed effects
inlafit6$summary.fixedinlafit6C$summary.fixedRandom effects
plot(inlafit6$summary.random[[1]][,2],
inlafit6C$summary.random[[1]][,2],
main = "Random effect mean")
abline(a = 0, b = 1, col = "red")
plot(inlafit6$summary.random[[1]][,3],
inlafit6C$summary.random[[1]][,3],
main = "Random effect standard deviation")
abline(a = 0, b = 1, col = "red")
7.2 TMB
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_time);
DATA_SPARSE_MATRIX(R_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time * sigma_space_time);
val += SEPARABLE(GMRF(R_time), GMRF(R_space))(u_raw_space_time);
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_space = as(R_space, "dgTMatrix"),
R_time = R_time_adj)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = array(0, c(nrow(tmbdata$R_space), nrow(tmbdata$R_time))))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)## Warning in getParameterOrder(data, parameters, new.env(), DLL = DLL): Expected
## sparse matrix of class 'dgTMatrix'.## Error in getParameterOrder(data, parameters, new.env(), DLL = DLL): Error when reading the variable: 'R_time'. Please check data and parameters.tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr6 <- TMB::sdreport(obj)
summary(sdr6, "all")## Estimate Std. Error
## log_prec_space_time 3.875661e+00 1.69378108
## beta0 -5.221459e-03 0.03087566
## u_raw_space_time -2.896786e-02 0.49002883
## u_raw_space_time 1.117100e-01 0.53582481
## u_raw_space_time -8.877444e-02 0.59474099
## u_raw_space_time 9.327017e-02 0.49013512
## u_raw_space_time 1.662071e-01 0.71131738
## u_raw_space_time 7.612835e-02 0.47365455
## u_raw_space_time 1.057382e-01 0.45291800
## u_raw_space_time -1.508944e-01 0.53754828
## u_raw_space_time -3.983940e-02 0.51205428
## u_raw_space_time 6.331016e-02 0.70261911
## u_raw_space_time 5.276750e-02 0.40952776
## u_raw_space_time -2.046752e-02 0.70584993
## u_raw_space_time 1.040578e-01 0.46992149
## u_raw_space_time 1.254558e-01 0.56131317
## u_raw_space_time -1.378408e-02 0.60364491
## u_raw_space_time -6.799266e-02 0.54046659
## u_raw_space_time 4.281944e-02 0.65596208
## u_raw_space_time -1.293708e-02 0.80311525
## u_raw_space_time -3.548134e-02 0.53336628
## u_raw_space_time -9.496102e-02 0.71299924
## u_raw_space_time -1.533059e-01 0.72586155
## u_raw_space_time -1.117008e-01 0.71024056
## u_raw_space_time 8.187905e-03 0.50659036
## u_raw_space_time 3.481731e-02 0.57389435
## u_raw_space_time -1.169782e-01 0.93547937
## u_raw_space_time -1.042277e-01 0.72481393
## u_raw_space_time -3.656632e-02 0.44488580
## u_raw_space_time 1.494457e-01 0.69758261
## u_raw_space_time -6.052097e-02 0.72307000
## u_raw_space_time 7.882757e-02 0.70509371
## u_raw_space_time 7.363387e-02 0.53588237
## u_raw_space_time -1.072031e-01 0.56510612
## u_raw_space_time 9.760119e-03 0.48332324
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## u_space_time -6.118224e-02 0.13805938
## u_space_time 2.510864e-02 0.10556567
## u_space_time 1.090785e-02 0.07639250
## u_space_time -3.097224e-02 0.09046941
## u_space_time 3.042067e-03 0.06797993
## u_space_time 1.477823e-02 0.07900390
## u_space_time -3.439273e-03 0.08181500
## u_space_time 2.503341e-02 0.07992859
## u_space_time 6.472406e-03 0.10089764
## u_space_time 2.540249e-02 0.07769397
## u_space_time 3.612001e-02 0.08485736
## u_space_time -3.231937e-02 0.08688438
## u_space_time 7.412620e-03 0.07397856
## u_space_time 3.230928e-03 0.10012288
## u_space_time 2.281343e-02 0.06924325
## u_space_time -3.787574e-03 0.10003588
## u_space_time 2.789333e-02 0.08014692
## u_space_time 3.782561e-02 0.09397594
## u_space_time -9.965863e-03 0.08683065
## u_space_time -2.047661e-02 0.08158412
## u_space_time 1.435304e-02 0.09554556
## u_space_time -2.478534e-02 0.12084494
## u_space_time -8.474014e-03 0.07603514
## u_space_time -2.292633e-02 0.10467521
## u_space_time -9.404371e-03 0.10209262
## u_space_time -2.292500e-02 0.10573428
## u_space_time 9.338094e-04 0.07128417
## u_space_time 1.211910e-02 0.08323690
## u_space_time -1.487029e-02 0.13424895
## u_space_time -2.629284e-02 0.10956078
## u_space_time -9.495982e-03 0.06358512
## u_space_time 1.947469e-02 0.10155683
## u_space_time -2.402123e-02 0.10815227
## u_space_time 6.121633e-03 0.09968446
## u_space_time 1.788013e-02 0.07989757
## u_space_time -1.200573e-02 0.08017005
## u_space_time 5.062297e-03 0.06836055
## u_space_time 2.075914e-02 0.08281235
## u_space_time -1.792915e-02 0.08622098
## u_space_time 1.632804e-02 0.07600010
## u_space_time 9.717789e-03 0.10149419
## u_space_time 1.241464e-02 0.06943531
## u_space_time 1.906571e-02 0.07185194
## u_space_time -1.450571e-02 0.07554558
## u_space_time -6.741789e-03 0.07366161
## u_space_time 3.103382e-02 0.11161350
## u_space_time 7.198670e-03 0.06114329
## u_space_time 9.615493e-03 0.10098604
## u_space_time 4.916623e-03 0.06906328
## u_space_time 3.819871e-03 0.07682847
## u_space_time 2.347824e-03 0.08544227
## u_space_time 1.878509e-02 0.08129718
## u_space_time -1.599658e-02 0.09638272
## u_space_time 1.421276e-02 0.11651543
## u_space_time 5.442588e-03 0.07548181
## u_space_time 1.143537e-02 0.10061021
## u_space_time 1.316678e-02 0.10338218
## u_space_time 1.827712e-03 0.09970276
## u_space_time -3.081848e-03 0.07145022
## u_space_time -1.282185e-02 0.08358453
## u_space_time 1.038361e-03 0.13257224
## u_space_time 1.500287e-02 0.10470843
## u_space_time 5.001103e-03 0.06241086
## u_space_time -2.104651e-02 0.10312083
## u_space_time 1.327726e-02 0.10399017
## u_space_time 2.217949e-02 0.10655591
## u_space_time -1.737563e-02 0.08000413
## u_space_time 8.466008e-03 0.07949897
## u_space_time 2.145108e-03 0.06779703
## u_space_time -1.516601e-02 0.08031409
## u_space_time 1.434149e-02 0.08493793
## u_space_time -4.073476e-03 0.07264960
## u_space_time -1.423749e-02 0.10309181
## u_space_time -1.191425e-02 0.06935991
## u_space_time -1.707507e-02 0.07153706
## u_space_time 8.957015e-03 0.07374778
## u_space_time 3.242099e-03 0.07298560
## u_space_time -1.962978e-02 0.10486174
## u_space_time -4.704203e-03 0.06079527
## u_space_time -1.288934e-02 0.10197529
## u_space_time 5.323245e-03 0.06948513Hyper parameter comparison
The standard devation from TMB is somewhat larger than INLA.
summary(sdr6, "fixed")## Estimate Std. Error
## log_prec_space_time 3.875661 1.693781cbind("mean" = inlafit6$misc$theta.mode,
"se" = sqrt(diag(inlafit6$misc$cov.intern)))## mean se
## [1,] 5.386291 1.276547Fixed effects (Intercept)
summary(sdr6, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.005221459 0.03087566inlafit6$summary.fixed[ , 1:2]inlafit6C$summary.fixed[ , 1:2]Random effects mean and standard deviation
plot(inlafit6$summary.random[[1]][,2], summary(sdr6, "report")[,1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1)
plot(inlafit6$summary.random[[1]][,3], summary(sdr6, "report")[,2],
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation")
abline(0, 1)
7.3 Model 6b: Express GMRF as a Kronecker product
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_space_time);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time * sigma_space_time);
val += GMRF(R_space_time)(u_raw_space_time);
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
## R_space_time_adj <- R_space_time + Matrix::Diagonal(ncol(R_space_time), diagval)
R_space_time_adj <- kronecker(R_time_adj, R_space)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_space_time = R_space_time_adj)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = array(0, c(nrow(tmbdata$R_space), nrow(tmbdata$R_time))))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr6b <- TMB::sdreport(obj)
summary(sdr6b, "all")## Estimate Std. Error
## log_prec_space_time 9.558215e+00 0.2596289
## beta0 -1.476750e-01 0.3478549
## u_raw_space_time 2.777543e+01 41.9670255
## u_raw_space_time -2.248671e+01 86.3536077
## u_raw_space_time 3.805117e+01 44.0952263
## u_raw_space_time -1.214896e+01 53.8202427
## u_raw_space_time 2.024617e+01 45.5889354
## u_raw_space_time -2.830052e+01 86.1829076
## u_raw_space_time 1.179770e+01 43.3376305
## u_raw_space_time -1.241089e+01 45.4971490
## u_raw_space_time -3.025881e+01 50.0311401
## u_raw_space_time 5.189801e+00 69.5374671
## u_raw_space_time -1.466181e+02 155.2819607
## u_raw_space_time 4.689832e+00 65.1244306
## u_raw_space_time -2.956390e+01 46.0953632
## u_raw_space_time -4.822344e+01 59.6468240
## u_raw_space_time 3.670473e+01 49.2571397
## u_raw_space_time -4.490938e+01 54.0678828
## u_raw_space_time 3.439532e+01 45.9310570
## u_raw_space_time -7.291325e+01 85.1027599
## u_raw_space_time -3.154737e+01 47.5817547
## u_raw_space_time 2.563180e+01 52.2343838
## u_raw_space_time 3.559764e+01 46.6155827
## u_raw_space_time 1.953930e+01 50.3851646
## u_raw_space_time 2.464318e+01 47.6707198
## u_raw_space_time -1.218690e+01 47.0191422
## u_raw_space_time 1.647580e+01 44.1963040
## u_raw_space_time 3.081966e+01 43.4987549
## u_raw_space_time 4.249013e+01 50.1282846
## u_raw_space_time 4.859942e+01 51.2194822
## u_raw_space_time -5.303429e-01 50.3410887
## u_raw_space_time 4.916402e-01 59.5315476
## u_raw_space_time 2.645714e+01 62.1310198
## u_raw_space_time 4.107684e+01 44.3680458
## u_raw_space_time 2.777958e+01 41.9562157
## u_raw_space_time -2.248593e+01 86.3483679
## u_raw_space_time 3.805745e+01 44.0848147
## u_raw_space_time -1.214454e+01 53.8118734
## u_raw_space_time 2.024459e+01 45.5790677
## u_raw_space_time -2.830802e+01 86.1776885
## u_raw_space_time 1.182237e+01 43.3268748
## u_raw_space_time -1.241401e+01 45.4871040
## u_raw_space_time -3.026107e+01 50.0219953
## u_raw_space_time 5.191598e+00 69.5310455
## u_raw_space_time -1.466181e+02 155.2791022
## u_raw_space_time 4.683221e+00 65.1174199
## u_raw_space_time -2.956845e+01 46.0854112
## u_raw_space_time -4.822123e+01 59.6391233
## u_raw_space_time 3.671116e+01 49.2477547
## u_raw_space_time -4.490582e+01 54.0594200
## u_raw_space_time 3.439143e+01 45.9211887
## u_raw_space_time -7.291236e+01 85.0975169
## u_raw_space_time -3.155541e+01 47.5720406
## u_raw_space_time 2.563730e+01 52.2257455
## u_raw_space_time 3.559202e+01 46.6058867
## u_raw_space_time 1.954610e+01 50.3762202
## u_raw_space_time 2.465307e+01 47.6612049
## u_raw_space_time -1.219753e+01 47.0090523
## u_raw_space_time 1.645954e+01 44.1859142
## u_raw_space_time 3.081191e+01 43.4882439
## u_raw_space_time 4.249638e+01 50.1191197
## u_raw_space_time 4.860450e+01 51.2104515
## u_raw_space_time -5.328857e-01 50.3320316
## u_raw_space_time 4.855319e-01 59.5238133
## u_raw_space_time 2.645986e+01 62.1238264
## u_raw_space_time 4.106091e+01 44.3578412
## u_raw_space_time 2.779327e+01 41.9464493
## u_raw_space_time -2.248436e+01 86.3436883
## u_raw_space_time 3.806220e+01 44.0754466
## u_raw_space_time -1.214402e+01 53.8043922
## u_raw_space_time 2.024163e+01 45.5703075
## u_raw_space_time -2.831462e+01 86.1730958
## u_raw_space_time 1.184745e+01 43.3169905
## u_raw_space_time -1.241161e+01 45.4780683
## u_raw_space_time -3.025707e+01 50.0138102
## u_raw_space_time 5.195185e+00 69.5253891
## u_raw_space_time -1.466181e+02 155.2766003
## u_raw_space_time 4.678529e+00 65.1111194
## u_raw_space_time -2.957732e+01 46.0763690
## u_raw_space_time -4.821667e+01 59.6321459
## u_raw_space_time 3.671566e+01 49.2392279
## u_raw_space_time -4.491531e+01 54.0515302
## u_raw_space_time 3.439218e+01 45.9122514
## u_raw_space_time -7.291058e+01 85.0929025
## u_raw_space_time -3.156284e+01 47.5632509
## u_raw_space_time 2.563991e+01 52.2180974
## u_raw_space_time 3.557262e+01 46.5973251
## u_raw_space_time 1.955141e+01 50.3683083
## u_raw_space_time 2.466446e+01 47.6527170
## u_raw_space_time -1.220143e+01 46.9996611
## u_raw_space_time 1.643620e+01 44.1764371
## u_raw_space_time 3.078050e+01 43.4788958
## u_raw_space_time 4.250886e+01 50.1107604
## u_raw_space_time 4.860635e+01 51.2022797
## u_raw_space_time -5.295596e-01 50.3238637
## u_raw_space_time 4.733819e-01 59.5167605
## u_raw_space_time 2.646531e+01 62.1174897
## u_raw_space_time 4.102115e+01 44.3488240
## u_raw_space_time 2.782015e+01 41.9376732
## u_raw_space_time -2.248202e+01 86.3395562
## u_raw_space_time 3.806633e+01 44.0671405
## u_raw_space_time -1.214737e+01 53.7978090
## u_raw_space_time 2.022925e+01 45.5627182
## u_raw_space_time -2.832033e+01 86.1691199
## u_raw_space_time 1.188258e+01 43.3079833
## u_raw_space_time -1.240309e+01 45.4700826
## u_raw_space_time -3.024656e+01 50.0066201
## u_raw_space_time 5.200557e+00 69.5204964
## u_raw_space_time -1.466180e+02 155.2744513
## u_raw_space_time 4.667508e+00 65.1055799
## u_raw_space_time -2.958997e+01 46.0682802
## u_raw_space_time -4.822647e+01 59.6256790
## u_raw_space_time 3.672704e+01 49.2315125
## u_raw_space_time -4.492080e+01 54.0445137
## u_raw_space_time 3.438967e+01 45.9043620
## u_raw_space_time -7.290793e+01 85.0889148
## u_raw_space_time -3.157780e+01 47.5553039
## u_raw_space_time 2.564810e+01 52.2113953
## u_raw_space_time 3.555657e+01 46.5897365
## u_raw_space_time 1.956357e+01 50.3613778
## u_raw_space_time 2.468585e+01 47.6452073
## u_raw_space_time -1.221486e+01 46.9909155
## u_raw_space_time 1.643201e+01 44.1678016
## u_raw_space_time 3.074335e+01 43.4705605
## u_raw_space_time 4.251943e+01 50.1033389
## u_raw_space_time 4.861358e+01 51.1948770
## u_raw_space_time -5.286200e-01 50.3165898
## u_raw_space_time 4.636442e-01 59.5104132
## u_raw_space_time 2.645664e+01 62.1120853
## u_raw_space_time 4.098372e+01 44.3406811
## u_raw_space_time 2.786470e+01 41.9298360
## u_raw_space_time -2.247887e+01 86.3359966
## u_raw_space_time 3.808725e+01 44.0596941
## u_raw_space_time -1.215448e+01 53.7921436
## u_raw_space_time 2.020702e+01 45.5562700
## u_raw_space_time -2.832516e+01 86.1657571
## u_raw_space_time 1.193754e+01 43.2998633
## u_raw_space_time -1.238774e+01 45.4631947
## u_raw_space_time -3.023793e+01 50.0003283
## u_raw_space_time 5.207742e+00 69.5163776
## u_raw_space_time -1.466180e+02 155.2726587
## u_raw_space_time 4.658678e+00 65.1008424
## u_raw_space_time -2.958924e+01 46.0613826
## u_raw_space_time -4.823375e+01 59.6199787
## u_raw_space_time 3.674546e+01 49.2246267
## u_raw_space_time -4.493050e+01 54.0382812
## u_raw_space_time 3.438475e+01 45.8975689
## u_raw_space_time -7.290440e+01 85.0855541
## u_raw_space_time -3.158336e+01 47.5484271
## u_raw_space_time 2.565348e+01 52.2056914
## u_raw_space_time 3.554409e+01 46.5831345
## u_raw_space_time 1.957417e+01 50.3554678
## u_raw_space_time 2.471727e+01 47.6386761
## u_raw_space_time -1.222038e+01 46.9829780
## u_raw_space_time 1.642279e+01 44.1601547
## u_raw_space_time 3.071018e+01 43.4631768
## u_raw_space_time 4.252000e+01 50.0969934
## u_raw_space_time 4.862641e+01 51.1882709
## u_raw_space_time -5.216228e-01 50.3102299
## u_raw_space_time 4.564333e-01 59.5048017
## u_raw_space_time 2.645068e+01 62.1075289
## u_raw_space_time 4.095826e+01 44.3333328
## u_raw_space_time 2.794136e+01 41.9228069
## u_raw_space_time -2.247487e+01 86.3330296
## u_raw_space_time 3.812586e+01 44.0531272
## u_raw_space_time -1.215696e+01 53.7874332
## u_raw_space_time 2.016686e+01 45.5510253
## u_raw_space_time -2.832911e+01 86.1630048
## u_raw_space_time 1.198844e+01 43.2927328
## u_raw_space_time -1.238211e+01 45.4573155
## u_raw_space_time -3.023096e+01 49.9949676
## u_raw_space_time 5.216742e+00 69.5130332
## u_raw_space_time -1.466179e+02 155.2712168
## u_raw_space_time 4.643639e+00 65.0969064
## u_raw_space_time -2.958345e+01 46.0555741
## u_raw_space_time -4.823843e+01 59.6150694
## u_raw_space_time 3.677115e+01 49.2185941
## u_raw_space_time -4.493597e+01 54.0329679
## u_raw_space_time 3.439438e+01 45.8917012
## u_raw_space_time -7.289999e+01 85.0828190
## u_raw_space_time -3.158766e+01 47.5425382
## u_raw_space_time 2.565603e+01 52.2009836
## u_raw_space_time 3.553538e+01 46.5775310
## u_raw_space_time 1.957486e+01 50.3506225
## u_raw_space_time 2.475876e+01 47.6331258
## u_raw_space_time -1.223409e+01 46.9758129
## u_raw_space_time 1.640112e+01 44.1535695
## u_raw_space_time 3.069899e+01 43.4565963
## u_raw_space_time 4.251913e+01 50.0916424
## u_raw_space_time 4.864502e+01 51.1824861
## u_raw_space_time -5.168652e-01 50.3047829
## u_raw_space_time 4.518039e-01 59.4999406
## u_raw_space_time 2.643903e+01 62.1038663
## u_raw_space_time 4.092087e+01 44.3271564
## u_raw_space_time 2.802968e+01 41.9167813
## u_raw_space_time -2.246997e+01 86.3306827
## u_raw_space_time 3.818259e+01 44.0474479
## u_raw_space_time -1.215484e+01 53.7836739
## u_raw_space_time 2.011785e+01 45.5469801
## u_raw_space_time -2.833214e+01 86.1608897
## u_raw_space_time 1.202842e+01 43.2866538
## u_raw_space_time -1.236897e+01 45.4525782
## u_raw_space_time -3.021707e+01 49.9906577
## u_raw_space_time 5.227537e+00 69.5104559
## u_raw_space_time -1.466179e+02 155.2701213
## u_raw_space_time 4.622405e+00 65.0937758
## u_raw_space_time -2.960591e+01 46.0504599
## u_raw_space_time -4.824045e+01 59.6109638
## u_raw_space_time 3.679602e+01 49.2135380
## u_raw_space_time -4.494545e+01 54.0284790
## u_raw_space_time 3.441036e+01 45.8868580
## u_raw_space_time -7.290310e+01 85.0806064
## u_raw_space_time -3.159039e+01 47.5376712
## u_raw_space_time 2.565580e+01 52.1972795
## u_raw_space_time 3.553070e+01 46.5729422
## u_raw_space_time 1.958245e+01 50.3467634
## u_raw_space_time 2.478506e+01 47.6287218
## u_raw_space_time -1.223855e+01 46.9695820
## u_raw_space_time 1.638496e+01 44.1480168
## u_raw_space_time 3.071052e+01 43.4508286
## u_raw_space_time 4.251687e+01 50.0872893
## u_raw_space_time 4.866134e+01 51.1776836
## u_raw_space_time -5.227457e-01 50.3002333
## u_raw_space_time 4.498350e-01 59.4958505
## u_raw_space_time 2.643010e+01 62.1010567
## u_raw_space_time 4.088956e+01 44.3219589
## u_raw_space_time 2.812731e+01 41.9117952
## u_raw_space_time -2.246417e+01 86.3289582
## u_raw_space_time 3.822453e+01 44.0430991
## u_raw_space_time -1.215646e+01 53.7808345
## u_raw_space_time 2.007677e+01 45.5440397
## u_raw_space_time -2.833427e+01 86.1593977
## u_raw_space_time 1.206681e+01 43.2816241
## u_raw_space_time -1.235641e+01 45.4489500
## u_raw_space_time -3.020456e+01 49.9873152
## u_raw_space_time 5.240139e+00 69.5086502
## u_raw_space_time -1.466178e+02 155.2693728
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## u_space_time -4.051051e-01 0.4979536
## u_space_time 3.105930e-01 0.4121680
## u_space_time -3.776415e-01 0.4509797
## u_space_time 2.892299e-01 0.3843168
## u_space_time -6.127337e-01 0.7119354
## u_space_time -2.655501e-01 0.3975108
## u_space_time 2.159362e-01 0.4381262
## u_space_time 2.985661e-01 0.3900072
## u_space_time 1.639135e-01 0.4228332
## u_space_time 2.091757e-01 0.3996976
## u_space_time -1.025895e-01 0.3941674
## u_space_time 1.385994e-01 0.3707420
## u_space_time 2.599748e-01 0.3639922
## u_space_time 3.573438e-01 0.4190542
## u_space_time 4.097194e-01 0.4276103
## u_space_time -4.243518e-03 0.4226213
## u_space_time 3.780883e-03 0.4999112
## u_space_time 2.217863e-01 0.5214888
## u_space_time 3.437347e-01 0.3703903
## u_space_time 2.401699e-01 0.3512954
## u_space_time -1.884202e-01 0.7250829
## u_space_time 3.235832e-01 0.3683844
## u_space_time -1.018161e-01 0.4516200
## u_space_time 1.676341e-01 0.3824453
## u_space_time -2.384369e-01 0.7234897
## u_space_time 1.031515e-01 0.3636203
## u_space_time -1.028795e-01 0.3815082
## u_space_time -2.533221e-01 0.4184191
## u_space_time 4.478627e-02 0.5841653
## u_space_time -1.232099e+00 1.3121854
## u_space_time 3.816268e-02 0.5470284
## u_space_time -2.489923e-01 0.3851746
## u_space_time -4.049645e-01 0.4979643
## u_space_time 3.108034e-01 0.4121825
## u_space_time -3.775461e-01 0.4509925
## u_space_time 2.890872e-01 0.3843345
## u_space_time -6.127865e-01 0.7119483
## u_space_time -2.654563e-01 0.3975245
## u_space_time 2.160786e-01 0.4381457
## u_space_time 2.984006e-01 0.3900227
## u_space_time 1.636238e-01 0.4228519
## u_space_time 2.093513e-01 0.3997167
## u_space_time -1.025771e-01 0.3941664
## u_space_time 1.389992e-01 0.3707577
## u_space_time 2.604929e-01 0.3640049
## u_space_time 3.572793e-01 0.4190699
## u_space_time 4.099089e-01 0.4276245
## u_space_time -4.159490e-03 0.4226344
## u_space_time 3.773540e-03 0.4999199
## u_space_time 2.217792e-01 0.5215089
## u_space_time 3.439993e-01 0.3704039
## u_space_time 2.407757e-01 0.3513228
## u_space_time -1.883964e-01 0.7250996
## u_space_time 3.240130e-01 0.3684162
## u_space_time -1.017886e-01 0.4516469
## u_space_time 1.675534e-01 0.3824766
## u_space_time -2.384801e-01 0.7235084
## u_space_time 1.032277e-01 0.3636395
## u_space_time -1.027209e-01 0.3815391
## u_space_time -2.532489e-01 0.4184447
## u_space_time 4.498120e-02 0.5841888
## u_space_time -1.232098e+00 1.3121958
## u_space_time 3.811721e-02 0.5470505
## u_space_time -2.490134e-01 0.3852052
## u_space_time -4.048637e-01 0.4979837
## u_space_time 3.110904e-01 0.4122071
## u_space_time -3.774770e-01 0.4510141
## u_space_time 2.890140e-01 0.3843629
## u_space_time -6.128314e-01 0.7119665
## u_space_time -2.653319e-01 0.3975484
## u_space_time 2.162732e-01 0.4381744
## u_space_time 2.982116e-01 0.3900482
## u_space_time 1.633948e-01 0.4228799
## u_space_time 2.094798e-01 0.3997454
## u_space_time -1.026628e-01 0.3941756
## u_space_time 1.393544e-01 0.3707844
## u_space_time 2.611269e-01 0.3640289
## u_space_time 3.572811e-01 0.4190951
## u_space_time 4.101586e-01 0.4276484
## u_space_time -4.080336e-03 0.4226572
## u_space_time 3.722916e-03 0.4999362
## u_space_time 2.217245e-01 0.5215361
## u_space_time 3.440699e-01 0.3704274
## u_space_time 2.411492e-01 0.3513602
## u_space_time -1.884355e-01 0.7251215
## u_space_time 3.245571e-01 0.3684594
## u_space_time -1.017876e-01 0.4516827
## u_space_time 1.674730e-01 0.3825178
## u_space_time -2.385160e-01 0.7235323
## u_space_time 1.030298e-01 0.3636692
## u_space_time -1.024774e-01 0.3815810
## u_space_time -2.531092e-01 0.4184798
## u_space_time 4.512013e-02 0.5842183
## u_space_time -1.232098e+00 1.3122090
## u_space_time 3.809103e-02 0.5470796
## u_space_time -2.489614e-01 0.3852470
## u_space_time -4.047321e-01 0.4980119
## u_space_time 3.113148e-01 0.4122415
## u_space_time -3.774338e-01 0.4510446
## u_space_time 2.890111e-01 0.3844020
## u_space_time -6.128687e-01 0.7119901
## u_space_time -2.653854e-01 0.3975821
## u_space_time 2.165194e-01 0.4382123
## u_space_time 2.981430e-01 0.3900843
## u_space_time 1.632276e-01 0.4229171
## u_space_time 2.093501e-01 0.3997829
## u_space_time -1.028371e-01 0.3941959
## u_space_time 1.394603e-01 0.3708215
## u_space_time 2.616810e-01 0.3640639
## u_space_time 3.572794e-01 0.4191296
## u_space_time 4.102586e-01 0.4276811
## u_space_time -3.934246e-03 0.4226900
## u_space_time 3.630475e-03 0.4999606
## u_space_time 2.216926e-01 0.5215707
## u_space_time 3.442373e-01 0.3704622
## u_space_time 2.420435e-01 0.3514117
## u_space_time -1.884669e-01 0.7251488
## u_space_time 3.251411e-01 0.3685134
## u_space_time -1.018140e-01 0.4517270
## u_space_time 1.673232e-01 0.3825688
## u_space_time -2.385446e-01 0.7235613
## u_space_time 1.029976e-01 0.3637117
## u_space_time -1.021533e-01 0.3816336
## u_space_time -2.529724e-01 0.4185243
## u_space_time 4.520284e-02 0.5842539
## u_space_time -1.232098e+00 1.3122248
## u_space_time 3.808425e-02 0.5471159
## u_space_time -2.489860e-01 0.3852987
## u_space_time -4.046405e-01 0.4980487
## u_space_time 3.113367e-01 0.4122850
## u_space_time -3.774156e-01 0.4510842
## u_space_time 2.891504e-01 0.3844521
## u_space_time -6.128985e-01 0.7120189
## u_space_time -2.654024e-01 0.3976267
## u_space_time 2.167466e-01 0.4382593
## u_space_time 2.981265e-01 0.3901311
## u_space_time 1.631215e-01 0.4229636
## u_space_time 2.091744e-01 0.3998300
## u_space_time -1.030891e-01 0.3942284
## u_space_time 1.396681e-01 0.3708704
## u_space_time 2.619587e-01 0.3641097
## u_space_time 3.572755e-01 0.4191738
## u_space_time 4.103505e-01 0.4277235
## u_space_time -3.862139e-03 0.4227324
## u_space_time 3.569140e-03 0.4999938
## u_space_time 2.216115e-01 0.5216120
## u_space_time 3.441568e-01 0.3705073
## u_space_time 2.428561e-01 0.3514750
## u_space_time -1.884904e-01 0.7251814
## u_space_time 3.255544e-01 0.3685773
## u_space_time -1.018681e-01 0.4517799
## u_space_time 1.673150e-01 0.3826303
## u_space_time -2.385661e-01 0.7235951
## u_space_time 1.027253e-01 0.3637661
## u_space_time -1.018235e-01 0.3816963
## u_space_time -2.530487e-01 0.4185780
## u_space_time 4.530015e-02 0.5842957
## u_space_time -1.232097e+00 1.3122433
## u_space_time 3.809677e-02 0.5471593
## u_space_time -2.490201e-01 0.3853605
## u_space_time -4.046602e-01 0.4980940
## u_space_time 3.112988e-01 0.4123387
## u_space_time -3.773509e-01 0.4511332
## u_space_time 2.894288e-01 0.3845132
## u_space_time -6.129208e-01 0.7120529
## u_space_time -2.653811e-01 0.3976822
## u_space_time 2.169534e-01 0.4383149
## u_space_time 2.980930e-01 0.3901884
## u_space_time 1.630765e-01 0.4230194
## u_space_time 2.090945e-01 0.3998875
## u_space_time -1.032689e-01 0.3942744
## u_space_time 1.399084e-01 0.3709310
## u_space_time 2.619014e-01 0.3641663
## u_space_time 3.573404e-01 0.4192278
## u_space_time 4.104363e-01 0.4277759
## u_space_time -3.721926e-03 0.4227850
## u_space_time 3.469070e-03 0.5000360
## u_space_time 2.215516e-01 0.5216601
## u_space_time 3.441188e-01 0.3705643
## u_space_time 2.430624e-01 0.3515484
## u_space_time -1.885061e-01 0.7252193
## u_space_time 3.255198e-01 0.3686502
## u_space_time -1.018806e-01 0.4518412
## u_space_time 1.672357e-01 0.3827016
## u_space_time -2.385806e-01 0.7236338
## u_space_time 1.027189e-01 0.3638349
## u_space_time -1.016306e-01 0.3817686
## u_space_time -2.531243e-01 0.4186416
## u_space_time 4.527103e-02 0.5843436
## u_space_time -1.232097e+00 1.3122647
## u_space_time 3.812861e-02 0.5472097
## u_space_time -2.489947e-01 0.3854324
## u_space_time -4.046503e-01 0.4981476
## u_space_time 3.112024e-01 0.4124026
## u_space_time -3.772384e-01 0.4511918
## u_space_time 2.896339e-01 0.3845842
## u_space_time -6.129357e-01 0.7120921
## u_space_time -2.652496e-01 0.3977491
## u_space_time 2.172095e-01 0.4383794
## u_space_time 2.981153e-01 0.3902567
## u_space_time 1.630921e-01 0.4230843
## u_space_time 2.091103e-01 0.3999554
## u_space_time -1.033715e-01 0.3943344
## u_space_time 1.399016e-01 0.3710022
## u_space_time 2.616635e-01 0.3642350
## u_space_time 3.574054e-01 0.4192917
## u_space_time 4.103756e-01 0.4278377
## u_space_time -3.583315e-03 0.4228479
## u_space_time 3.401765e-03 0.5000874
## u_space_time 2.215118e-01 0.5217148
## u_space_time 3.439926e-01 0.3706333
## u_space_time 2.430351e-01 0.3516341
## u_space_time -1.885138e-01 0.7252627
## u_space_time 3.256765e-01 0.3687352
## u_space_time -1.018513e-01 0.4519111
## u_space_time 1.672974e-01 0.3827833
## u_space_time -2.385879e-01 0.7236775
## u_space_time 1.027768e-01 0.3639177
## u_space_time -1.015721e-01 0.3818506
## u_space_time -2.531274e-01 0.4187156
## u_space_time 4.525649e-02 0.5843977
## u_space_time -1.232097e+00 1.3122888
## u_space_time 3.810918e-02 0.5472671
## u_space_time -2.489114e-01 0.3855143
## u_space_time -4.046110e-01 0.4982096
## u_space_time 3.111192e-01 0.4124773
## u_space_time -3.770774e-01 0.4512602
## u_space_time 2.899098e-01 0.3846661
## u_space_time -6.129432e-01 0.7121365
## u_space_time -2.652199e-01 0.3978268
## u_space_time 2.173027e-01 0.4384519
## u_space_time 2.981256e-01 0.3903359
## u_space_time 1.631692e-01 0.4231585
## u_space_time 2.091523e-01 0.4000335
## u_space_time -1.034620e-01 0.3944086
## u_space_time 1.398587e-01 0.3710850
## u_space_time 2.614671e-01 0.3643172
## u_space_time 3.574723e-01 0.4193658
## u_space_time 4.103803e-01 0.4279100
## u_space_time -3.585876e-03 0.4229206
## u_space_time 3.367402e-03 0.5001483
## u_space_time 2.214918e-01 0.5217760
## u_space_time 3.439966e-01 0.3707155Hyper parameter comparison
The standard devation from TMB is somewhat larger than INLA.
summary(sdr6, "fixed")## Estimate Std. Error
## log_prec_space_time 3.875661 1.693781summary(sdr6b, "fixed")## Estimate Std. Error
## log_prec_space_time 9.558215 0.2596289cbind("mean" = inlafit6$misc$theta.mode,
"se" = sqrt(diag(inlafit6$misc$cov.intern)))## mean se
## [1,] 5.386291 1.276547Fixed effects (Intercept)
summary(sdr6, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.005221459 0.03087566summary(sdr6b, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.147675 0.3478549Random effects mean and standard deviation
plot(summary(sdr6, "report")[,1], summary(sdr6b, "report")[,1],
xlab = "TMB SEPARABLE()", ylab = "TMB Kronecker()", main = "Random effect point estimates")
abline(0, 1)
plot(summary(sdr6, "report")[,2], summary(sdr6b, "report")[,2],
xlab = "TMB SEPARABLE()", ylab = "TMB Kronecker()", main = "Random effect standard devation")
abline(0, 1)
8 Model 7: IID x RW1 with constraints
By changing the order of the main and group terms in the R-INLA smooth formula, we can implement the same model with constraints on the RW1 terms.
First fit the unconstrained model and confirm it matches the previous version.
inlafit7_unconstr <- inla(Observed ~
f(ID.Year, model = "rw1", hyper = prec.prior,
scale.model = TRUE, diagonal = diagval, constr = FALSE,
group = ID, control.group = list(model = "iid")),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
summary(inlafit6)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.81, Running = 0.283, Post = 0.0147, Total = 3.11
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.265 4.067 -8.236 -0.265 7.706 NA 0
##
## Random effects:
## Name Model
## as.integer(IDf) IID model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for as.integer(IDf) 463.12 695.03 42.18 261.51 2170.08 NA
##
## Marginal log-Likelihood: -1089.57
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafit7_unconstr)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.64, Running = 0.282, Post = 0.0147, Total = 2.93
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.265 4.067 -8.236 -0.265 7.706 NA 0
##
## Random effects:
## Name Model
## ID.Year RW1 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for ID.Year 463.12 695.01 42.18 261.51 2170.03 NA
##
## Marginal log-Likelihood: -1089.57
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Fit the model with default sum-to-zero constraints on each group.
inlafit7 <- inla(Observed ~
f(ID.Year, model = "rw1", hyper = prec.prior,
scale.model = TRUE, diagonal = diagval, constr = TRUE,
group = ID, control.group = list(model = "iid")),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep(".*rank.*", inlafit7_unconstr$logfile, value = TRUE)## [1] " computed/guessed rank-deficiency = [1]"8.1 R-INLA with custom Cmatrix
R_space <- diag(ncol(adj.mat))
D_time <- diff(diag(length(levels(data$Yearf))), differences = 1)
R_time <- Matrix::Matrix(t(D_time) %*% D_time)
R_time_adj <- R_time + Matrix::Diagonal(ncol(R_time), 1e-6)
R_time_scaled <- inla.scale.model(R_time, constr = list(A = matrix(1, ncol = ncol(R_time)), e = 0))
R_space_time <- kronecker(R_time_scaled, R_space)
Aconstr <- t(model.matrix(~0+factor(IDf), data[order(data$id.area.year), ]))
inlafit7C <- inla(Observed ~
f(id.area.year, model = "generic0", Cmatrix = R_space_time, hyper = prec.prior,
extraconstr = list(A = Aconstr, e = numeric(nrow(Aconstr))),
diagonal = diagval),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep(".*rank.*", inlafit7C$logfile, value = TRUE)## [1] " computed/guessed rank-deficiency = [32]"These results match:
summary(inlafit7)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.57, Running = 0.32, Post = 0.0141, Total = 2.9
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.004 0.029 -0.061 -0.004 0.054 NA 0
##
## Random effects:
## Name Model
## ID.Year RW1 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for ID.Year 453.97 768.35 40.68 238.75 2237.85 NA
##
## Marginal log-Likelihood: -1150.48
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafit7C)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.8, Running = 0.323, Post = 0.0151, Total = 3.13
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.004 0.029 -0.061 -0.004 0.054 NA 0
##
## Random effects:
## Name Model
## id.area.year Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for id.area.year 453.97 768.35 40.68 238.75 2237.85 NA
##
## Marginal log-Likelihood: -1150.48
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Marginal likelihood
inlafit7$mlik## [,1]
## log marginal-likelihood (integration) -1151.690
## log marginal-likelihood (Gaussian) -1150.483inlafit7C$mlik## [,1]
## log marginal-likelihood (integration) -1151.690
## log marginal-likelihood (Gaussian) -1150.483Same constraints
dim(inlafit7$misc$configs$constr$A)## [1] 32 1217dim(inlafit7C$misc$configs$constr$A)## [1] 32 1217Hyper parameters
inlafit7$internal.summary.hyperpar[, 1:2]inlafit7C$internal.summary.hyperpar[, 1:2]Fixed effects
inlafit7$summary.fixedinlafit7C$summary.fixedRandom effects
plot(inlafit7$summary.random[[1]][order(inlafit7$summary.random[[1]]$ID), 2],
inlafit7C$summary.random[[1]][,2],
main = "Random effect mean")
abline(a = 0, b = 1, col = "red")
plot(inlafit7$summary.random[[1]][order(inlafit7$summary.random[[1]]$ID), 3],
inlafit7C$summary.random[[1]][,3],
main = "Random effect standard deviation")
abline(a = 0, b = 1, col = "red")
8.2 TMB
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(Aconstr);
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_time);
DATA_SPARSE_MATRIX(R_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_raw_space_time_v(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time_v * sigma_space_time);
val += SEPARABLE(GMRF(R_time), GMRF(R_space))(u_raw_space_time);
val -= dnorm(Aconstr * u_raw_space_time_v, Type(0), Type(0.001) * u_raw_space_time.cols(), true).sum(); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Aconstr = Aconstr,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_space = as(R_space, "dgTMatrix"),
R_time = R_time_adj)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = array(0, c(nrow(tmbdata$R_space), nrow(tmbdata$R_time))))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)## Warning in getParameterOrder(data, parameters, new.env(), DLL = DLL): Expected
## sparse matrix of class 'dgTMatrix'.## Error in getParameterOrder(data, parameters, new.env(), DLL = DLL): Error when reading the variable: 'R_time'. Please check data and parameters.tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr7 <- TMB::sdreport(obj)
summary(sdr7, "all")## Estimate Std. Error
## log_prec_space_time 9.558215e+00 0.2596289
## beta0 -1.476750e-01 0.3478549
## u_raw_space_time 2.777543e+01 41.9670255
## u_raw_space_time -2.248671e+01 86.3536077
## u_raw_space_time 3.805117e+01 44.0952263
## u_raw_space_time -1.214896e+01 53.8202427
## u_raw_space_time 2.024617e+01 45.5889354
## u_raw_space_time -2.830052e+01 86.1829076
## u_raw_space_time 1.179770e+01 43.3376305
## u_raw_space_time -1.241089e+01 45.4971490
## u_raw_space_time -3.025881e+01 50.0311401
## u_raw_space_time 5.189801e+00 69.5374671
## u_raw_space_time -1.466181e+02 155.2819607
## u_raw_space_time 4.689832e+00 65.1244306
## u_raw_space_time -2.956390e+01 46.0953632
## u_raw_space_time -4.822344e+01 59.6468240
## u_raw_space_time 3.670473e+01 49.2571397
## u_raw_space_time -4.490938e+01 54.0678828
## u_raw_space_time 3.439532e+01 45.9310570
## u_raw_space_time -7.291325e+01 85.1027599
## u_raw_space_time -3.154737e+01 47.5817547
## u_raw_space_time 2.563180e+01 52.2343838
## u_raw_space_time 3.559764e+01 46.6155827
## u_raw_space_time 1.953930e+01 50.3851646
## u_raw_space_time 2.464318e+01 47.6707198
## u_raw_space_time -1.218690e+01 47.0191422
## u_raw_space_time 1.647580e+01 44.1963040
## u_raw_space_time 3.081966e+01 43.4987549
## u_raw_space_time 4.249013e+01 50.1282846
## u_raw_space_time 4.859942e+01 51.2194822
## u_raw_space_time -5.303429e-01 50.3410887
## u_raw_space_time 4.916402e-01 59.5315476
## u_raw_space_time 2.645714e+01 62.1310198
## u_raw_space_time 4.107684e+01 44.3680458
## u_raw_space_time 2.777958e+01 41.9562157
## u_raw_space_time -2.248593e+01 86.3483679
## u_raw_space_time 3.805745e+01 44.0848147
## u_raw_space_time -1.214454e+01 53.8118734
## u_raw_space_time 2.024459e+01 45.5790677
## u_raw_space_time -2.830802e+01 86.1776885
## u_raw_space_time 1.182237e+01 43.3268748
## u_raw_space_time -1.241401e+01 45.4871040
## u_raw_space_time -3.026107e+01 50.0219953
## u_raw_space_time 5.191598e+00 69.5310455
## u_raw_space_time -1.466181e+02 155.2791022
## u_raw_space_time 4.683221e+00 65.1174199
## u_raw_space_time -2.956845e+01 46.0854112
## u_raw_space_time -4.822123e+01 59.6391233
## u_raw_space_time 3.671116e+01 49.2477547
## u_raw_space_time -4.490582e+01 54.0594200
## u_raw_space_time 3.439143e+01 45.9211887
## u_raw_space_time -7.291236e+01 85.0975169
## u_raw_space_time -3.155541e+01 47.5720406
## u_raw_space_time 2.563730e+01 52.2257455
## u_raw_space_time 3.559202e+01 46.6058867
## u_raw_space_time 1.954610e+01 50.3762202
## u_raw_space_time 2.465307e+01 47.6612049
## u_raw_space_time -1.219753e+01 47.0090523
## u_raw_space_time 1.645954e+01 44.1859142
## u_raw_space_time 3.081191e+01 43.4882439
## u_raw_space_time 4.249638e+01 50.1191197
## u_raw_space_time 4.860450e+01 51.2104515
## u_raw_space_time -5.328857e-01 50.3320316
## u_raw_space_time 4.855319e-01 59.5238133
## u_raw_space_time 2.645986e+01 62.1238264
## u_raw_space_time 4.106091e+01 44.3578412
## u_raw_space_time 2.779327e+01 41.9464493
## u_raw_space_time -2.248436e+01 86.3436883
## u_raw_space_time 3.806220e+01 44.0754466
## u_raw_space_time -1.214402e+01 53.8043922
## u_raw_space_time 2.024163e+01 45.5703075
## u_raw_space_time -2.831462e+01 86.1730958
## u_raw_space_time 1.184745e+01 43.3169905
## u_raw_space_time -1.241161e+01 45.4780683
## u_raw_space_time -3.025707e+01 50.0138102
## u_raw_space_time 5.195185e+00 69.5253891
## u_raw_space_time -1.466181e+02 155.2766003
## u_raw_space_time 4.678529e+00 65.1111194
## u_raw_space_time -2.957732e+01 46.0763690
## u_raw_space_time -4.821667e+01 59.6321459
## u_raw_space_time 3.671566e+01 49.2392279
## u_raw_space_time -4.491531e+01 54.0515302
## u_raw_space_time 3.439218e+01 45.9122514
## u_raw_space_time -7.291058e+01 85.0929025
## u_raw_space_time -3.156284e+01 47.5632509
## u_raw_space_time 2.563991e+01 52.2180974
## u_raw_space_time 3.557262e+01 46.5973251
## u_raw_space_time 1.955141e+01 50.3683083
## u_raw_space_time 2.466446e+01 47.6527170
## u_raw_space_time -1.220143e+01 46.9996611
## u_raw_space_time 1.643620e+01 44.1764371
## u_raw_space_time 3.078050e+01 43.4788958
## u_raw_space_time 4.250886e+01 50.1107604
## u_raw_space_time 4.860635e+01 51.2022797
## u_raw_space_time -5.295596e-01 50.3238637
## u_raw_space_time 4.733819e-01 59.5167605
## u_raw_space_time 2.646531e+01 62.1174897
## u_raw_space_time 4.102115e+01 44.3488240
## u_raw_space_time 2.782015e+01 41.9376732
## u_raw_space_time -2.248202e+01 86.3395562
## u_raw_space_time 3.806633e+01 44.0671405
## u_raw_space_time -1.214737e+01 53.7978090
## u_raw_space_time 2.022925e+01 45.5627182
## u_raw_space_time -2.832033e+01 86.1691199
## u_raw_space_time 1.188258e+01 43.3079833
## u_raw_space_time -1.240309e+01 45.4700826
## u_raw_space_time -3.024656e+01 50.0066201
## u_raw_space_time 5.200557e+00 69.5204964
## u_raw_space_time -1.466180e+02 155.2744513
## u_raw_space_time 4.667508e+00 65.1055799
## u_raw_space_time -2.958997e+01 46.0682802
## u_raw_space_time -4.822647e+01 59.6256790
## u_raw_space_time 3.672704e+01 49.2315125
## u_raw_space_time -4.492080e+01 54.0445137
## u_raw_space_time 3.438967e+01 45.9043620
## u_raw_space_time -7.290793e+01 85.0889148
## u_raw_space_time -3.157780e+01 47.5553039
## u_raw_space_time 2.564810e+01 52.2113953
## u_raw_space_time 3.555657e+01 46.5897365
## u_raw_space_time 1.956357e+01 50.3613778
## u_raw_space_time 2.468585e+01 47.6452073
## u_raw_space_time -1.221486e+01 46.9909155
## u_raw_space_time 1.643201e+01 44.1678016
## u_raw_space_time 3.074335e+01 43.4705605
## u_raw_space_time 4.251943e+01 50.1033389
## u_raw_space_time 4.861358e+01 51.1948770
## u_raw_space_time -5.286200e-01 50.3165898
## u_raw_space_time 4.636442e-01 59.5104132
## u_raw_space_time 2.645664e+01 62.1120853
## u_raw_space_time 4.098372e+01 44.3406811
## u_raw_space_time 2.786470e+01 41.9298360
## u_raw_space_time -2.247887e+01 86.3359966
## u_raw_space_time 3.808725e+01 44.0596941
## u_raw_space_time -1.215448e+01 53.7921436
## u_raw_space_time 2.020702e+01 45.5562700
## u_raw_space_time -2.832516e+01 86.1657571
## u_raw_space_time 1.193754e+01 43.2998633
## u_raw_space_time -1.238774e+01 45.4631947
## u_raw_space_time -3.023793e+01 50.0003283
## u_raw_space_time 5.207742e+00 69.5163776
## u_raw_space_time -1.466180e+02 155.2726587
## u_raw_space_time 4.658678e+00 65.1008424
## u_raw_space_time -2.958924e+01 46.0613826
## u_raw_space_time -4.823375e+01 59.6199787
## u_raw_space_time 3.674546e+01 49.2246267
## u_raw_space_time -4.493050e+01 54.0382812
## u_raw_space_time 3.438475e+01 45.8975689
## u_raw_space_time -7.290440e+01 85.0855541
## u_raw_space_time -3.158336e+01 47.5484271
## u_raw_space_time 2.565348e+01 52.2056914
## u_raw_space_time 3.554409e+01 46.5831345
## u_raw_space_time 1.957417e+01 50.3554678
## u_raw_space_time 2.471727e+01 47.6386761
## u_raw_space_time -1.222038e+01 46.9829780
## u_raw_space_time 1.642279e+01 44.1601547
## u_raw_space_time 3.071018e+01 43.4631768
## u_raw_space_time 4.252000e+01 50.0969934
## u_raw_space_time 4.862641e+01 51.1882709
## u_raw_space_time -5.216228e-01 50.3102299
## u_raw_space_time 4.564333e-01 59.5048017
## u_raw_space_time 2.645068e+01 62.1075289
## u_raw_space_time 4.095826e+01 44.3333328
## u_raw_space_time 2.794136e+01 41.9228069
## u_raw_space_time -2.247487e+01 86.3330296
## u_raw_space_time 3.812586e+01 44.0531272
## u_raw_space_time -1.215696e+01 53.7874332
## u_raw_space_time 2.016686e+01 45.5510253
## u_raw_space_time -2.832911e+01 86.1630048
## u_raw_space_time 1.198844e+01 43.2927328
## u_raw_space_time -1.238211e+01 45.4573155
## u_raw_space_time -3.023096e+01 49.9949676
## u_raw_space_time 5.216742e+00 69.5130332
## u_raw_space_time -1.466179e+02 155.2712168
## u_raw_space_time 4.643639e+00 65.0969064
## u_raw_space_time -2.958345e+01 46.0555741
## u_raw_space_time -4.823843e+01 59.6150694
## u_raw_space_time 3.677115e+01 49.2185941
## u_raw_space_time -4.493597e+01 54.0329679
## u_raw_space_time 3.439438e+01 45.8917012
## u_raw_space_time -7.289999e+01 85.0828190
## u_raw_space_time -3.158766e+01 47.5425382
## u_raw_space_time 2.565603e+01 52.2009836
## u_raw_space_time 3.553538e+01 46.5775310
## u_raw_space_time 1.957486e+01 50.3506225
## u_raw_space_time 2.475876e+01 47.6331258
## u_raw_space_time -1.223409e+01 46.9758129
## u_raw_space_time 1.640112e+01 44.1535695
## u_raw_space_time 3.069899e+01 43.4565963
## u_raw_space_time 4.251913e+01 50.0916424
## u_raw_space_time 4.864502e+01 51.1824861
## u_raw_space_time -5.168652e-01 50.3047829
## u_raw_space_time 4.518039e-01 59.4999406
## u_raw_space_time 2.643903e+01 62.1038663
## u_raw_space_time 4.092087e+01 44.3271564
## u_raw_space_time 2.802968e+01 41.9167813
## u_raw_space_time -2.246997e+01 86.3306827
## u_raw_space_time 3.818259e+01 44.0474479
## u_raw_space_time -1.215484e+01 53.7836739
## u_raw_space_time 2.011785e+01 45.5469801
## u_raw_space_time -2.833214e+01 86.1608897
## u_raw_space_time 1.202842e+01 43.2866538
## u_raw_space_time -1.236897e+01 45.4525782
## u_raw_space_time -3.021707e+01 49.9906577
## u_raw_space_time 5.227537e+00 69.5104559
## u_raw_space_time -1.466179e+02 155.2701213
## u_raw_space_time 4.622405e+00 65.0937758
## u_raw_space_time -2.960591e+01 46.0504599
## u_raw_space_time -4.824045e+01 59.6109638
## u_raw_space_time 3.679602e+01 49.2135380
## u_raw_space_time -4.494545e+01 54.0284790
## u_raw_space_time 3.441036e+01 45.8868580
## u_raw_space_time -7.290310e+01 85.0806064
## u_raw_space_time -3.159039e+01 47.5376712
## u_raw_space_time 2.565580e+01 52.1972795
## u_raw_space_time 3.553070e+01 46.5729422
## u_raw_space_time 1.958245e+01 50.3467634
## u_raw_space_time 2.478506e+01 47.6287218
## u_raw_space_time -1.223855e+01 46.9695820
## u_raw_space_time 1.638496e+01 44.1480168
## u_raw_space_time 3.071052e+01 43.4508286
## u_raw_space_time 4.251687e+01 50.0872893
## u_raw_space_time 4.866134e+01 51.1776836
## u_raw_space_time -5.227457e-01 50.3002333
## u_raw_space_time 4.498350e-01 59.4958505
## u_raw_space_time 2.643010e+01 62.1010567
## u_raw_space_time 4.088956e+01 44.3219589
## u_raw_space_time 2.812731e+01 41.9117952
## u_raw_space_time -2.246417e+01 86.3289582
## u_raw_space_time 3.822453e+01 44.0430991
## u_raw_space_time -1.215646e+01 53.7808345
## u_raw_space_time 2.007677e+01 45.5440397
## u_raw_space_time -2.833427e+01 86.1593977
## u_raw_space_time 1.206681e+01 43.2816241
## u_raw_space_time -1.235641e+01 45.4489500
## u_raw_space_time -3.020456e+01 49.9873152
## u_raw_space_time 5.240139e+00 69.5086502
## u_raw_space_time -1.466178e+02 155.2693728
## u_raw_space_time 4.603322e+00 65.0914332
## u_raw_space_time -2.961420e+01 46.0465970
## u_raw_space_time -4.823974e+01 59.6076841
## u_raw_space_time 3.682894e+01 49.2094215
## u_raw_space_time -4.495033e+01 54.0249817
## u_raw_space_time 3.440808e+01 45.8833407
## u_raw_space_time -7.290535e+01 85.0790119
## u_raw_space_time -3.159977e+01 47.5337373
## u_raw_space_time 2.566112e+01 52.1945134
## u_raw_space_time 3.553861e+01 46.5692876
## u_raw_space_time 1.958008e+01 50.3439610
## u_raw_space_time 2.482131e+01 47.6252858
## u_raw_space_time -1.224160e+01 46.9642872
## u_raw_space_time 1.639225e+01 44.1434666
## u_raw_space_time 3.072935e+01 43.4460683
## u_raw_space_time 4.251332e+01 50.0839453
## u_raw_space_time 4.868400e+01 51.1737747
## u_raw_space_time -5.224690e-01 50.2966093
## u_raw_space_time 4.506763e-01 59.4925716
## u_raw_space_time 2.642382e+01 62.0990789
## u_raw_space_time 4.087387e+01 44.3176581
## u_raw_space_time 2.821506e+01 41.9080469
## u_raw_space_time -2.245747e+01 86.3278508
## u_raw_space_time 3.827781e+01 44.0397835
## u_raw_space_time -1.215330e+01 53.7789791
## u_raw_space_time 2.005192e+01 45.5421497
## u_raw_space_time -2.834391e+01 86.1584881
## u_raw_space_time 1.210502e+01 43.2776795
## u_raw_space_time -1.232727e+01 45.4465625
## u_raw_space_time -3.020150e+01 49.9848934
## u_raw_space_time 5.254508e+00 69.5075999
## u_raw_space_time -1.466177e+02 155.2689668
## u_raw_space_time 4.586387e+00 65.0898777
## u_raw_space_time -2.962471e+01 46.0438089
## u_raw_space_time -4.823613e+01 59.6052748
## u_raw_space_time 3.686164e+01 49.2063465
## u_raw_space_time -4.495049e+01 54.0224978
## u_raw_space_time 3.442105e+01 45.8807851
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## u_space_time 1.636238e-01 0.4228519
## u_space_time 2.093513e-01 0.3997167
## u_space_time -1.025771e-01 0.3941664
## u_space_time 1.389992e-01 0.3707577
## u_space_time 2.604929e-01 0.3640049
## u_space_time 3.572793e-01 0.4190699
## u_space_time 4.099089e-01 0.4276245
## u_space_time -4.159490e-03 0.4226344
## u_space_time 3.773540e-03 0.4999199
## u_space_time 2.217792e-01 0.5215089
## u_space_time 3.439993e-01 0.3704039
## u_space_time 2.407757e-01 0.3513228
## u_space_time -1.883964e-01 0.7250996
## u_space_time 3.240130e-01 0.3684162
## u_space_time -1.017886e-01 0.4516469
## u_space_time 1.675534e-01 0.3824766
## u_space_time -2.384801e-01 0.7235084
## u_space_time 1.032277e-01 0.3636395
## u_space_time -1.027209e-01 0.3815391
## u_space_time -2.532489e-01 0.4184447
## u_space_time 4.498120e-02 0.5841888
## u_space_time -1.232098e+00 1.3121958
## u_space_time 3.811721e-02 0.5470505
## u_space_time -2.490134e-01 0.3852052
## u_space_time -4.048637e-01 0.4979837
## u_space_time 3.110904e-01 0.4122071
## u_space_time -3.774770e-01 0.4510141
## u_space_time 2.890140e-01 0.3843629
## u_space_time -6.128314e-01 0.7119665
## u_space_time -2.653319e-01 0.3975484
## u_space_time 2.162732e-01 0.4381744
## u_space_time 2.982116e-01 0.3900482
## u_space_time 1.633948e-01 0.4228799
## u_space_time 2.094798e-01 0.3997454
## u_space_time -1.026628e-01 0.3941756
## u_space_time 1.393544e-01 0.3707844
## u_space_time 2.611269e-01 0.3640289
## u_space_time 3.572811e-01 0.4190951
## u_space_time 4.101586e-01 0.4276484
## u_space_time -4.080336e-03 0.4226572
## u_space_time 3.722916e-03 0.4999362
## u_space_time 2.217245e-01 0.5215361
## u_space_time 3.440699e-01 0.3704274
## u_space_time 2.411492e-01 0.3513602
## u_space_time -1.884355e-01 0.7251215
## u_space_time 3.245571e-01 0.3684594
## u_space_time -1.017876e-01 0.4516827
## u_space_time 1.674730e-01 0.3825178
## u_space_time -2.385160e-01 0.7235323
## u_space_time 1.030298e-01 0.3636692
## u_space_time -1.024774e-01 0.3815810
## u_space_time -2.531092e-01 0.4184798
## u_space_time 4.512013e-02 0.5842183
## u_space_time -1.232098e+00 1.3122090
## u_space_time 3.809103e-02 0.5470796
## u_space_time -2.489614e-01 0.3852470
## u_space_time -4.047321e-01 0.4980119
## u_space_time 3.113148e-01 0.4122415
## u_space_time -3.774338e-01 0.4510446
## u_space_time 2.890111e-01 0.3844020
## u_space_time -6.128687e-01 0.7119901
## u_space_time -2.653854e-01 0.3975821
## u_space_time 2.165194e-01 0.4382123
## u_space_time 2.981430e-01 0.3900843
## u_space_time 1.632276e-01 0.4229171
## u_space_time 2.093501e-01 0.3997829
## u_space_time -1.028371e-01 0.3941959
## u_space_time 1.394603e-01 0.3708215
## u_space_time 2.616810e-01 0.3640639
## u_space_time 3.572794e-01 0.4191296
## u_space_time 4.102586e-01 0.4276811
## u_space_time -3.934246e-03 0.4226900
## u_space_time 3.630475e-03 0.4999606
## u_space_time 2.216926e-01 0.5215707
## u_space_time 3.442373e-01 0.3704622
## u_space_time 2.420435e-01 0.3514117
## u_space_time -1.884669e-01 0.7251488
## u_space_time 3.251411e-01 0.3685134
## u_space_time -1.018140e-01 0.4517270
## u_space_time 1.673232e-01 0.3825688
## u_space_time -2.385446e-01 0.7235613
## u_space_time 1.029976e-01 0.3637117
## u_space_time -1.021533e-01 0.3816336
## u_space_time -2.529724e-01 0.4185243
## u_space_time 4.520284e-02 0.5842539
## u_space_time -1.232098e+00 1.3122248
## u_space_time 3.808425e-02 0.5471159
## u_space_time -2.489860e-01 0.3852987
## u_space_time -4.046405e-01 0.4980487
## u_space_time 3.113367e-01 0.4122850
## u_space_time -3.774156e-01 0.4510842
## u_space_time 2.891504e-01 0.3844521
## u_space_time -6.128985e-01 0.7120189
## u_space_time -2.654024e-01 0.3976267
## u_space_time 2.167466e-01 0.4382593
## u_space_time 2.981265e-01 0.3901311
## u_space_time 1.631215e-01 0.4229636
## u_space_time 2.091744e-01 0.3998300
## u_space_time -1.030891e-01 0.3942284
## u_space_time 1.396681e-01 0.3708704
## u_space_time 2.619587e-01 0.3641097
## u_space_time 3.572755e-01 0.4191738
## u_space_time 4.103505e-01 0.4277235
## u_space_time -3.862139e-03 0.4227324
## u_space_time 3.569140e-03 0.4999938
## u_space_time 2.216115e-01 0.5216120
## u_space_time 3.441568e-01 0.3705073
## u_space_time 2.428561e-01 0.3514750
## u_space_time -1.884904e-01 0.7251814
## u_space_time 3.255544e-01 0.3685773
## u_space_time -1.018681e-01 0.4517799
## u_space_time 1.673150e-01 0.3826303
## u_space_time -2.385661e-01 0.7235951
## u_space_time 1.027253e-01 0.3637661
## u_space_time -1.018235e-01 0.3816963
## u_space_time -2.530487e-01 0.4185780
## u_space_time 4.530015e-02 0.5842957
## u_space_time -1.232097e+00 1.3122433
## u_space_time 3.809677e-02 0.5471593
## u_space_time -2.490201e-01 0.3853605
## u_space_time -4.046602e-01 0.4980940
## u_space_time 3.112988e-01 0.4123387
## u_space_time -3.773509e-01 0.4511332
## u_space_time 2.894288e-01 0.3845132
## u_space_time -6.129208e-01 0.7120529
## u_space_time -2.653811e-01 0.3976822
## u_space_time 2.169534e-01 0.4383149
## u_space_time 2.980930e-01 0.3901884
## u_space_time 1.630765e-01 0.4230194
## u_space_time 2.090945e-01 0.3998875
## u_space_time -1.032689e-01 0.3942744
## u_space_time 1.399084e-01 0.3709310
## u_space_time 2.619014e-01 0.3641663
## u_space_time 3.573404e-01 0.4192278
## u_space_time 4.104363e-01 0.4277759
## u_space_time -3.721926e-03 0.4227850
## u_space_time 3.469070e-03 0.5000360
## u_space_time 2.215516e-01 0.5216601
## u_space_time 3.441188e-01 0.3705643
## u_space_time 2.430624e-01 0.3515484
## u_space_time -1.885061e-01 0.7252193
## u_space_time 3.255198e-01 0.3686502
## u_space_time -1.018806e-01 0.4518412
## u_space_time 1.672357e-01 0.3827016
## u_space_time -2.385806e-01 0.7236338
## u_space_time 1.027189e-01 0.3638349
## u_space_time -1.016306e-01 0.3817686
## u_space_time -2.531243e-01 0.4186416
## u_space_time 4.527103e-02 0.5843436
## u_space_time -1.232097e+00 1.3122647
## u_space_time 3.812861e-02 0.5472097
## u_space_time -2.489947e-01 0.3854324
## u_space_time -4.046503e-01 0.4981476
## u_space_time 3.112024e-01 0.4124026
## u_space_time -3.772384e-01 0.4511918
## u_space_time 2.896339e-01 0.3845842
## u_space_time -6.129357e-01 0.7120921
## u_space_time -2.652496e-01 0.3977491
## u_space_time 2.172095e-01 0.4383794
## u_space_time 2.981153e-01 0.3902567
## u_space_time 1.630921e-01 0.4230843
## u_space_time 2.091103e-01 0.3999554
## u_space_time -1.033715e-01 0.3943344
## u_space_time 1.399016e-01 0.3710022
## u_space_time 2.616635e-01 0.3642350
## u_space_time 3.574054e-01 0.4192917
## u_space_time 4.103756e-01 0.4278377
## u_space_time -3.583315e-03 0.4228479
## u_space_time 3.401765e-03 0.5000874
## u_space_time 2.215118e-01 0.5217148
## u_space_time 3.439926e-01 0.3706333
## u_space_time 2.430351e-01 0.3516341
## u_space_time -1.885138e-01 0.7252627
## u_space_time 3.256765e-01 0.3687352
## u_space_time -1.018513e-01 0.4519111
## u_space_time 1.672974e-01 0.3827833
## u_space_time -2.385879e-01 0.7236775
## u_space_time 1.027768e-01 0.3639177
## u_space_time -1.015721e-01 0.3818506
## u_space_time -2.531274e-01 0.4187156
## u_space_time 4.525649e-02 0.5843977
## u_space_time -1.232097e+00 1.3122888
## u_space_time 3.810918e-02 0.5472671
## u_space_time -2.489114e-01 0.3855143
## u_space_time -4.046110e-01 0.4982096
## u_space_time 3.111192e-01 0.4124773
## u_space_time -3.770774e-01 0.4512602
## u_space_time 2.899098e-01 0.3846661
## u_space_time -6.129432e-01 0.7121365
## u_space_time -2.652199e-01 0.3978268
## u_space_time 2.173027e-01 0.4384519
## u_space_time 2.981256e-01 0.3903359
## u_space_time 1.631692e-01 0.4231585
## u_space_time 2.091523e-01 0.4000335
## u_space_time -1.034620e-01 0.3944086
## u_space_time 1.398587e-01 0.3710850
## u_space_time 2.614671e-01 0.3643172
## u_space_time 3.574723e-01 0.4193658
## u_space_time 4.103803e-01 0.4279100
## u_space_time -3.585876e-03 0.4229206
## u_space_time 3.367402e-03 0.5001483
## u_space_time 2.214918e-01 0.5217760
## u_space_time 3.439966e-01 0.3707155Hyper parameter comparison
The standard devation from TMB is somewhat larger than INLA.
summary(sdr7, "fixed")## Estimate Std. Error
## log_prec_space_time 9.558215 0.2596289cbind("mean" = inlafit7C$misc$theta.mode,
"se" = sqrt(diag(inlafit7C$misc$cov.intern)))## mean se
## [1,] 5.187209 1.178537Fixed effects (Intercept)
summary(sdr7, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.147675 0.3478549inlafit7$summary.fixed[ , 1:2]inlafit7C$summary.fixed[ , 1:2]Random effects mean and standard deviation
plot(inlafit7C$summary.random[[1]][,2], summary(sdr7, "report")[,1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(inlafit7C$summary.random[[1]][,3], summary(sdr7, "report")[,2],
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation")
abline(0, 1, col = "red")
8.3 Model 7b: Explicit sum-to-zero constraint
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(L_space_time)
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(LRL_space_time);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_VECTOR(u_raw_space_time);
vector<Type> u_space_time(L_space_time * u_raw_space_time * sigma_space_time);
val += GMRF(LRL_space_time)(u_raw_space_time);
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
qrc <- qr(t(Aconstr))
L_space_time <- qr.Q(qrc, complete=TRUE)[ , (nrow(Aconstr)+1):ncol(Aconstr)]
tmbdata <- list(y = data$Observed,
E = data$Expected,
L_space_time = L_space_time,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
LRL_space_time = as(t(L_space_time) %*% R_space_time %*% L_space_time, "dgCMatrix"))
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = numeric(ncol(tmbdata$L_space_time)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr7b <- TMB::sdreport(obj)
summary(sdr7b, "all")## Estimate Std. Error
## log_prec_space_time 5.187216e+00 1.18582076
## beta0 -3.722469e-03 0.02944028
## u_raw_space_time -7.060124e-01 0.81226878
## u_raw_space_time -8.113415e-02 1.07396708
## u_raw_space_time -6.841674e-01 1.04289438
## u_raw_space_time -6.955986e-03 1.05691721
## u_raw_space_time 4.255400e-01 1.04592457
## u_raw_space_time 1.040200e-01 1.07419372
## u_raw_space_time -5.072628e-01 0.99715253
## u_raw_space_time -2.281876e-01 1.01949252
## u_raw_space_time -1.338802e-01 1.04480063
## u_raw_space_time -2.271769e-01 1.07830340
## u_raw_space_time -4.653864e-03 1.07502399
## u_raw_space_time 2.226836e-01 1.07661509
## u_raw_space_time 1.678756e-01 1.01627524
## u_raw_space_time 5.748908e-03 1.06017694
## u_raw_space_time -4.368179e-01 1.08014594
## u_raw_space_time 1.022142e-01 1.05309465
## u_raw_space_time -7.011092e-02 1.03699335
## u_raw_space_time 1.164380e-02 1.07012985
## u_raw_space_time 1.425252e-01 1.02935391
## u_raw_space_time -1.651308e-01 1.06307420
## u_raw_space_time 1.285608e-01 1.04370277
## u_raw_space_time 8.723741e-02 1.05513568
## u_raw_space_time -3.817783e-01 1.06413415
## u_raw_space_time 1.994751e-01 1.03749127
## u_raw_space_time -6.514972e-02 1.00791929
## u_raw_space_time -2.055218e-01 0.98967065
## u_raw_space_time -6.433440e-02 1.05848594
## u_raw_space_time -3.038383e-01 1.07407116
## u_raw_space_time -6.415935e-02 1.05153231
## u_raw_space_time 1.038974e-01 1.06726745
## u_raw_space_time 1.252057e-01 1.07123156
## u_raw_space_time 1.815722e-01 1.02345493
## u_raw_space_time -6.955209e-01 0.75792313
## u_raw_space_time -7.656194e-02 0.99773201
## u_raw_space_time -6.724300e-01 0.97409582
## u_raw_space_time -6.124616e-03 0.98205213
## u_raw_space_time 4.201326e-01 0.97520243
## u_raw_space_time 8.367761e-02 0.99755732
## u_raw_space_time -4.415974e-01 0.92291673
## u_raw_space_time -2.282320e-01 0.94820571
## u_raw_space_time -1.261574e-01 0.97080736
## u_raw_space_time -2.162722e-01 1.00216371
## u_raw_space_time -4.261788e-03 0.99868959
## u_raw_space_time 2.081480e-01 1.00023299
## u_raw_space_time 1.343250e-01 0.94381159
## u_raw_space_time 1.621490e-02 0.98490785
## u_raw_space_time -4.224851e-01 1.00547778
## u_raw_space_time 6.796575e-02 0.97751934
## u_raw_space_time -6.611044e-02 0.96353955
## u_raw_space_time 1.716982e-02 0.99423950
## u_raw_space_time 1.139591e-01 0.95585032
## u_raw_space_time -1.574244e-01 0.98794719
## u_raw_space_time 7.086141e-02 0.96841644
## u_raw_space_time 1.040456e-01 0.98110481
## u_raw_space_time -3.482259e-01 0.98821740
## u_raw_space_time 1.793124e-01 0.96281975
## u_raw_space_time -1.385138e-01 0.93727914
## u_raw_space_time -3.000398e-01 0.92174993
## u_raw_space_time -2.427308e-02 0.98286371
## u_raw_space_time -2.966636e-01 0.99907867
## u_raw_space_time -5.529068e-02 0.97689357
## u_raw_space_time 6.565263e-02 0.99060749
## u_raw_space_time 1.422556e-01 0.99642237
## u_raw_space_time 6.847828e-02 0.94644387
## u_raw_space_time -6.626638e-01 0.72685274
## u_raw_space_time -6.975889e-02 0.93142561
## u_raw_space_time -6.648766e-01 0.92052478
## u_raw_space_time -1.757137e-02 0.91811193
## u_raw_space_time 3.873432e-01 0.91399865
## u_raw_space_time 6.612703e-02 0.93107473
## u_raw_space_time -3.502427e-01 0.86240618
## u_raw_space_time -2.124740e-01 0.88925758
## u_raw_space_time -1.001073e-01 0.90785699
## u_raw_space_time -1.999349e-01 0.93570875
## u_raw_space_time -3.733723e-03 0.93224424
## u_raw_space_time 1.740579e-01 0.93273737
## u_raw_space_time 8.688532e-02 0.88453215
## u_raw_space_time -1.939074e-02 0.92029757
## u_raw_space_time -3.868556e-01 0.93896640
## u_raw_space_time 4.447891e-02 0.91361336
## u_raw_space_time -7.164366e-02 0.90227781
## u_raw_space_time 2.542182e-02 0.92854590
## u_raw_space_time 5.971213e-02 0.89449895
## u_raw_space_time -1.325757e-01 0.92273406
## u_raw_space_time 2.447668e-02 0.90586752
## u_raw_space_time 1.422889e-01 0.91921242
## u_raw_space_time -2.844362e-01 0.92080487
## u_raw_space_time 1.261998e-01 0.89907933
## u_raw_space_time -1.532616e-01 0.87981166
## u_raw_space_time -4.120759e-01 0.87455883
## u_raw_space_time 1.040822e-02 0.91855866
## u_raw_space_time -2.722550e-01 0.93304686
## u_raw_space_time -5.447987e-02 0.91361183
## u_raw_space_time 3.464685e-02 0.92481584
## u_raw_space_time 1.155669e-01 0.92986668
## u_raw_space_time -3.326710e-02 0.88629114
## u_raw_space_time -5.946402e-01 0.71348707
## u_raw_space_time -6.053632e-02 0.87720089
## u_raw_space_time -6.077567e-01 0.87355056
## u_raw_space_time -4.089322e-02 0.86706288
## u_raw_space_time 3.257345e-01 0.86308381
## u_raw_space_time 5.133386e-02 0.87684661
## u_raw_space_time -2.022418e-01 0.81625838
## u_raw_space_time -1.781159e-01 0.84345730
## u_raw_space_time -8.172561e-02 0.85838411
## u_raw_space_time -1.780670e-01 0.88101728
## u_raw_space_time -3.035253e-03 0.87787186
## u_raw_space_time 1.468949e-01 0.87807342
## u_raw_space_time 7.917071e-02 0.84045052
## u_raw_space_time -4.851902e-02 0.86869948
## u_raw_space_time -3.294392e-01 0.88250680
## u_raw_space_time 6.572634e-03 0.86274636
## u_raw_space_time -8.447476e-02 0.85470489
## u_raw_space_time 3.640642e-02 0.87515160
## u_raw_space_time 3.245430e-02 0.84775883
## u_raw_space_time -1.166560e-01 0.87052903
## u_raw_space_time -1.009808e-02 0.85710673
## u_raw_space_time 1.759696e-01 0.87054874
## u_raw_space_time -1.902151e-01 0.86488919
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## u_space_time 1.316009e-03 0.05876903
## u_space_time 3.124071e-03 0.05898938
## u_space_time -1.828450e-02 0.06099847
## u_space_time -9.321923e-03 0.05032058
## u_space_time -1.181421e-03 0.05635667
## u_space_time -1.759089e-02 0.05798024
## u_space_time -3.709077e-03 0.05625337
## u_space_time -1.257610e-03 0.05565759
## u_space_time 1.757250e-03 0.05637654
## u_space_time 4.830992e-03 0.05487250
## u_space_time -8.327368e-03 0.05600160
## u_space_time -3.155089e-03 0.05592178
## u_space_time -6.636622e-03 0.05684225
## u_space_time -3.652075e-05 0.05637751
## u_space_time -1.492109e-03 0.05630878
## u_space_time -3.125825e-03 0.05543802
## u_space_time -6.880291e-03 0.05668873
## u_space_time -5.591438e-03 0.05635488
## u_space_time -5.594479e-03 0.05634544
## u_space_time 1.886703e-03 0.05577087
## u_space_time 2.315151e-03 0.05636881
## u_space_time -4.413852e-03 0.05580542
## u_space_time -9.835815e-03 0.05727923
## u_space_time 2.858383e-03 0.05591232
## u_space_time 1.327558e-02 0.05812357
## u_space_time 4.606673e-03 0.05618132
## u_space_time 2.091197e-04 0.05556796
## u_space_time -2.147283e-02 0.05998730
## u_space_time -3.186725e-02 0.06438024
## u_space_time -1.765923e-04 0.05608394
## u_space_time -5.718624e-03 0.05650517
## u_space_time -5.409324e-03 0.05631781
## u_space_time 1.192164e-04 0.05620978
## u_space_time 3.552324e-04 0.05628714
## u_space_time -1.837763e-02 0.05914009
## u_space_time 1.145627e-03 0.05035725
## u_space_time 5.318197e-04 0.05544276
## u_space_time -6.003765e-03 0.05493800
## u_space_time -2.016191e-03 0.05529228
## u_space_time -2.439087e-03 0.05493615
## u_space_time -2.486825e-04 0.05543702
## u_space_time 9.463973e-03 0.05503748
## u_space_time -3.001936e-03 0.05473147
## u_space_time -1.594313e-03 0.05507182
## u_space_time -2.953194e-03 0.05552276
## u_space_time 3.739087e-05 0.05546669
## u_space_time -4.876988e-03 0.05569063
## u_space_time -6.950282e-03 0.05518553
## u_space_time -6.148066e-03 0.05574250
## u_space_time 1.908823e-03 0.05525680
## u_space_time -5.382426e-03 0.05551839
## u_space_time 8.407330e-04 0.05498036
## u_space_time 2.172694e-03 0.05546850
## u_space_time -5.062411e-03 0.05516157
## u_space_time -8.092899e-03 0.05608967
## u_space_time 5.128494e-03 0.05537641
## u_space_time 1.218438e-02 0.05703065
## u_space_time 7.668658e-03 0.05581112
## u_space_time 6.431791e-04 0.05492232
## u_space_time -1.476605e-02 0.05694920
## u_space_time -2.247386e-02 0.05967210
## u_space_time 7.203180e-04 0.05525632
## u_space_time -1.177556e-03 0.05530562
## u_space_time -4.648501e-03 0.05543225
## u_space_time -4.064405e-04 0.05535193
## u_space_time -1.796064e-03 0.05543174
## u_space_time -1.446584e-02 0.05709944
## u_space_time 1.267109e-02 0.05137613
## u_space_time 2.453334e-03 0.05641739
## u_space_time 6.421129e-03 0.05562532
## u_space_time -1.103480e-03 0.05610719
## u_space_time -1.156335e-02 0.05704219
## u_space_time -2.044561e-03 0.05638786
## u_space_time 1.468602e-02 0.05676693
## u_space_time -3.169576e-03 0.05553021
## u_space_time -8.563091e-05 0.05586798
## u_space_time 1.122579e-03 0.05632068
## u_space_time 1.071853e-04 0.05637646
## u_space_time -7.734352e-03 0.05700917
## u_space_time -6.762669e-03 0.05581094
## u_space_time -4.695143e-03 0.05641432
## u_space_time 9.438543e-03 0.05709669
## u_space_time -5.975615e-03 0.05644385
## u_space_time 1.433654e-03 0.05579396
## u_space_time 2.225232e-03 0.05636298
## u_space_time -5.235438e-03 0.05593774
## u_space_time -5.056254e-03 0.05646515
## u_space_time 6.515267e-03 0.05640537
## u_space_time 8.841141e-03 0.05703610
## u_space_time 1.311999e-02 0.05787986
## u_space_time 1.590975e-03 0.05581418
## u_space_time -7.784535e-03 0.05600035
## u_space_time -8.926288e-03 0.05611481
## u_space_time 1.436954e-03 0.05613912
## u_space_time 2.966890e-03 0.05626530
## u_space_time -4.240533e-03 0.05623719
## u_space_time -2.321849e-04 0.05624137
## u_space_time -3.281907e-03 0.05640702
## u_space_time -1.042085e-02 0.05672632
## u_space_time 1.695322e-02 0.05169867
## u_space_time 4.594299e-03 0.05921790
## u_space_time 1.805735e-02 0.05991901
## u_space_time 1.017802e-03 0.05857716
## u_space_time -1.880486e-02 0.06133143
## u_space_time -3.628052e-03 0.05911262
## u_space_time 2.485527e-02 0.06201664
## u_space_time 1.048393e-03 0.05748448
## u_space_time 3.283771e-03 0.05832604
## u_space_time 5.614055e-03 0.05926803
## u_space_time 1.714668e-04 0.05902339
## u_space_time -1.004848e-02 0.06003269
## u_space_time -8.205364e-03 0.05806902
## u_space_time -2.408542e-03 0.05871548
## u_space_time 1.701107e-02 0.06162913
## u_space_time -5.391245e-03 0.05877494
## u_space_time 1.772021e-03 0.05808811
## u_space_time 5.399115e-04 0.05890376
## u_space_time -6.690131e-03 0.05830596
## u_space_time -2.658018e-03 0.05873264
## u_space_time 1.236619e-03 0.05822506
## u_space_time -6.207194e-04 0.05851063
## u_space_time 1.714840e-02 0.06141305
## u_space_time 3.181094e-03 0.05820885
## u_space_time -2.248755e-03 0.05740462
## u_space_time 7.182573e-03 0.05719066
## u_space_time 1.970514e-03 0.05864502
## u_space_time 8.695877e-03 0.05951020
## u_space_time -2.151615e-03 0.05849502
## u_space_time 6.575701e-04 0.05880739
## u_space_time -4.114545e-03 0.05905715
## u_space_time -6.191521e-03 0.05804893
## u_space_time 3.249736e-02 0.05508980
## u_space_time 6.959348e-03 0.06363806
## u_space_time 2.897412e-02 0.06673710
## u_space_time 4.346425e-03 0.06268712
## u_space_time -2.225119e-02 0.06581003
## u_space_time -5.003625e-03 0.06336447
## u_space_time 3.468760e-02 0.06928337
## u_space_time 5.726379e-03 0.06103465
## u_space_time 6.576987e-03 0.06235853
## u_space_time 1.052877e-02 0.06419679
## u_space_time 2.312708e-04 0.06319116
## u_space_time -1.182302e-02 0.06444819
## u_space_time -7.459928e-03 0.06121591
## u_space_time 8.045223e-04 0.06260248
## u_space_time 2.275167e-02 0.06748168
## u_space_time -3.552526e-03 0.06240764
## u_space_time -1.972989e-03 0.06174022
## u_space_time -9.283029e-04 0.06300896
## u_space_time -5.401457e-03 0.06159563
## u_space_time 1.144150e-03 0.06259264
## u_space_time -2.828260e-03 0.06197322
## u_space_time -8.410540e-03 0.06310952
## u_space_time 2.178938e-02 0.06671387
## u_space_time 1.827570e-03 0.06166200
## u_space_time 7.814393e-03 0.06098629
## u_space_time 2.035906e-02 0.06275288
## u_space_time 4.098552e-04 0.06252750
## u_space_time 1.408824e-02 0.06467145
## u_space_time -2.199849e-04 0.06226545
## u_space_time 3.356947e-04 0.06283284
## u_space_time -4.296200e-03 0.06314367
## u_space_time 2.201921e-03 0.06102447
## u_space_time 3.931484e-02 0.05686184
## u_space_time 7.585901e-03 0.06906515
## u_space_time 3.923312e-02 0.07518176
## u_space_time 5.000385e-03 0.06786124
## u_space_time -2.385936e-02 0.07064239
## u_space_time -6.166937e-03 0.06884854
## u_space_time 3.493708e-02 0.07214758
## u_space_time 8.961421e-03 0.06578970
## u_space_time 7.867300e-03 0.06737256
## u_space_time 1.586054e-02 0.07082182
## u_space_time 2.774269e-04 0.06860469
## u_space_time -1.304980e-02 0.06994383
## u_space_time -8.351771e-03 0.06575465
## u_space_time 3.020090e-03 0.06784676
## u_space_time 3.061605e-02 0.07583293
## u_space_time -2.379602e-03 0.06734797
## u_space_time -3.842723e-03 0.06666122
## u_space_time -2.169774e-03 0.06838513
## u_space_time -3.169144e-03 0.06601926
## u_space_time 6.381121e-03 0.06811282
## u_space_time -7.571647e-03 0.06730423
## u_space_time -1.453691e-02 0.06948184
## u_space_time 2.516760e-02 0.07270407
## u_space_time -2.110689e-03 0.06621079
## u_space_time 1.669204e-02 0.06702264
## u_space_time 3.677842e-02 0.07320920
## u_space_time 6.640378e-04 0.06766062
## u_space_time 2.112857e-02 0.07200406
## u_space_time 1.602721e-03 0.06729928
## u_space_time -1.174786e-03 0.06808870
## u_space_time -5.795635e-03 0.06860104
## u_space_time 5.178012e-03 0.06566290
## u_space_time 4.062326e-02 0.05758349
## u_space_time 6.475291e-03 0.07522718
## u_space_time 5.276426e-02 0.08731889
## u_space_time 4.954580e-03 0.07395327
## u_space_time -2.554570e-02 0.07647423
## u_space_time -7.128736e-03 0.07528946
## u_space_time 2.798011e-02 0.07256227
## u_space_time 1.469478e-02 0.07213423
## u_space_time 1.106340e-02 0.07370261
## u_space_time 1.965221e-02 0.07813043
## u_space_time 3.131876e-04 0.07499739
## u_space_time -1.374821e-02 0.07626542
## u_space_time -7.039563e-03 0.07099723
## u_space_time 6.185100e-03 0.07424640
## u_space_time 3.676296e-02 0.08445392
## u_space_time -1.858329e-03 0.07331710
## u_space_time -3.827856e-03 0.07243023
## u_space_time -3.191801e-03 0.07475207
## u_space_time -5.698183e-03 0.07178937
## u_space_time 1.304341e-02 0.07535755
## u_space_time -9.043704e-03 0.07331719
## u_space_time -1.898173e-02 0.07667781
## u_space_time 2.146411e-02 0.07666208
## u_space_time -8.313808e-03 0.07219064
## u_space_time 1.878352e-02 0.07251577
## u_space_time 5.112456e-02 0.08540206
## u_space_time 7.992714e-04 0.07377092
## u_space_time 2.401693e-02 0.07889144
## u_space_time 5.308502e-03 0.07350845
## u_space_time -3.828740e-03 0.07442519
## u_space_time -6.669833e-03 0.07497952
## u_space_time 1.074461e-02 0.07182440
## u_space_time 5.718615e-02 0.06487750
## u_space_time 5.584856e-03 0.08221749
## u_space_time 6.760847e-02 0.10207962
## u_space_time 4.177928e-03 0.08078647
## u_space_time -2.924199e-02 0.08397973
## u_space_time -7.893386e-03 0.08246094
## u_space_time 2.592146e-02 0.07681367
## u_space_time 2.279418e-02 0.08042115
## u_space_time 1.426486e-02 0.08093010
## u_space_time 2.189819e-02 0.08567413
## u_space_time 3.364062e-04 0.08214369
## u_space_time -1.391621e-02 0.08321958
## u_space_time -7.757728e-03 0.07738338
## u_space_time 8.342798e-03 0.08140440
## u_space_time 3.734145e-02 0.09066436
## u_space_time -1.938212e-03 0.08010781
## u_space_time 5.738255e-05 0.07886357
## u_space_time -4.008548e-03 0.08187830
## u_space_time -7.043865e-03 0.07834128
## u_space_time 1.919064e-02 0.08375038
## u_space_time -9.145836e-03 0.07993275
## u_space_time -2.177112e-02 0.08417452
## u_space_time 1.652715e-02 0.08159737
## u_space_time -1.642236e-02 0.07996714
## u_space_time 2.380791e-02 0.07985305
## u_space_time 5.807160e-02 0.09392398
## u_space_time 8.488605e-04 0.08066740
## u_space_time 2.666504e-02 0.08642900
## u_space_time 7.015052e-03 0.08044384
## u_space_time -5.605394e-03 0.08154422
## u_space_time -8.902502e-03 0.08231076
## u_space_time 9.368997e-03 0.07771208
## u_space_time 7.304420e-02 0.07528431
## u_space_time 4.919038e-03 0.08984172
## u_space_time 7.802773e-02 0.11462035
## u_space_time 2.657433e-03 0.08825125
## u_space_time -2.913433e-02 0.09064476
## u_space_time -8.472275e-03 0.09019023
## u_space_time 1.756130e-02 0.08075107
## u_space_time 3.120808e-02 0.09018281
## u_space_time 1.170268e-02 0.08765541
## u_space_time 2.455621e-02 0.09393717
## u_space_time 3.473747e-04 0.08986711
## u_space_time -1.355641e-02 0.09067189
## u_space_time -8.702792e-03 0.08458501
## u_space_time 7.522189e-03 0.08881250
## u_space_time 3.628256e-02 0.09675124
## u_space_time -6.318224e-04 0.08753202
## u_space_time 7.747931e-03 0.08658364
## u_space_time -4.619128e-03 0.08958683
## u_space_time -7.137529e-03 0.08551303
## u_space_time 2.479604e-02 0.09286583
## u_space_time -9.789706e-03 0.08732713
## u_space_time -2.290642e-02 0.09173885
## u_space_time 1.426300e-02 0.08834918
## u_space_time -2.227651e-02 0.08844639
## u_space_time 2.987793e-02 0.08855723
## u_space_time 5.603975e-02 0.09721605
## u_space_time 2.769626e-03 0.08825164
## u_space_time 2.912210e-02 0.09449102
## u_space_time 1.065355e-02 0.08833965
## u_space_time -8.432004e-03 0.08942933
## u_space_time -1.055558e-02 0.09017530
## u_space_time 9.029340e-03 0.08460668
## u_space_time 7.438432e-02 0.07966266
## u_space_time 4.475355e-03 0.09795333
## u_space_time 7.642098e-02 0.11855556
## u_space_time 2.295508e-03 0.09631873
## u_space_time -3.109513e-02 0.09882830
## u_space_time -8.860895e-03 0.09834920
## u_space_time 1.680544e-02 0.08793341
## u_space_time 3.601876e-02 0.09941547
## u_space_time 9.298352e-03 0.09520792
## u_space_time 2.373371e-02 0.10144856
## u_space_time 3.551379e-04 0.09803557
## u_space_time -1.266823e-02 0.09853313
## u_space_time -7.996980e-03 0.09233905
## u_space_time 7.594716e-03 0.09686161
## u_space_time 3.361985e-02 0.10287746
## u_space_time 2.123364e-03 0.09558166
## u_space_time 1.340631e-02 0.09530310
## u_space_time -5.026801e-03 0.09774739
## u_space_time -3.975658e-03 0.09314919
## u_space_time 3.179054e-02 0.10321554
## u_space_time -8.979047e-03 0.09513922
## u_space_time -2.240485e-02 0.09932928
## u_space_time 1.465998e-02 0.09631662
## u_space_time -2.572994e-02 0.09699739
## u_space_time 2.931630e-02 0.09552741
## u_space_time 4.925129e-02 0.09932303
## u_space_time 4.654626e-03 0.09640818
## u_space_time 2.750479e-02 0.10146591
## u_space_time 1.431109e-02 0.09691402
## u_space_time -1.033345e-02 0.09773614
## u_space_time -1.165166e-02 0.09842156
## u_space_time 6.058688e-03 0.09201605
## u_space_time 7.153770e-02 0.08507375
## u_space_time 4.256433e-03 0.10644623
## u_space_time 8.041602e-02 0.12785549
## u_space_time 3.094876e-03 0.10487488
## u_space_time -2.927281e-02 0.10628648
## u_space_time -9.059717e-03 0.10684517
## u_space_time 1.809009e-02 0.09653403
## u_space_time 3.735155e-02 0.10785888
## u_space_time 9.044854e-03 0.10366280
## u_space_time 2.332303e-02 0.10954408
## u_space_time 3.584852e-04 0.10655042
## u_space_time -1.320006e-02 0.10705267
## u_space_time -5.701310e-03 0.10071228
## u_space_time 8.565295e-03 0.10544743
## u_space_time 3.132453e-02 0.10996370
## u_space_time 6.362937e-03 0.10433980
## u_space_time 2.102131e-02 0.10539440
## u_space_time -5.230193e-03 0.10626625
## u_space_time -3.403103e-03 0.10165858
## u_space_time 3.432365e-02 0.11223957
## u_space_time -8.593678e-03 0.10358527
## u_space_time -2.023948e-02 0.10701598
## u_space_time 1.580129e-02 0.10494853
## u_space_time -2.855710e-02 0.10604140
## u_space_time 2.794511e-02 0.10317344
## u_space_time 4.375913e-02 0.10429531
## u_space_time 6.551446e-03 0.10505337
## u_space_time 2.766716e-02 0.10964811
## u_space_time 1.414944e-02 0.10528726
## u_space_time -1.130600e-02 0.10632559
## u_space_time -1.220212e-02 0.10695321
## u_space_time 6.442697e-03 0.10054263Hyper parameter comparison
summary(sdr7, "fixed")## Estimate Std. Error
## log_prec_space_time 9.558215 0.2596289summary(sdr7b, "fixed")## Estimate Std. Error
## log_prec_space_time 5.187216 1.185821cbind("mean" = inlafit7C$misc$theta.mode,
"se" = sqrt(diag(inlafit7C$misc$cov.intern)))## mean se
## [1,] 5.187209 1.178537Fixed effects (Intercept)
summary(sdr7, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.147675 0.3478549summary(sdr7b, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.003722469 0.02944028inlafit7C$summary.fixed[ , 1:2]Random effects mean and standard deviation
plot(summary(sdr7, "report")[,1], summary(sdr7b, "report")[,1],
xlab = "TMB soft constraint)", ylab = "TMB hard constraint",
main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(summary(sdr7, "report")[,2], summary(sdr7b, "report")[,2],
xlab = "TMB soft constraint", ylab = "TMB hard constraint",
main = "Random effect standard devation")
abline(0, 1, col = "red")
plot(inlafit7C$summary.random[[1]][ , 2], summary(sdr7b, "report")[,1],
xlab = "TMB soft constraint)", ylab = "TMB hard constraint",
main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(inlafit7C$summary.random[[1]][ , 3], summary(sdr7b, "report")[,2],
xlab = "TMB soft constraint", ylab = "TMB hard constraint",
main = "Random effect standard devation")
abline(0, 1, col = "red")
9 Model 8: ICAR space x RW1
Change the time trend to a RW1. The TMB code remains the same. The only change is the structure matrix for the time trend. If we were using a scaled parameterisation, we would need to adjust the penalty for the additional rank deficiency of the structure matrix.
For the Cmatrix version of the of the R-INLA model, the rank deficiency is \(\text{rows} + \text{columns} - 1\) since both are rank deficient one.
R-INLA automatically applies a sum-to-zero constraint for the spatial field at each time.
This is too many constraints for an intercept plus interaction only model because it imposes that the average is the intercept at each time, and hence there is no time trend in the model.
The correct specification for this model should probably be a single sum-to-zero constraint to account for the lost degree of freedom from the intercept term.
inlafit8 <- inla(Observed ~
f(Year, model = "rw1", scale.model = TRUE,
hyper = prec.prior, diagonal = diagval, constr = TRUE,
group = ID,
control.group = list(model = "besag", graph = adj.mat, scale.model = TRUE)),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))Only the main effect (ICAR) rank deficiency is reflected in the log file.
grep(".*rank.*", inlafit8$logfile, value = TRUE)## [1] " computed/guessed rank-deficiency = [1]"9.1 R-INLA with custom Cmatrix
R_space <- diag(rowSums(adj.mat)) - adj.mat
R_space_scaled <- inla.scale.model(R_space, constr = list(A = matrix(1, ncol = ncol(R_space)), e = 0))
R_space_scaled_adj <- R_space_scaled + Matrix::Diagonal(ncol(R_space_scaled), diagval)
D_time <- diff(diag(length(levels(data$Yearf))), differences = 1)
R_time <- Matrix::Matrix(t(D_time) %*% D_time)
R_time_scaled <- inla.scale.model(R_time, constr = list(A = matrix(1, ncol = ncol(R_time)), e = 0))
R_time_scaled_adj <- R_time_scaled + Matrix::Diagonal(ncol(R_time_scaled), diagval)
R_space_time <- kronecker(R_time_scaled, R_space_scaled)
rankdef <- nrow(R_space) + ncol(R_time) - 1
Aconstr <- t(model.matrix(~0+IDf, data[order(data$id.area.year), ]))
inlafit8C <- inla(Observed ~
f(id.area.year, model = "generic0", Cmatrix = R_space_time,
hyper = prec.prior, diagonal = diagval, rankdef = rankdef,
extraconstr = list(A = Aconstr, e = numeric(nrow(Aconstr)))),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep(".*rank.*", inlafit8C$logfile, value = TRUE)## [1] " rank-deficiency is *defined* [50]"The precision parameter estimates match
summary(inlafit8)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.62, Running = 0.443, Post = 0.0145, Total = 3.08
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.019 0.03 -0.078 -0.019 0.039 NA 0
##
## Random effects:
## Name Model
## Year RW1 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for Year 569.88 667.75 50.16 368.25 2327.11 NA
##
## Marginal log-Likelihood: -1201.32
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')summary(inlafit8C)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.76, Running = 0.415, Post = 0.0276, Total = 3.2
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.019 0.03 -0.078 -0.019 0.039 NA 0
##
## Random effects:
## Name Model
## id.area.year Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for id.area.year 569.88 667.77 50.16 368.25 2327.14 NA
##
## Marginal log-Likelihood: -1201.32
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Marginal likelihood
inlafit8$mlik## [,1]
## log marginal-likelihood (integration) -1202.529
## log marginal-likelihood (Gaussian) -1201.322inlafit8C$mlik## [,1]
## log marginal-likelihood (integration) -1202.529
## log marginal-likelihood (Gaussian) -1201.322Same number of constraints. Different order since we switched the main and group effect order.
nrow(inlafit8$misc$configs$constr$A)## [1] 32nrow(inlafit8C$misc$configs$constr$A)## [1] 32Hyper parameters
inlafit8$internal.summary.hyperpar[, 1:2]inlafit8C$internal.summary.hyperpar[, 1:2]Fixed effects
inlafit8$summary.fixed[ , 1:2]inlafit8C$summary.fixed[ , 1:2]Random effects
plot(inlafit8$summary.random[[1]][order(inlafit8$summary.random[[1]]$ID), 2],
inlafit8C$summary.random[[1]][,2],
main = "Random effect mean")
abline(a = 0, b = 1, col = "red")
plot(inlafit8$summary.random[[1]][order(inlafit8$summary.random[[1]]$ID), 3],
inlafit8C$summary.random[[1]][,3],
main = "Random effect standard deviation")
abline(a = 0, b = 1, col = "red")
9.2 TMB
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(Aconstr);
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_time);
DATA_SPARSE_MATRIX(R_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_raw_space_time_v(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time * sigma_space_time);
val += SEPARABLE(GMRF(R_time), GMRF(R_space))(u_raw_space_time);
val -= dnorm(Aconstr * u_raw_space_time_v, Type(0), Type(0.001) * u_raw_space_time.cols(), true).sum(); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
R_space_time_adj <- R_space_time + Matrix::Diagonal(ncol(R_space_time), 1e-6)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Aconstr = Aconstr,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_space = R_space_scaled_adj,
R_time = R_time_scaled_adj)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = array(0, c(nrow(tmbdata$R_space), nrow(tmbdata$R_time))))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr8 <- TMB::sdreport(obj)
summary(sdr8, "all")## Estimate Std. Error
## log_prec_space_time 8.096334574 0.44518321
## beta0 -0.013667418 0.02981683
## u_raw_space_time 2.480252874 6.59982062
## u_raw_space_time 2.588862844 6.69775648
## u_raw_space_time 2.589800875 6.66282071
## u_raw_space_time 2.683932371 6.73485033
## u_raw_space_time 2.976495938 6.82149850
## u_raw_space_time 2.683109497 6.66739530
## u_raw_space_time 2.524070856 6.75495366
## u_raw_space_time 2.633107219 6.75762167
## u_raw_space_time 2.640546116 6.76608505
## u_raw_space_time 2.619034487 6.65116240
## u_raw_space_time 2.686584652 6.69033334
## u_raw_space_time 2.759014993 6.93300204
## u_raw_space_time 2.759595379 6.76048415
## u_raw_space_time 2.618563887 6.62311370
## u_raw_space_time 2.462040272 6.72761570
## u_raw_space_time 2.662542766 6.77624379
## u_raw_space_time 2.629416273 6.72982183
## u_raw_space_time 2.649293272 6.66052972
## u_raw_space_time 2.642930180 6.67209602
## u_raw_space_time 2.712856907 6.64596728
## u_raw_space_time 2.628496461 6.65371474
## u_raw_space_time 2.764027329 6.68814715
## u_raw_space_time 2.541641754 6.62710766
## u_raw_space_time 2.705119566 6.74028684
## u_raw_space_time 2.657703268 6.62998222
## u_raw_space_time 2.567652155 6.61023730
## u_raw_space_time 2.599886841 6.65856018
## u_raw_space_time 2.558025125 6.64978236
## u_raw_space_time 2.653222162 6.75668821
## u_raw_space_time 2.606371079 6.60100746
## u_raw_space_time 2.730836032 6.76704053
## u_raw_space_time 2.619418105 6.61673892
## u_raw_space_time 3.158004996 5.73713327
## u_raw_space_time 3.269554993 5.83518610
## u_raw_space_time 3.271337719 5.80128438
## u_raw_space_time 3.359296062 5.87315111
## u_raw_space_time 3.654473436 5.96193920
## u_raw_space_time 3.359002636 5.80753836
## u_raw_space_time 3.227699088 5.88887755
## u_raw_space_time 3.310892807 5.89388804
## u_raw_space_time 3.323649763 5.90216971
## u_raw_space_time 3.294566861 5.79063078
## u_raw_space_time 3.360460816 5.82995078
## u_raw_space_time 3.438659772 6.06577283
## u_raw_space_time 3.435985983 5.89883633
## u_raw_space_time 3.298508208 5.76320788
## u_raw_space_time 3.137100904 5.86196124
## u_raw_space_time 3.352489982 5.91223212
## u_raw_space_time 3.294833936 5.86664731
## u_raw_space_time 3.320538889 5.80017463
## u_raw_space_time 3.325378247 5.81100668
## u_raw_space_time 3.390150200 5.78729344
## u_raw_space_time 3.296632469 5.79290471
## u_raw_space_time 3.444728954 5.82908220
## u_raw_space_time 3.214230757 5.76543984
## u_raw_space_time 3.365362612 5.87773462
## u_raw_space_time 3.325773567 5.77043849
## u_raw_space_time 3.238075912 5.74946620
## u_raw_space_time 3.287576674 5.79717764
## u_raw_space_time 3.238516737 5.78806234
## u_raw_space_time 3.323000430 5.89365612
## u_raw_space_time 3.279139671 5.74134326
## u_raw_space_time 3.408229319 5.90529964
## u_raw_space_time 3.289493889 5.75666880
## u_raw_space_time -2.946681378 5.78425692
## u_raw_space_time -2.835064545 5.86658714
## u_raw_space_time -2.829728285 5.83854241
## u_raw_space_time -2.754810536 5.90000237
## u_raw_space_time -2.453259468 5.97974993
## u_raw_space_time -2.749459122 5.84519854
## u_raw_space_time -2.851431370 5.91074436
## u_raw_space_time -2.789695866 5.91654674
## u_raw_space_time -2.773887339 5.92334513
## u_raw_space_time -2.816446602 5.82994785
## u_raw_space_time -2.754315696 5.86385349
## u_raw_space_time -2.663096062 6.06224915
## u_raw_space_time -2.672121641 5.92278230
## u_raw_space_time -2.805809737 5.80715574
## u_raw_space_time -2.981786521 5.88682630
## u_raw_space_time -2.745793512 5.93208267
## u_raw_space_time -2.824957306 5.89337747
## u_raw_space_time -2.798943962 5.83826746
## u_raw_space_time -2.773746430 5.84745443
## u_raw_space_time -2.719914491 5.82890735
## u_raw_space_time -2.829929769 5.83168475
## u_raw_space_time -2.659549394 5.86488974
## u_raw_space_time -2.902672161 5.80773594
## u_raw_space_time -2.759801580 5.90347890
## u_raw_space_time -2.798114005 5.81337780
## u_raw_space_time -2.888189290 5.79458594
## u_raw_space_time -2.806874216 5.83520502
## u_raw_space_time -2.869011328 5.82677829
## u_raw_space_time -2.794100692 5.91649265
## u_raw_space_time -2.838738259 5.78863180
## u_raw_space_time -2.700251059 5.92785673
## u_raw_space_time -2.834490708 5.80144147
## u_raw_space_time -9.419055134 5.90799520
## u_raw_space_time -9.312195317 5.97515942
## u_raw_space_time -9.312323093 5.95246931
## u_raw_space_time -9.253451258 6.00312352
## u_raw_space_time -8.951441571 6.07309097
## u_raw_space_time -9.237355380 5.95864066
## u_raw_space_time -9.302060657 6.01186656
## u_raw_space_time -9.269730383 6.01706606
## u_raw_space_time -9.243694628 6.02262635
## u_raw_space_time -9.303428807 5.94529407
## u_raw_space_time -9.249620483 5.97354610
## u_raw_space_time -9.144144150 6.13781956
## u_raw_space_time -9.164632542 6.02341899
## u_raw_space_time -9.291607023 5.92674838
## u_raw_space_time -9.468722051 5.99005131
## u_raw_space_time -9.217156446 6.03000504
## u_raw_space_time -9.321810954 5.99697198
## u_raw_space_time -9.295410280 5.95205972
## u_raw_space_time -9.254605063 5.96033241
## u_raw_space_time -9.209595540 5.94567588
## u_raw_space_time -9.330705566 5.94624878
## u_raw_space_time -9.144210860 5.97624672
## u_raw_space_time -9.390126074 5.92632096
## u_raw_space_time -9.266883261 6.00555772
## u_raw_space_time -9.292322929 5.93176904
## u_raw_space_time -9.387843594 5.91533968
## u_raw_space_time -9.276904366 5.94997309
## u_raw_space_time -9.348960593 5.94214237
## u_raw_space_time -9.291805177 6.01627583
## u_raw_space_time -9.329192044 5.91126174
## u_raw_space_time -9.200053927 6.02661794
## u_raw_space_time -9.329729760 5.92164026
## u_raw_space_time -6.245643618 5.88234630
## u_raw_space_time -6.144764782 5.93987505
## u_raw_space_time -6.141120408 5.92142787
## u_raw_space_time -6.115034130 5.96348153
## u_raw_space_time -5.824478772 6.02935507
## u_raw_space_time -6.081337984 5.92749948
## u_raw_space_time -6.098062729 5.97288190
## u_raw_space_time -6.094203858 5.97705737
## u_raw_space_time -6.073898872 5.98151360
## u_raw_space_time -6.148218773 5.91457842
## u_raw_space_time -6.107370444 5.93890711
## u_raw_space_time -5.984471885 6.08017561
## u_raw_space_time -6.001253033 5.98445831
## u_raw_space_time -6.130461218 5.89964493
## u_raw_space_time -6.304784702 5.94862919
## u_raw_space_time -6.047692560 5.98834585
## u_raw_space_time -6.172570731 5.95734082
## u_raw_space_time -6.150603119 5.91997049
## u_raw_space_time -6.082618131 5.92936455
## u_raw_space_time -6.062736636 5.91686902
## u_raw_space_time -6.184254891 5.91440482
## u_raw_space_time -5.987428993 5.94479455
## u_raw_space_time -6.227593205 5.89717143
## u_raw_space_time -6.123733698 5.96539264
## u_raw_space_time -6.148539709 5.90299257
## u_raw_space_time -6.242301437 5.88714543
## u_raw_space_time -6.103800839 5.91997797
## u_raw_space_time -6.181457346 5.91136674
## u_raw_space_time -6.145309405 5.97403943
## u_raw_space_time -6.174445415 5.88553552
## u_raw_space_time -6.062117521 5.98442925
## u_raw_space_time -6.175414650 5.89419981
## u_raw_space_time -4.776981166 5.82127035
## u_raw_space_time -4.706697085 5.86853181
## u_raw_space_time -4.704960277 5.85323923
## u_raw_space_time -4.701741736 5.88763716
## u_raw_space_time -4.461608536 5.94430833
## u_raw_space_time -4.662136257 5.85788812
## u_raw_space_time -4.637347209 5.89858775
## u_raw_space_time -4.665384649 5.90070351
## u_raw_space_time -4.642990099 5.90434088
## u_raw_space_time -4.724281507 5.84636955
## u_raw_space_time -4.695686235 5.86674749
## u_raw_space_time -4.570237444 5.98814065
## u_raw_space_time -4.579451223 5.90772116
## u_raw_space_time -4.700671224 5.83429828
## u_raw_space_time -4.845307482 5.87548810
## u_raw_space_time -4.617365716 5.91042420
## u_raw_space_time -4.728050116 5.88367532
## u_raw_space_time -4.726863847 5.85095604
## u_raw_space_time -4.649406357 5.86036179
## u_raw_space_time -4.656400768 5.84837968
## u_raw_space_time -4.749752943 5.84655088
## u_raw_space_time -4.578544111 5.87323246
## u_raw_space_time -4.772236342 5.83270597
## u_raw_space_time -4.695669585 5.89004696
## u_raw_space_time -4.735306778 5.83601857
## u_raw_space_time -4.805384280 5.82301638
## u_raw_space_time -4.666323315 5.85222177
## u_raw_space_time -4.739014203 5.84424621
## u_raw_space_time -4.720246254 5.89695758
## u_raw_space_time -4.740628789 5.82212750
## u_raw_space_time -4.660183403 5.90530570
## u_raw_space_time -4.740922615 5.82952196
## u_raw_space_time -7.475825895 5.78843651
## u_raw_space_time -7.445543059 5.82613617
## u_raw_space_time -7.446747298 5.81334066
## u_raw_space_time -7.459623410 5.84152662
## u_raw_space_time -7.287994249 5.88736118
## u_raw_space_time -7.419577302 5.81627737
## u_raw_space_time -7.371926687 5.85268586
## u_raw_space_time -7.419972722 5.85357570
## u_raw_space_time -7.394816276 5.85634397
## u_raw_space_time -7.473384731 5.80670904
## u_raw_space_time -7.456221424 5.82351228
## u_raw_space_time -7.348799607 5.92681513
## u_raw_space_time -7.359309543 5.85839813
## u_raw_space_time -7.447265325 5.79661761
## u_raw_space_time -7.553055136 5.83357561
## u_raw_space_time -7.377878979 5.86121044
## u_raw_space_time -7.446010688 5.84027212
## u_raw_space_time -7.475832317 5.81062038
## u_raw_space_time -7.399389167 5.81902179
## u_raw_space_time -7.427142278 5.80748780
## u_raw_space_time -7.482608936 5.80773857
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## u_space_time -0.053272865 0.09572072
## u_space_time 0.137639334 0.08991353
## u_space_time 0.136908923 0.09060752
## u_space_time 0.135768886 0.09025822
## u_space_time 0.136053619 0.09085309
## u_space_time 0.132172412 0.09131403
## u_space_time 0.135265077 0.09025898
## u_space_time 0.137327229 0.09109322
## u_space_time 0.135168943 0.09097232
## u_space_time 0.136189422 0.09110194
## u_space_time 0.136199594 0.09019264
## u_space_time 0.135817839 0.09047985
## u_space_time 0.135055961 0.09235828
## u_space_time 0.134118671 0.09093016
## u_space_time 0.136147254 0.08996496
## u_space_time 0.138505195 0.09096877
## u_space_time 0.135971879 0.09116598
## u_space_time 0.136852288 0.09086971
## u_space_time 0.136449716 0.09028983
## u_space_time 0.135767918 0.09032967
## u_space_time 0.135007293 0.09007551
## u_space_time 0.137037393 0.09028208
## u_space_time 0.134103325 0.09035417
## u_space_time 0.137766022 0.09012482
## u_space_time 0.136918043 0.09095759
## u_space_time 0.136275737 0.09003298
## u_space_time 0.137491566 0.08997128
## u_space_time 0.136431689 0.09026305
## u_space_time 0.137041473 0.09023731
## u_space_time 0.136331058 0.09104585
## u_space_time 0.136795960 0.08984246
## u_space_time 0.135339005 0.09105851
## u_space_time 0.137001776 0.08998400
## u_space_time 0.149802248 0.08838652
## u_space_time 0.149114853 0.08926952
## u_space_time 0.148083206 0.08890182
## u_space_time 0.148351390 0.08960204
## u_space_time 0.143904252 0.09027063
## u_space_time 0.147489325 0.08892099
## u_space_time 0.149116164 0.08979596
## u_space_time 0.147300724 0.08976533
## u_space_time 0.148193570 0.08987541
## u_space_time 0.148719489 0.08882001
## u_space_time 0.148169381 0.08916910
## u_space_time 0.146891953 0.09141024
## u_space_time 0.145987981 0.08974440
## u_space_time 0.148475273 0.08853827
## u_space_time 0.151240906 0.08966179
## u_space_time 0.147897777 0.08995399
## u_space_time 0.148724977 0.08955766
## u_space_time 0.148805447 0.08891396
## u_space_time 0.147882069 0.08897357
## u_space_time 0.147321369 0.08871531
## u_space_time 0.149209979 0.08886638
## u_space_time 0.145965028 0.08906049
## u_space_time 0.150035711 0.08865242
## u_space_time 0.148654643 0.08964853
## u_space_time 0.148852782 0.08862475
## u_space_time 0.150166706 0.08850280
## u_space_time 0.148531625 0.08886851
## u_space_time 0.149482900 0.08883390
## u_space_time 0.148637065 0.08981880
## u_space_time 0.149164757 0.08836442
## u_space_time 0.147561719 0.08987397
## u_space_time 0.149215266 0.08851354
## u_space_time 0.070371408 0.08972749
## u_space_time 0.069732131 0.09084218
## u_space_time 0.069382737 0.09049734
## u_space_time 0.069318498 0.09130308
## u_space_time 0.064588906 0.09255758
## u_space_time 0.068524500 0.09059497
## u_space_time 0.069069437 0.09147184
## u_space_time 0.068517692 0.09159557
## u_space_time 0.068750505 0.09164799
## u_space_time 0.069822325 0.09034144
## u_space_time 0.069171411 0.09081393
## u_space_time 0.067276575 0.09358644
## u_space_time 0.067054087 0.09171763
## u_space_time 0.069432968 0.09004156
## u_space_time 0.072346911 0.09105439
## u_space_time 0.068245213 0.09177814
## u_space_time 0.069137083 0.09122719
## u_space_time 0.069755131 0.09044655
## u_space_time 0.068535694 0.09062538
## u_space_time 0.068416278 0.09036775
## u_space_time 0.069956104 0.09035155
## u_space_time 0.066837834 0.09092719
## u_space_time 0.070619480 0.09001702
## u_space_time 0.068788309 0.09134706
## u_space_time 0.069864676 0.09010023
## u_space_time 0.071271244 0.08980614
## u_space_time 0.069079289 0.09043466
## u_space_time 0.070297630 0.09028430
## u_space_time 0.069677376 0.09152103
## u_space_time 0.069995513 0.08976711
## u_space_time 0.068530462 0.09170684
## u_space_time 0.069937294 0.08993810
## u_space_time 0.109040863 0.08817647
## u_space_time 0.108042475 0.08953847
## u_space_time 0.108424229 0.08911355
## u_space_time 0.107536759 0.09008707
## u_space_time 0.102654276 0.09137837
## u_space_time 0.107154253 0.08918852
## u_space_time 0.107131043 0.09025423
## u_space_time 0.107614575 0.09040496
## u_space_time 0.107111905 0.09046675
## u_space_time 0.108437686 0.08894263
## u_space_time 0.107526543 0.08948547
## u_space_time 0.105525404 0.09274478
## u_space_time 0.105766972 0.09048106
## u_space_time 0.108051014 0.08855465
## u_space_time 0.110517937 0.08991859
## u_space_time 0.106460956 0.09059685
## u_space_time 0.107195018 0.08996432
## u_space_time 0.108028628 0.08906014
## u_space_time 0.107066644 0.08921627
## u_space_time 0.106957823 0.08890607
## u_space_time 0.108023392 0.08894452
## u_space_time 0.105321523 0.08950814
## u_space_time 0.108650221 0.08856486
## u_space_time 0.106296316 0.09008758
## u_space_time 0.108383627 0.08864439
## u_space_time 0.109692843 0.08834332
## u_space_time 0.107430597 0.08900867
## u_space_time 0.108758847 0.08888850
## u_space_time 0.107907575 0.09036295
## u_space_time 0.108450511 0.08823809
## u_space_time 0.106770115 0.09054264
## u_space_time 0.108173533 0.08843559
## u_space_time 0.166858396 0.08786401
## u_space_time 0.165202930 0.08942272
## u_space_time 0.166324894 0.08900988
## u_space_time 0.164921522 0.09007002
## u_space_time 0.160210986 0.09129142
## u_space_time 0.164806710 0.08898988
## u_space_time 0.163660102 0.09016415
## u_space_time 0.165645450 0.09050895
## u_space_time 0.163984102 0.09045214
## u_space_time 0.166097936 0.08878187
## u_space_time 0.165035620 0.08935780
## u_space_time 0.162400953 0.09306733
## u_space_time 0.163482012 0.09044964
## u_space_time 0.165528080 0.08827379
## u_space_time 0.167741134 0.09004309
## u_space_time 0.163332060 0.09056056
## u_space_time 0.164763120 0.08988484
## u_space_time 0.165446771 0.08886779
## u_space_time 0.164428508 0.08898845
## u_space_time 0.164657798 0.08864040
## u_space_time 0.165274985 0.08870509
## u_space_time 0.162968564 0.08924471
## u_space_time 0.165952313 0.08829050
## u_space_time 0.163267702 0.08992331
## u_space_time 0.166016925 0.08841066
## u_space_time 0.167054826 0.08810530
## u_space_time 0.164554475 0.08874372
## u_space_time 0.166145204 0.08871014
## u_space_time 0.165411485 0.09042595
## u_space_time 0.165910788 0.08790619
## u_space_time 0.164200816 0.09056945
## u_space_time 0.165567320 0.08810902
## u_space_time 0.043684489 0.09099919
## u_space_time 0.042174794 0.09300149
## u_space_time 0.043553895 0.09234700
## u_space_time 0.041909297 0.09378998
## u_space_time 0.037210758 0.09586118
## u_space_time 0.041945817 0.09255796
## u_space_time 0.040804911 0.09405628
## u_space_time 0.043172892 0.09414701
## u_space_time 0.041033476 0.09434956
## u_space_time 0.043116492 0.09213431
## u_space_time 0.042045256 0.09295610
## u_space_time 0.039627662 0.09754057
## u_space_time 0.040803927 0.09441446
## u_space_time 0.042619837 0.09163099
## u_space_time 0.044344082 0.09336463
## u_space_time 0.040578554 0.09455246
## u_space_time 0.041789345 0.09359741
## u_space_time 0.042345505 0.09234206
## u_space_time 0.041741853 0.09259771
## u_space_time 0.041856816 0.09217700
## u_space_time 0.042064895 0.09218918
## u_space_time 0.040159141 0.09312014
## u_space_time 0.042723703 0.09161458
## u_space_time 0.039915764 0.09388565
## u_space_time 0.042930812 0.09173167
## u_space_time 0.043705713 0.09124128
## u_space_time 0.041702808 0.09230696
## u_space_time 0.043071260 0.09204879
## u_space_time 0.042491717 0.09411661
## u_space_time 0.042760190 0.09117335
## u_space_time 0.041216564 0.09446409
## u_space_time 0.042371597 0.09146078
## u_space_time 0.115614549 0.09675448
## u_space_time 0.114672864 0.09885556
## u_space_time 0.116400400 0.09829553
## u_space_time 0.114094719 0.09970406
## u_space_time 0.109944174 0.10156144
## u_space_time 0.114557649 0.09835256
## u_space_time 0.113567948 0.09990031
## u_space_time 0.116050106 0.10025983
## u_space_time 0.113688095 0.10024992
## u_space_time 0.115516792 0.09799007
## u_space_time 0.114286326 0.09879131
## u_space_time 0.112322274 0.10365992
## u_space_time 0.113718966 0.10032513
## u_space_time 0.115199959 0.09738466
## u_space_time 0.116207142 0.09945474
## u_space_time 0.113427779 0.10043272
## u_space_time 0.114276950 0.09946891
## u_space_time 0.114430633 0.09813031
## u_space_time 0.114489959 0.09837956
## u_space_time 0.114366977 0.09791733
## u_space_time 0.114062891 0.09792457
## u_space_time 0.112938675 0.09880497
## u_space_time 0.114805881 0.09734995
## u_space_time 0.111915982 0.09958863
## u_space_time 0.115104413 0.09751407
## u_space_time 0.115616100 0.09703199
## u_space_time 0.114386695 0.09804385
## u_space_time 0.115517769 0.09788608
## u_space_time 0.114554821 0.10011750
## u_space_time 0.114976897 0.09687094
## u_space_time 0.113490068 0.10038797
## u_space_time 0.114638716 0.09714818Hyper parameter comparison
summary(sdr8, "fixed")## Estimate Std. Error
## log_prec_space_time 8.096335 0.4451832cbind("mean" = inlafit8$misc$theta.mode,
"se" = sqrt(diag(inlafit8$misc$cov.intern)))## mean se
## [1,] 5.997606 1.160329cbind("mean" = inlafit8C$misc$theta.mode,
"se" = sqrt(diag(inlafit8C$misc$cov.intern)))## mean se
## [1,] 5.997604 1.16031Fixed effects (Intercept)
summary(sdr8, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.01366742 0.02981683inlafit8$summary.fixed[ , 1:2]inlafit8C$summary.fixed[ , 1:2]Random effects mean and standard deviation
plot(inlafit8C$summary.random[[1]][,2], summary(sdr8, "report")[,1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(inlafit8C$summary.random[[1]][,3], summary(sdr8, "report")[,2],
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation")
abline(0, 1, col = "red")
9.3 Model 8b: Explicit sum-to-zero constraint
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(L_space_time)
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(LRL_space_time);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_VECTOR(u_raw_space_time);
vector<Type> u_space_time(L_space_time * u_raw_space_time * sigma_space_time);
val += GMRF(LRL_space_time)(u_raw_space_time);
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
qrc <- qr(t(Aconstr))
L_space_time <- qr.Q(qrc, complete=TRUE)[ , (nrow(Aconstr)+1):ncol(Aconstr)]
LRL <- as(t(L_space_time) %*% R_space_time %*% L_space_time, "dgCMatrix")
LRL_adj <- LRL + Matrix::Diagonal(ncol(LRL), diagval)
tmbdata <- list(y = data$Observed,
E = data$Expected,
L_space_time = L_space_time,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
LRL_space_time = LRL_adj)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = numeric(ncol(tmbdata$L_space_time)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr8b <- TMB::sdreport(obj)
summary(sdr8b, "all")## Estimate Std. Error
## log_prec_space_time 8.518427968 0.35982238
## beta0 -0.013457167 0.02973430
## u_raw_space_time 6.279347381 8.54869251
## u_raw_space_time 6.396576201 8.58619240
## u_raw_space_time 6.398119356 8.57147410
## u_raw_space_time 6.460478670 8.60198232
## u_raw_space_time 6.655301830 8.64077249
## u_raw_space_time 6.459504142 8.57238111
## u_raw_space_time 6.367871279 8.61268478
## u_raw_space_time 6.421987066 8.61407529
## u_raw_space_time 6.437471547 8.61600448
## u_raw_space_time 6.413919024 8.56555191
## u_raw_space_time 6.460972896 8.58217523
## u_raw_space_time 6.517791657 8.69000448
## u_raw_space_time 6.505693673 8.61492126
## u_raw_space_time 6.415945367 8.55330923
## u_raw_space_time 6.297273762 8.60126326
## u_raw_space_time 6.457055460 8.62046658
## u_raw_space_time 6.396951229 8.60227790
## u_raw_space_time 6.427994822 8.56980704
## u_raw_space_time 6.432732631 8.57531441
## u_raw_space_time 6.480649090 8.56273823
## u_raw_space_time 6.403817339 8.56780518
## u_raw_space_time 6.516600784 8.58164201
## u_raw_space_time 6.342388611 8.55718373
## u_raw_space_time 6.443624952 8.60640247
## u_raw_space_time 6.428306974 8.55670994
## u_raw_space_time 6.358939350 8.54956885
## u_raw_space_time 6.410834205 8.56875875
## u_raw_space_time 6.373222984 8.56543619
## u_raw_space_time 6.430885893 8.61244514
## u_raw_space_time 6.391990022 8.54468740
## u_raw_space_time 6.496433136 8.61579554
## u_raw_space_time 6.393104311 8.55220959
## u_raw_space_time -2.906662573 8.82160805
## u_raw_space_time -2.791214091 8.85266823
## u_raw_space_time -2.786794135 8.84044016
## u_raw_space_time -2.735214320 8.86686194
## u_raw_space_time -2.535233387 8.90254802
## u_raw_space_time -2.731584774 8.84203891
## u_raw_space_time -2.799077351 8.87448679
## u_raw_space_time -2.762477948 8.87650549
## u_raw_space_time -2.744782749 8.87800637
## u_raw_space_time -2.779110461 8.83570745
## u_raw_space_time -2.735264361 8.85025313
## u_raw_space_time -2.668089123 8.94102784
## u_raw_space_time -2.684924393 8.87851237
## u_raw_space_time -2.771611359 8.82550881
## u_raw_space_time -2.901604317 8.86421333
## u_raw_space_time -2.725783922 8.88200988
## u_raw_space_time -2.802553378 8.86674857
## u_raw_space_time -2.771794847 8.83960298
## u_raw_space_time -2.750562565 8.84418898
## u_raw_space_time -2.711754057 8.83434143
## u_raw_space_time -2.801378187 8.83780674
## u_raw_space_time -2.671093258 8.85078475
## u_raw_space_time -2.854656703 8.82832494
## u_raw_space_time -2.760279762 8.87080728
## u_raw_space_time -2.774948728 8.82866968
## u_raw_space_time -2.845737240 8.82211145
## u_raw_space_time -2.768772019 8.83823864
## u_raw_space_time -2.816801757 8.83506736
## u_raw_space_time -2.767033864 8.87530662
## u_raw_space_time -2.806128588 8.81835294
## u_raw_space_time -2.694844768 8.87889922
## u_raw_space_time -2.809751116 8.82481236
## u_raw_space_time -14.936684270 9.32406830
## u_raw_space_time -14.828031480 9.34927092
## u_raw_space_time -14.828027734 9.33934767
## u_raw_space_time -14.789801991 9.36147086
## u_raw_space_time -14.589440329 9.39510600
## u_raw_space_time -14.777414899 9.34134439
## u_raw_space_time -14.814120008 9.36787897
## u_raw_space_time -14.801533656 9.36969400
## u_raw_space_time -14.775990994 9.37101858
## u_raw_space_time -14.824000338 9.33533918
## u_raw_space_time -14.787152718 9.34782187
## u_raw_space_time -14.708739688 9.42401437
## u_raw_space_time -14.734182468 9.37259608
## u_raw_space_time -14.815461014 9.32713540
## u_raw_space_time -14.945607805 9.35717162
## u_raw_space_time -14.758222989 9.37463254
## u_raw_space_time -14.854416927 9.36098053
## u_raw_space_time -14.824207956 9.33862267
## u_raw_space_time -14.790322584 9.34313136
## u_raw_space_time -14.759055086 9.33535407
## u_raw_space_time -14.856754900 9.33688410
## u_raw_space_time -14.714287790 9.34989302
## u_raw_space_time -14.898718902 9.32871260
## u_raw_space_time -14.820509853 9.36485451
## u_raw_space_time -14.825369323 9.32971074
## u_raw_space_time -14.899759510 9.32346057
## u_raw_space_time -14.799921062 9.33788856
## u_raw_space_time -14.855706125 9.33437583
## u_raw_space_time -14.820544185 9.36810458
## u_raw_space_time -14.853076902 9.32099487
## u_raw_space_time -14.750600101 9.37198996
## u_raw_space_time -14.860238033 9.32637644
## u_raw_space_time -5.219071798 8.73122367
## u_raw_space_time -5.115556659 8.75578117
## u_raw_space_time -5.112648859 8.74663387
## u_raw_space_time -5.100897125 8.76694192
## u_raw_space_time -4.910022172 8.79645065
## u_raw_space_time -5.074321853 8.74788690
## u_raw_space_time -5.071911882 8.77358321
## u_raw_space_time -5.082695634 8.77513871
## u_raw_space_time -5.061617037 8.77613425
## u_raw_space_time -5.121497625 8.74253769
## u_raw_space_time -5.095155444 8.75399002
## u_raw_space_time -5.002581842 8.82571757
## u_raw_space_time -5.025240650 8.77707879
## u_raw_space_time -5.108134162 8.73469152
## u_raw_space_time -5.235867944 8.76409607
## u_raw_space_time -5.044095711 8.77937716
## u_raw_space_time -5.156596446 8.76636182
## u_raw_space_time -5.130067573 8.74542781
## u_raw_space_time -5.074213006 8.74969082
## u_raw_space_time -5.063379537 8.74191913
## u_raw_space_time -5.161203090 8.74379654
## u_raw_space_time -5.010643842 8.75531910
## u_raw_space_time -5.190090534 8.73631348
## u_raw_space_time -5.127557494 8.76957248
## u_raw_space_time -5.132069794 8.73686275
## u_raw_space_time -5.204891446 8.73126491
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## u_space_time -0.006749703 0.11131300
## u_space_time -0.006853481 0.11128320
## u_space_time -0.006979607 0.11163901
## u_space_time -0.006894828 0.11113426
## u_space_time -0.007121865 0.11166725
## u_space_time -0.006835649 0.11118957
## u_space_time 0.034600237 0.10842305
## u_space_time 0.033729444 0.10870122
## u_space_time 0.033391048 0.10858620
## u_space_time 0.033487908 0.10883178
## u_space_time 0.032558486 0.10914740
## u_space_time 0.033302710 0.10859644
## u_space_time 0.033996799 0.10891420
## u_space_time 0.033149948 0.10891808
## u_space_time 0.033397838 0.10893688
## u_space_time 0.033458767 0.10854016
## u_space_time 0.033420730 0.10867519
## u_space_time 0.032957475 0.10951672
## u_space_time 0.033000621 0.10893039
## u_space_time 0.033458792 0.10844473
## u_space_time 0.034563958 0.10882487
## u_space_time 0.033326518 0.10897312
## u_space_time 0.034293839 0.10883930
## u_space_time 0.033748687 0.10858090
## u_space_time 0.033397131 0.10861888
## u_space_time 0.033166944 0.10852383
## u_space_time 0.034243006 0.10857124
## u_space_time 0.033258965 0.10867607
## u_space_time 0.034478633 0.10848635
## u_space_time 0.034375609 0.10887673
## u_space_time 0.033456993 0.10847310
## u_space_time 0.033996652 0.10842137
## u_space_time 0.033599433 0.10856715
## u_space_time 0.033761531 0.10853799
## u_space_time 0.033673918 0.10891367
## u_space_time 0.033801416 0.10838061
## u_space_time 0.033241360 0.10893841
## u_space_time 0.033918799 0.10844133
## u_space_time -0.138436430 0.11508324
## u_space_time -0.139249498 0.11539059
## u_space_time -0.139854888 0.11529498
## u_space_time -0.139722340 0.11554025
## u_space_time -0.141604396 0.11593709
## u_space_time -0.140100355 0.11531275
## u_space_time -0.138940081 0.11559653
## u_space_time -0.140167709 0.11565077
## u_space_time -0.139665512 0.11564766
## u_space_time -0.139687884 0.11524095
## u_space_time -0.139845239 0.11538371
## u_space_time -0.140292704 0.11626858
## u_space_time -0.140632897 0.11567987
## u_space_time -0.139678764 0.11514244
## u_space_time -0.138176725 0.11547850
## u_space_time -0.139781470 0.11568900
## u_space_time -0.138782093 0.11551590
## u_space_time -0.139401942 0.11527068
## u_space_time -0.139835268 0.11532764
## u_space_time -0.140278225 0.11524441
## u_space_time -0.138867491 0.11524046
## u_space_time -0.140545260 0.11540959
## u_space_time -0.138466728 0.11514083
## u_space_time -0.138724151 0.11555067
## u_space_time -0.139645140 0.11517212
## u_space_time -0.138822839 0.11509071
## u_space_time -0.139469704 0.11525934
## u_space_time -0.139182802 0.11522099
## u_space_time -0.139465605 0.11561574
## u_space_time -0.139226125 0.11506214
## u_space_time -0.140123968 0.11566492
## u_space_time -0.139066633 0.11512027
## u_space_time 0.171463683 0.10203522
## u_space_time 0.170614696 0.10241025
## u_space_time 0.169866067 0.10223107
## u_space_time 0.170006644 0.10256415
## u_space_time 0.167462851 0.10289885
## u_space_time 0.169497250 0.10223450
## u_space_time 0.170926799 0.10269301
## u_space_time 0.169486619 0.10265933
## u_space_time 0.170100931 0.10270759
## u_space_time 0.170136167 0.10218169
## u_space_time 0.169852543 0.10235142
## u_space_time 0.169314207 0.10345523
## u_space_time 0.168785152 0.10265115
## u_space_time 0.170112126 0.10205221
## u_space_time 0.171756900 0.10260060
## u_space_time 0.169961769 0.10274951
## u_space_time 0.170696619 0.10257680
## u_space_time 0.170320107 0.10223557
## u_space_time 0.169874028 0.10227284
## u_space_time 0.169324769 0.10213225
## u_space_time 0.170772931 0.10222754
## u_space_time 0.168736650 0.10231049
## u_space_time 0.171317574 0.10212952
## u_space_time 0.170730076 0.10262550
## u_space_time 0.170229020 0.10209277
## u_space_time 0.171127587 0.10204502
## u_space_time 0.170290200 0.10221908
## u_space_time 0.170726275 0.10219579
## u_space_time 0.170229361 0.10267566
## u_space_time 0.170623690 0.10197780
## u_space_time 0.169509210 0.10269280
## u_space_time 0.170803732 0.10206066
## u_space_time 0.153586645 0.10183376
## u_space_time 0.152595121 0.10228529
## u_space_time 0.151916637 0.10208573
## u_space_time 0.152024532 0.10247388
## u_space_time 0.149110094 0.10288410
## u_space_time 0.151470685 0.10209138
## u_space_time 0.152652520 0.10260074
## u_space_time 0.151424415 0.10258982
## u_space_time 0.151929558 0.10263500
## u_space_time 0.152315677 0.10202904
## u_space_time 0.151905997 0.10222666
## u_space_time 0.151012301 0.10351988
## u_space_time 0.150545076 0.10258249
## u_space_time 0.152170937 0.10187210
## u_space_time 0.154129017 0.10250906
## u_space_time 0.151741016 0.10268334
## u_space_time 0.152514223 0.10247175
## u_space_time 0.152401366 0.10208481
## u_space_time 0.151798007 0.10212878
## u_space_time 0.151355294 0.10197400
## u_space_time 0.152765864 0.10206550
## u_space_time 0.150472817 0.10218076
## u_space_time 0.153405151 0.10194724
## u_space_time 0.152455008 0.10252297
## u_space_time 0.152466321 0.10192344
## u_space_time 0.153476443 0.10185958
## u_space_time 0.152195917 0.10205997
## u_space_time 0.152872524 0.10203725
## u_space_time 0.152271993 0.10260299
## u_space_time 0.152749268 0.10177978
## u_space_time 0.151464251 0.10262992
## u_space_time 0.152848787 0.10187148
## u_space_time 0.017354656 0.10604548
## u_space_time 0.016217186 0.10657668
## u_space_time 0.015984934 0.10638220
## u_space_time 0.015852265 0.10681312
## u_space_time 0.012753035 0.10738408
## u_space_time 0.015346913 0.10639811
## u_space_time 0.015858922 0.10693068
## u_space_time 0.015445807 0.10697885
## u_space_time 0.015491731 0.10699555
## u_space_time 0.016249009 0.10629789
## u_space_time 0.015756952 0.10653476
## u_space_time 0.014440405 0.10803606
## u_space_time 0.014461623 0.10700115
## u_space_time 0.016014461 0.10612277
## u_space_time 0.018117673 0.10678725
## u_space_time 0.015167920 0.10705585
## u_space_time 0.016059818 0.10679714
## u_space_time 0.016244120 0.10635654
## u_space_time 0.015447821 0.10642888
## u_space_time 0.015268805 0.10626650
## u_space_time 0.016508268 0.10632254
## u_space_time 0.014243034 0.10653584
## u_space_time 0.017073276 0.10616774
## u_space_time 0.015814754 0.10685275
## u_space_time 0.016361190 0.10617293
## u_space_time 0.017481231 0.10606787
## u_space_time 0.015766949 0.10633111
## u_space_time 0.016635926 0.10628643
## u_space_time 0.016167948 0.10695402
## u_space_time 0.016551209 0.10599944
## u_space_time 0.015278838 0.10701067
## u_space_time 0.016601627 0.10610150
## u_space_time 0.084247452 0.10311100
## u_space_time 0.082777709 0.10376658
## u_space_time 0.083024387 0.10354141
## u_space_time 0.082339052 0.10405729
## u_space_time 0.079139715 0.10472245
## u_space_time 0.082105868 0.10354777
## u_space_time 0.082267158 0.10418544
## u_space_time 0.082525475 0.10427080
## u_space_time 0.082074834 0.10427375
## u_space_time 0.083005431 0.10343284
## u_space_time 0.082333696 0.10371581
## u_space_time 0.080938178 0.10553947
## u_space_time 0.081285943 0.10428153
## u_space_time 0.082775672 0.10321275
## u_space_time 0.084611754 0.10404190
## u_space_time 0.081656596 0.10434048
## u_space_time 0.082486075 0.10402643
## u_space_time 0.082781622 0.10349669
## u_space_time 0.082155117 0.10357984
## u_space_time 0.081968744 0.10338331
## u_space_time 0.082928638 0.10344945
## u_space_time 0.080906416 0.10370254
## u_space_time 0.083487298 0.10326135
## u_space_time 0.081878466 0.10408048
## u_space_time 0.083065863 0.10327549
## u_space_time 0.084149096 0.10315014
## u_space_time 0.082351431 0.10345901
## u_space_time 0.083301792 0.10341624
## u_space_time 0.082673633 0.10423112
## u_space_time 0.083228179 0.10305899
## u_space_time 0.081771893 0.10429635
## u_space_time 0.083147353 0.10317794
## u_space_time 0.194414459 0.09895185
## u_space_time 0.192457074 0.09976023
## u_space_time 0.193189168 0.09951256
## u_space_time 0.192157721 0.10012957
## u_space_time 0.189071241 0.10091415
## u_space_time 0.192101541 0.09950040
## u_space_time 0.191538688 0.10024954
## u_space_time 0.192777967 0.10041692
## u_space_time 0.191558789 0.10037464
## u_space_time 0.193010192 0.09936791
## u_space_time 0.192234009 0.09970788
## u_space_time 0.190417515 0.10193147
## u_space_time 0.191329091 0.10040318
## u_space_time 0.192662088 0.09908344
## u_space_time 0.194346920 0.10012419
## u_space_time 0.191140401 0.10045015
## u_space_time 0.192451066 0.10007981
## u_space_time 0.192630682 0.09943308
## u_space_time 0.191967868 0.09952685
## u_space_time 0.191994835 0.09929520
## u_space_time 0.192679068 0.09936369
## u_space_time 0.190898695 0.09967577
## u_space_time 0.193281154 0.09913331
## u_space_time 0.191450880 0.10011874
## u_space_time 0.193059643 0.09916712
## u_space_time 0.193980883 0.09900758
## u_space_time 0.192005111 0.09937087
## u_space_time 0.193133070 0.09933760
## u_space_time 0.192576941 0.10034872
## u_space_time 0.193116844 0.09888831
## u_space_time 0.191616141 0.10042261
## u_space_time 0.192999601 0.09902715
## u_space_time -0.030591718 0.10649372
## u_space_time -0.032560918 0.10741732
## u_space_time -0.031662882 0.10710896
## u_space_time -0.032865326 0.10783480
## u_space_time -0.035944904 0.10886136
## u_space_time -0.032820397 0.10714428
## u_space_time -0.033353087 0.10800967
## u_space_time -0.031874555 0.10813237
## u_space_time -0.033420509 0.10813455
## u_space_time -0.031982647 0.10695521
## u_space_time -0.032774544 0.10736521
## u_space_time -0.034458183 0.10990307
## u_space_time -0.033462165 0.10819190
## u_space_time -0.032279329 0.10665979
## u_space_time -0.030885130 0.10773990
## u_space_time -0.033708172 0.10823898
## u_space_time -0.032496022 0.10779117
## u_space_time -0.032434310 0.10704969
## u_space_time -0.032824791 0.10718750
## u_space_time -0.032889009 0.10692257
## u_space_time -0.032436617 0.10698129
## u_space_time -0.033989789 0.10740117
## u_space_time -0.031827063 0.10670767
## u_space_time -0.033745368 0.10788304
## u_space_time -0.031988906 0.10673901
## u_space_time -0.031209437 0.10653070
## u_space_time -0.032900517 0.10700792
## u_space_time -0.031908052 0.10692039
## u_space_time -0.032372721 0.10806112
## u_space_time -0.031957761 0.10643627
## u_space_time -0.033397457 0.10818186
## u_space_time -0.032090588 0.10660264
## u_space_time 0.119266364 0.10006576
## u_space_time 0.117676950 0.10123010
## u_space_time 0.118804775 0.10088762
## u_space_time 0.117168444 0.10174954
## u_space_time 0.114449067 0.10294046
## u_space_time 0.117493546 0.10089184
## u_space_time 0.117057631 0.10194205
## u_space_time 0.118613284 0.10216230
## u_space_time 0.116921006 0.10211169
## u_space_time 0.118191727 0.10067212
## u_space_time 0.117295811 0.10116385
## u_space_time 0.115910386 0.10429272
## u_space_time 0.117050700 0.10219058
## u_space_time 0.118012854 0.10029260
## u_space_time 0.118934000 0.10166273
## u_space_time 0.116760792 0.10223378
## u_space_time 0.117732028 0.10167861
## u_space_time 0.117532475 0.10076645
## u_space_time 0.117577577 0.10094029
## u_space_time 0.117358155 0.10061005
## u_space_time 0.117471368 0.10066651
## u_space_time 0.116434423 0.10117992
## u_space_time 0.118134844 0.10033213
## u_space_time 0.116162643 0.10174341
## u_space_time 0.118035766 0.10038823
## u_space_time 0.118640232 0.10012871
## u_space_time 0.117459608 0.10071112
## u_space_time 0.118295587 0.10062348
## u_space_time 0.117579767 0.10203794
## u_space_time 0.118093978 0.09999919
## u_space_time 0.116694848 0.10217261
## u_space_time 0.117993080 0.10019665Hyper parameter comparison
summary(sdr8, "fixed")## Estimate Std. Error
## log_prec_space_time 8.096335 0.4451832summary(sdr8b, "fixed")## Estimate Std. Error
## log_prec_space_time 8.518428 0.3598224cbind("mean" = inlafit8C$misc$theta.mode,
"se" = sqrt(diag(inlafit8C$misc$cov.intern)))## mean se
## [1,] 5.997604 1.16031Fixed effects (Intercept)
summary(sdr8, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.01366742 0.02981683summary(sdr8b, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.01345717 0.0297343inlafit8C$summary.fixed[ , 1:2]Random effects mean and standard deviation
plot(summary(sdr8, "report")[,1], summary(sdr8b, "report")[,1],
xlab = "TMB soft constraint)", ylab = "TMB hard constraint",
main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(summary(sdr8, "report")[,2], summary(sdr8b, "report")[,2],
xlab = "TMB soft constraint", ylab = "TMB hard constraint",
main = "Random effect standard devation")
abline(0, 1, col = "red")
plot(inlafit8C$summary.random[[1]][ , 2], summary(sdr8b, "report")[,1],
xlab = "TMB soft constraint)", ylab = "TMB hard constraint",
main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(inlafit8C$summary.random[[1]][ , 3], summary(sdr8b, "report")[,2],
xlab = "TMB soft constraint", ylab = "TMB hard constraint",
main = "Random effect standard devation")
abline(0, 1, col = "red")
10 Model 9: ICAR space x RW1 time with full constraints
R_space <- diag(rowSums(adj.mat)) - adj.mat
R_space_scaled <- inla.scale.model(R_space, constr = list(A = matrix(1, ncol = ncol(R_space)), e = 0))
R_space_scaled_adj <- R_space_scaled + Matrix::Diagonal(ncol(R_space_scaled), diagval)
D_time <- diff(diag(length(levels(data$Yearf))), differences = 1)
R_time <- Matrix::Matrix(t(D_time) %*% D_time)
R_time_scaled <- inla.scale.model(R_time, constr = list(A = matrix(1, ncol = ncol(R_time)), e = 0))
R_time_scaled_adj <- R_time_scaled + Matrix::Diagonal(ncol(R_time_scaled), diagval)
R_space_time <- kronecker(R_time_scaled, R_space_scaled)
eig <- eigen(R_space_time)
eigval <- zapsmall(eig$values)
eigvec <- eig$vectors
rankdef <- nrow(R_space) + ncol(R_time) - 1
sum(eigval == 0)## [1] 50rankdef## [1] 50Aconstr <- t(eigvec[ , eigval == 0])
inlafit9C <- inla(Observed ~
f(id.area.year, model = "generic0", Cmatrix = R_space_time,
hyper = prec.prior, diagonal = diagval, rankdef = rankdef,
extraconstr = list(A = Aconstr, e = numeric(nrow(Aconstr)))),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep(".*rank.*", inlafit9C$logfile, value = TRUE)## [1] " rank-deficiency is *defined* [50]"summary(inlafit9C)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.03, Running = 0.49, Post = 0.0158, Total = 3.54
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.003 0.029 -0.06 -0.003 0.055 NA 0
##
## Random effects:
## Name Model
## id.area.year Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for id.area.year 505.78 700.19 48.34 298.97 2266.09 NA
##
## Marginal log-Likelihood: -1224.48
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')10.1 TMB
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(Aconstr);
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_time);
DATA_SPARSE_MATRIX(R_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_raw_space_time_v(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time * sigma_space_time);
val += SEPARABLE(GMRF(R_time), GMRF(R_space))(u_raw_space_time);
val -= dnorm(Aconstr * u_raw_space_time_v, Type(0), Type(0.001) * u_raw_space_time.cols(), true).sum(); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
R_space_time_adj <- R_space_time + Matrix::Diagonal(ncol(R_space_time), 1e-6)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Aconstr = Aconstr,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_space = R_space_scaled_adj,
R_time = R_time_scaled_adj)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = array(0, c(nrow(tmbdata$R_space), nrow(tmbdata$R_time))))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr9 <- TMB::sdreport(obj)
summary(sdr9, "all")## Estimate Std. Error
## log_prec_space_time 5.574152e+00 1.15582262
## beta0 -2.680206e-03 0.02933691
## u_raw_space_time -7.989248e-01 1.12667280
## u_raw_space_time -1.284828e-01 1.40810788
## u_raw_space_time -2.963552e-01 1.22621341
## u_raw_space_time 3.140558e-01 1.51628252
## u_raw_space_time 1.007087e+00 1.84148072
## u_raw_space_time 1.741425e-01 1.23505276
## u_raw_space_time -6.047064e-01 1.59035547
## u_raw_space_time -2.144584e-01 1.61059493
## u_raw_space_time 3.200018e-02 1.59955459
## u_raw_space_time -2.100527e-02 1.18385901
## u_raw_space_time 3.074912e-01 1.33420038
## u_raw_space_time 5.188217e-01 2.15639796
## u_raw_space_time 2.059059e-01 1.60574482
## u_raw_space_time -7.900866e-02 1.02435145
## u_raw_space_time -5.893576e-01 1.59127878
## u_raw_space_time 7.109861e-02 1.62942009
## u_raw_space_time -8.665258e-02 1.59889122
## u_raw_space_time 1.350900e-01 1.22515111
## u_raw_space_time -1.120157e-01 1.26287737
## u_raw_space_time 2.942985e-01 1.10782530
## u_raw_space_time -1.574810e-02 1.23579746
## u_raw_space_time 3.595275e-01 1.32014125
## u_raw_space_time -3.854470e-01 1.15746181
## u_raw_space_time 1.658342e-01 1.63025514
## u_raw_space_time 6.811454e-02 1.08419630
## u_raw_space_time -2.993495e-01 1.05502035
## u_raw_space_time -1.551094e-01 1.17570402
## u_raw_space_time -2.638146e-01 1.20950366
## u_raw_space_time 1.638048e-01 1.62557813
## u_raw_space_time -1.291233e-01 0.99608212
## u_raw_space_time 4.954295e-01 1.64880519
## u_raw_space_time -1.330951e-01 1.11634433
## u_raw_space_time -8.213341e-01 1.01032154
## u_raw_space_time -1.142814e-01 1.29270773
## u_raw_space_time -2.767173e-01 1.12131806
## u_raw_space_time 3.127926e-01 1.39069046
## u_raw_space_time 1.019981e+00 1.68905856
## u_raw_space_time 1.752976e-01 1.13377688
## u_raw_space_time -5.093354e-01 1.44038557
## u_raw_space_time -2.085321e-01 1.46999441
## u_raw_space_time 5.819039e-02 1.46162054
## u_raw_space_time -2.343795e-02 1.08837435
## u_raw_space_time 3.010534e-01 1.22505181
## u_raw_space_time 5.352384e-01 1.97663716
## u_raw_space_time 2.086132e-01 1.46358456
## u_raw_space_time -6.625044e-02 0.94043703
## u_raw_space_time -6.015238e-01 1.45993066
## u_raw_space_time 1.216866e-01 1.48747011
## u_raw_space_time -1.334279e-01 1.46061029
## u_raw_space_time 1.158257e-01 1.12265259
## u_raw_space_time -8.988367e-02 1.15356925
## u_raw_space_time 3.009642e-01 1.01648691
## u_raw_space_time -5.014637e-02 1.12881655
## u_raw_space_time 3.790879e-01 1.20896421
## u_raw_space_time -4.090634e-01 1.05737013
## u_raw_space_time 1.027569e-01 1.48657560
## u_raw_space_time 3.730688e-02 0.99253078
## u_raw_space_time -3.293414e-01 0.96273284
## u_raw_space_time -1.144912e-01 1.07418816
## u_raw_space_time -2.517582e-01 1.11144541
## u_raw_space_time 1.403880e-01 1.48870091
## u_raw_space_time -1.482295e-01 0.91421409
## u_raw_space_time 5.032227e-01 1.51592723
## u_raw_space_time -1.645010e-01 1.02045476
## u_raw_space_time -8.594779e-01 0.91501272
## u_raw_space_time -9.398795e-02 1.18349493
## u_raw_space_time -2.388925e-01 1.02336635
## u_raw_space_time 3.063995e-01 1.27343670
## u_raw_space_time 1.046207e+00 1.55415877
## u_raw_space_time 1.901790e-01 1.03830221
## u_raw_space_time -3.982844e-01 1.30479362
## u_raw_space_time -1.700549e-01 1.34064923
## u_raw_space_time 1.113091e-01 1.33477434
## u_raw_space_time -2.265459e-02 0.99726640
## u_raw_space_time 2.924959e-01 1.12232604
## u_raw_space_time 5.789768e-01 1.81198738
## u_raw_space_time 2.239985e-01 1.33552440
## u_raw_space_time -4.250638e-02 0.86061146
## u_raw_space_time -6.463956e-01 1.34183126
## u_raw_space_time 1.727847e-01 1.35822759
## u_raw_space_time -1.829717e-01 1.33371365
## u_raw_space_time 8.280076e-02 1.02643338
## u_raw_space_time -4.653653e-02 1.05245164
## u_raw_space_time 3.112230e-01 0.93136312
## u_raw_space_time -1.174232e-01 1.02995098
## u_raw_space_time 4.116138e-01 1.10732979
## u_raw_space_time -4.530174e-01 0.96789667
## u_raw_space_time 3.833510e-02 1.35448128
## u_raw_space_time -1.211668e-02 0.90716575
## u_raw_space_time -4.036125e-01 0.88097495
## u_raw_space_time -5.444768e-02 0.97965605
## u_raw_space_time -2.450408e-01 1.01878672
## u_raw_space_time 1.179663e-01 1.36143075
## u_raw_space_time -1.868965e-01 0.83691325
## u_raw_space_time 5.209133e-01 1.39223947
## u_raw_space_time -2.309266e-01 0.93109695
## u_raw_space_time -8.649576e-01 0.83981916
## u_raw_space_time -5.862909e-02 1.08140933
## u_raw_space_time -2.130820e-01 0.93614931
## u_raw_space_time 2.720885e-01 1.16415573
## u_raw_space_time 1.032264e+00 1.43160123
## u_raw_space_time 1.979654e-01 0.94953669
## u_raw_space_time -2.644620e-01 1.18393734
## u_raw_space_time -1.349693e-01 1.22546781
## u_raw_space_time 1.813479e-01 1.22078337
## u_raw_space_time -1.800598e-02 0.91151590
## u_raw_space_time 2.704596e-01 1.02633301
## u_raw_space_time 6.139369e-01 1.66215603
## u_raw_space_time 2.196259e-01 1.22204162
## u_raw_space_time -3.355644e-02 0.78665200
## u_raw_space_time -6.612653e-01 1.23308805
## u_raw_space_time 2.385609e-01 1.24290437
## u_raw_space_time -2.330333e-01 1.21928966
## u_raw_space_time 4.867151e-02 0.93781589
## u_raw_space_time -1.659387e-02 0.96169557
## u_raw_space_time 3.137393e-01 0.85292612
## u_raw_space_time -1.756721e-01 0.94159419
## u_raw_space_time 4.363040e-01 1.01565704
## u_raw_space_time -4.749288e-01 0.88821282
## u_raw_space_time -4.332926e-02 1.23448435
## u_raw_space_time -3.735670e-02 0.82922781
## u_raw_space_time -4.653069e-01 0.81067357
## u_raw_space_time 1.142572e-02 0.89395880
## u_raw_space_time -2.225655e-01 0.93121881
## u_raw_space_time 8.162304e-02 1.24436275
## u_raw_space_time -2.102325e-01 0.76542214
## u_raw_space_time 4.869390e-01 1.27387844
## u_raw_space_time -2.772823e-01 0.85113878
## u_raw_space_time -8.370946e-01 0.78243366
## u_raw_space_time -2.805990e-02 0.98846795
## u_raw_space_time -1.531409e-01 0.85473005
## u_raw_space_time 2.013171e-01 1.06396540
## u_raw_space_time 9.506737e-01 1.31688427
## u_raw_space_time 1.982334e-01 0.86871479
## u_raw_space_time -9.189185e-02 1.07978768
## u_raw_space_time -5.622588e-02 1.12219348
## u_raw_space_time 2.266513e-01 1.12073632
## u_raw_space_time -2.646395e-02 0.83297930
## u_raw_space_time 2.142061e-01 0.93728340
## u_raw_space_time 6.232390e-01 1.52739611
## u_raw_space_time 2.602492e-01 1.12398627
## u_raw_space_time -1.894076e-02 0.71884591
## u_raw_space_time -6.518276e-01 1.13480670
## u_raw_space_time 2.849354e-01 1.14223982
## u_raw_space_time -2.718502e-01 1.11881510
## u_raw_space_time -2.229187e-03 0.85771699
## u_raw_space_time 4.372130e-02 0.88060911
## u_raw_space_time 2.820880e-01 0.78064009
## u_raw_space_time -2.235303e-01 0.86454114
## u_raw_space_time 4.481951e-01 0.93466304
## u_raw_space_time -4.704298e-01 0.81796461
## u_raw_space_time -1.004882e-01 1.12998391
## u_raw_space_time -8.676585e-02 0.76000695
## u_raw_space_time -5.179494e-01 0.75290469
## u_raw_space_time 6.845607e-02 0.81867007
## u_raw_space_time -1.920765e-01 0.85054867
## u_raw_space_time 3.746799e-02 1.13918618
## u_raw_space_time -2.282873e-01 0.70075926
## u_raw_space_time 4.168092e-01 1.16334094
## u_raw_space_time -2.990839e-01 0.78077544
## u_raw_space_time -7.073573e-01 0.73650575
## u_raw_space_time -7.207298e-04 0.90661113
## u_raw_space_time -1.108487e-01 0.78628713
## u_raw_space_time 1.375290e-01 0.97706612
## u_raw_space_time 7.487453e-01 1.20427999
## u_raw_space_time 1.691468e-01 0.79696545
## u_raw_space_time 3.632376e-02 0.99960821
## u_raw_space_time -3.354553e-02 1.03786675
## u_raw_space_time 2.347694e-01 1.03487603
## u_raw_space_time -4.522414e-02 0.76349600
## u_raw_space_time 1.463547e-01 0.85866094
## u_raw_space_time 5.692107e-01 1.40588504
## u_raw_space_time 2.630576e-01 1.04191479
## u_raw_space_time -1.286090e-02 0.65913061
## u_raw_space_time -5.556767e-01 1.04060245
## u_raw_space_time 2.943813e-01 1.05614930
## u_raw_space_time -2.220381e-01 1.03161702
## u_raw_space_time -2.893565e-02 0.78844359
## u_raw_space_time 7.322879e-02 0.81233423
## u_raw_space_time 2.071392e-01 0.71556313
## u_raw_space_time -2.114257e-01 0.79777796
## u_raw_space_time 3.946330e-01 0.86101566
## u_raw_space_time -3.853450e-01 0.75324032
## u_raw_space_time -1.062027e-01 1.04210300
## u_raw_space_time -1.420227e-01 0.70158839
## u_raw_space_time -4.952046e-01 0.69951780
## u_raw_space_time 1.018838e-01 0.75463853
## u_raw_space_time -1.482684e-01 0.77837578
## u_raw_space_time 1.527274e-02 1.04816700
## u_raw_space_time -2.160180e-01 0.64329336
## u_raw_space_time 3.143511e-01 1.06374548
## u_raw_space_time -2.843759e-01 0.71993186
## u_raw_space_time -5.343087e-01 0.70334889
## u_raw_space_time 2.375798e-02 0.83837612
## u_raw_space_time -7.036608e-02 0.73051755
## u_raw_space_time 8.626113e-02 0.90569629
## u_raw_space_time 5.045981e-01 1.11022577
## u_raw_space_time 1.380709e-01 0.73703924
## u_raw_space_time 1.047687e-01 0.94052955
## u_raw_space_time -2.947663e-02 0.97089644
## u_raw_space_time 2.198118e-01 0.96482211
## u_raw_space_time -5.397333e-02 0.70471544
## u_raw_space_time 8.798129e-02 0.79374449
## u_raw_space_time 4.543024e-01 1.30097200
## u_raw_space_time 1.828887e-01 0.97486726
## u_raw_space_time -6.460594e-03 0.60909635
## u_raw_space_time -4.277716e-01 0.96008951
## u_raw_space_time 2.533742e-01 0.98485840
## u_raw_space_time -1.212275e-01 0.96066135
## u_raw_space_time -4.623267e-02 0.73137194
## u_raw_space_time 8.284873e-02 0.75740302
## u_raw_space_time 1.276537e-01 0.66200638
## u_raw_space_time -1.682195e-01 0.74275446
## u_raw_space_time 3.239189e-01 0.79961575
## u_raw_space_time -2.750321e-01 0.70017600
## u_raw_space_time -5.689921e-02 0.97197308
## u_raw_space_time -1.750325e-01 0.65416960
## u_raw_space_time -4.229191e-01 0.65194464
## u_raw_space_time 1.114824e-01 0.70256563
## u_raw_space_time -1.026123e-01 0.71783686
## u_raw_space_time -1.666161e-02 0.97331156
## u_raw_space_time -1.790506e-01 0.59452150
## u_raw_space_time 2.231798e-01 0.98151837
## u_raw_space_time -2.388729e-01 0.66947783
## u_raw_space_time -3.249439e-01 0.68274796
## u_raw_space_time 3.168843e-02 0.78654101
## u_raw_space_time -8.439700e-02 0.69226404
## u_raw_space_time 5.412679e-02 0.85227100
## u_raw_space_time 2.711791e-01 1.04380057
## u_raw_space_time 8.670081e-02 0.69117489
## u_raw_space_time 1.178268e-01 0.89951543
## u_raw_space_time -7.716462e-02 0.92376170
## u_raw_space_time 1.633882e-01 0.91175220
## u_raw_space_time -5.585298e-02 0.65942969
## u_raw_space_time 5.277123e-02 0.74502241
## u_raw_space_time 2.971436e-01 1.21979870
## u_raw_space_time 8.429975e-02 0.92618927
## u_raw_space_time -2.054235e-02 0.57142258
## u_raw_space_time -2.415024e-01 0.89644850
## u_raw_space_time 1.719313e-01 0.93111258
## u_raw_space_time -4.449645e-02 0.90888748
## u_raw_space_time -2.656824e-02 0.68829868
## u_raw_space_time 3.463512e-02 0.71698402
## u_raw_space_time 5.923516e-02 0.62222338
## u_raw_space_time -7.263390e-02 0.70147394
## u_raw_space_time 2.208618e-01 0.75169738
## u_raw_space_time -1.269411e-01 0.66160858
## u_raw_space_time 1.011857e-02 0.92146058
## u_raw_space_time -1.588830e-01 0.61613949
## u_raw_space_time -3.045393e-01 0.61233607
## u_raw_space_time 8.397777e-02 0.66333039
## u_raw_space_time -5.720577e-02 0.67180025
## u_raw_space_time -4.099237e-03 0.91724202
## u_raw_space_time -1.215743e-01 0.55683198
## u_raw_space_time 1.500597e-01 0.91981012
## u_raw_space_time -1.686138e-01 0.63122478
## u_raw_space_time -1.684860e-01 0.67264833
## u_raw_space_time 3.763355e-02 0.75404959
## u_raw_space_time -5.106446e-02 0.66618810
## u_raw_space_time 7.319006e-03 0.81898701
## u_raw_space_time 1.081335e-01 1.00614296
## u_raw_space_time 4.030952e-02 0.66247306
## u_raw_space_time 1.572697e-01 0.87764362
## u_raw_space_time -4.179164e-02 0.89231792
## u_raw_space_time 1.039554e-01 0.87923242
## u_raw_space_time -5.894147e-02 0.63075085
## u_raw_space_time 2.471904e-03 0.71455994
## u_raw_space_time 1.574423e-01 1.17054800
## u_raw_space_time 2.957248e-02 0.89683452
## u_raw_space_time -2.181812e-02 0.54747896
## u_raw_space_time -9.568844e-02 0.86017860
## u_raw_space_time 1.136935e-01 0.89915946
## u_raw_space_time 3.881655e-02 0.87799932
## u_raw_space_time -2.979909e-02 0.66161016
## u_raw_space_time 2.591808e-02 0.69228871
## u_raw_space_time -6.714077e-03 0.59786152
## u_raw_space_time -1.559710e-02 0.67694458
## u_raw_space_time 1.370420e-01 0.72254407
## u_raw_space_time -3.083732e-02 0.63967351
## u_raw_space_time 5.174565e-02 0.89105329
## u_raw_space_time -1.584494e-01 0.59365250
## u_raw_space_time -2.134245e-01 0.58981321
## u_raw_space_time 7.542434e-02 0.63962361
## u_raw_space_time -2.832804e-02 0.64316234
## u_raw_space_time -2.231158e-02 0.88259099
## u_raw_space_time -8.627366e-02 0.53368984
## u_raw_space_time 5.343373e-02 0.88063191
## u_raw_space_time -1.110173e-01 0.60814065
## u_raw_space_time -1.102100e-02 0.67095609
## u_raw_space_time 2.499375e-02 0.74296902
## u_raw_space_time -2.510943e-02 0.65745342
## u_raw_space_time -1.770195e-02 0.80792229
## u_raw_space_time 1.310060e-02 0.99400028
## u_raw_space_time -1.392147e-03 0.65278675
## u_raw_space_time 1.435729e-01 0.87164240
## u_raw_space_time -2.862341e-02 0.88222055
## u_raw_space_time 2.231791e-02 0.86853084
## u_raw_space_time -4.916884e-02 0.62046607
## u_raw_space_time -2.708423e-02 0.70445036
## u_raw_space_time -1.752047e-02 1.15471356
## u_raw_space_time -4.171386e-02 0.88733263
## u_raw_space_time -2.823643e-02 0.53937589
## u_raw_space_time 4.830627e-02 0.84962845
## u_raw_space_time 2.199622e-02 0.88882085
## u_raw_space_time 6.488893e-02 0.86719734
## u_raw_space_time -1.338169e-02 0.65268030
## u_raw_space_time 2.661331e-03 0.68446049
## u_raw_space_time -4.414185e-02 0.58959935
## u_raw_space_time 5.043194e-02 0.66959376
## u_raw_space_time 7.250919e-02 0.71289608
## u_raw_space_time 5.895272e-02 0.63327864
## u_raw_space_time 8.921620e-02 0.88203385
## u_raw_space_time -1.129750e-01 0.58353068
## u_raw_space_time -9.828643e-02 0.58026920
## u_raw_space_time 3.994533e-02 0.63166135
## u_raw_space_time -3.588769e-03 0.63347778
## u_raw_space_time -1.815218e-02 0.87112982
## u_raw_space_time -4.142069e-02 0.52570997
## u_raw_space_time -2.338337e-02 0.86775780
## u_raw_space_time -5.001767e-02 0.60065152
## u_raw_space_time 2.030833e-01 0.67762768
## u_raw_space_time 2.432368e-02 0.75438967
## u_raw_space_time -3.964235e-02 0.66569135
## u_raw_space_time -4.101659e-02 0.81961157
## u_raw_space_time -2.657874e-01 1.01077628
## u_raw_space_time -7.643377e-02 0.66315281
## u_raw_space_time 1.201762e-01 0.88201061
## u_raw_space_time -1.061054e-01 0.89353916
## u_raw_space_time -6.149730e-02 0.88072459
## u_raw_space_time -4.535543e-02 0.63003798
## u_raw_space_time -5.840446e-02 0.71531517
## u_raw_space_time -1.919864e-01 1.17493099
## u_raw_space_time -1.416239e-01 0.89871556
## u_raw_space_time -4.283835e-02 0.54772702
## u_raw_space_time 2.489308e-01 0.86878714
## u_raw_space_time -8.092840e-02 0.90160676
## u_raw_space_time 1.662981e-01 0.88066472
## u_raw_space_time 2.601555e-02 0.66228334
## u_raw_space_time -4.905632e-02 0.69385234
## u_raw_space_time -1.186648e-01 0.59947398
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## u_space_time 4.002757e-02 0.07813450
## u_space_time -1.382289e-02 0.07095725
## u_space_time 5.657484e-03 0.06881589
## u_space_time 5.398616e-04 0.05280298
## u_space_time -2.626900e-03 0.05415740
## u_space_time -1.496555e-02 0.04928433
## u_space_time 9.912736e-03 0.05350337
## u_space_time -2.844590e-02 0.06216179
## u_space_time 2.448685e-02 0.05320963
## u_space_time 1.343990e-03 0.06956704
## u_space_time 7.738664e-03 0.04742108
## u_space_time 3.219277e-02 0.05335442
## u_space_time -2.110282e-03 0.05038824
## u_space_time 1.248858e-02 0.05369956
## u_space_time -1.815576e-03 0.07015504
## u_space_time 1.289021e-02 0.04408016
## u_space_time -2.195095e-02 0.07385968
## u_space_time 1.670327e-02 0.04928639
## u_space_time 5.036240e-02 0.05884342
## u_space_time 1.903787e-03 0.06656761
## u_space_time 1.567954e-02 0.06055290
## u_space_time -1.404357e-02 0.07254266
## u_space_time -5.599200e-02 0.09878017
## u_space_time -1.033952e-02 0.05879059
## u_space_time 7.481898e-03 0.07223331
## u_space_time 1.047902e-02 0.07646206
## u_space_time -1.042757e-02 0.07524340
## u_space_time 5.295905e-03 0.05662212
## u_space_time -1.321724e-02 0.06399973
## u_space_time -3.637621e-02 0.10671001
## u_space_time -1.194364e-02 0.07533106
## u_space_time 4.568100e-03 0.04883531
## u_space_time 3.800759e-02 0.08208536
## u_space_time -1.600228e-02 0.07719460
## u_space_time 3.184494e-03 0.07497653
## u_space_time -1.696158e-03 0.05769140
## u_space_time -6.545947e-05 0.05898344
## u_space_time -1.413048e-02 0.05324275
## u_space_time 6.475105e-03 0.05782727
## u_space_time -2.761856e-02 0.06626623
## u_space_time 2.172177e-02 0.05599261
## u_space_time -7.842812e-03 0.07685079
## u_space_time 9.356440e-03 0.05186359
## u_space_time 3.409572e-02 0.05717802
## u_space_time -2.476663e-03 0.05484229
## u_space_time 1.344825e-02 0.05866091
## u_space_time -4.786957e-03 0.07664908
## u_space_time 1.447458e-02 0.04820785
## u_space_time -2.607000e-02 0.08136567
## u_space_time 1.650170e-02 0.05318665
## u_space_time 5.916904e-02 0.06624704
## u_space_time -1.419694e-03 0.07272954
## u_space_time 2.194352e-02 0.06819388
## u_space_time -1.574646e-02 0.07934841
## u_space_time -5.412022e-02 0.10419891
## u_space_time -8.081989e-03 0.06396031
## u_space_time -1.783819e-03 0.07753871
## u_space_time 1.872602e-02 0.08546522
## u_space_time -1.741346e-02 0.08297000
## u_space_time 7.774887e-03 0.06227140
## u_space_time -1.332281e-02 0.06972070
## u_space_time -4.397003e-02 0.11811667
## u_space_time -7.666140e-03 0.08191229
## u_space_time 5.201582e-03 0.05340629
## u_space_time 3.618682e-02 0.08725728
## u_space_time -2.271045e-02 0.08532423
## u_space_time 6.375733e-03 0.08170744
## u_space_time -2.231651e-03 0.06304363
## u_space_time -1.265269e-04 0.06432573
## u_space_time -1.139424e-02 0.05749364
## u_space_time 4.915099e-03 0.06306234
## u_space_time -2.503292e-02 0.07061209
## u_space_time 2.162632e-02 0.06051235
## u_space_time -1.194545e-02 0.08451546
## u_space_time 1.208668e-02 0.05707727
## u_space_time 3.459516e-02 0.06087659
## u_space_time -5.975648e-03 0.05994346
## u_space_time 1.316299e-02 0.06375034
## u_space_time -4.392983e-03 0.08364498
## u_space_time 1.562003e-02 0.05263808
## u_space_time -2.757106e-02 0.08864543
## u_space_time 1.754330e-02 0.05795340
## u_space_time 5.751562e-02 0.06888023
## u_space_time -2.150989e-03 0.07940493
## u_space_time 2.466767e-02 0.07469919
## u_space_time -1.675649e-02 0.08658645
## u_space_time -5.405738e-02 0.11180978
## u_space_time -6.875919e-03 0.06973236
## u_space_time 6.413218e-04 0.08489102
## u_space_time 2.532096e-02 0.09513600
## u_space_time -1.704671e-02 0.09035125
## u_space_time 7.446359e-03 0.06773333
## u_space_time -1.404092e-02 0.07603538
## u_space_time -4.162228e-02 0.12668538
## u_space_time -3.590171e-03 0.08971314
## u_space_time 5.995607e-03 0.05832320
## u_space_time 3.094738e-02 0.09228438
## u_space_time -1.973386e-02 0.09212643
## u_space_time 6.752556e-03 0.08928562
## u_space_time -4.118416e-03 0.06898220
## u_space_time 3.823653e-03 0.07067504
## u_space_time -9.477079e-03 0.06250690
## u_space_time 2.043266e-03 0.06901098
## u_space_time -2.292748e-02 0.07603104
## u_space_time 1.874996e-02 0.06520417
## u_space_time -1.615385e-02 0.09308264
## u_space_time 1.065354e-02 0.06176453
## u_space_time 3.016997e-02 0.06304088
## u_space_time -4.291601e-03 0.06538815
## u_space_time 1.190187e-02 0.06904578
## u_space_time -4.042628e-03 0.09131291
## u_space_time 1.348650e-02 0.05675886
## u_space_time -2.834499e-02 0.09626758
## u_space_time 1.510746e-02 0.06267734
## u_space_time 5.300235e-02 0.07244692
## u_space_time -1.303782e-03 0.08645630
## u_space_time 2.989628e-02 0.08308904
## u_space_time -2.018198e-02 0.09491395
## u_space_time -5.025204e-02 0.11897885
## u_space_time -4.856048e-03 0.07591412
## u_space_time 5.396831e-03 0.09376639
## u_space_time 3.103115e-02 0.10548314
## u_space_time -1.434178e-02 0.09826574
## u_space_time 6.929598e-03 0.07346600
## u_space_time -1.676956e-02 0.08317786
## u_space_time -3.867924e-02 0.13630599
## u_space_time 2.568605e-03 0.09890033
## u_space_time 7.843577e-03 0.06372776
## u_space_time 2.430126e-02 0.09842622
## u_space_time -1.451626e-02 0.09982023
## u_space_time 8.138836e-03 0.09763337
## u_space_time -8.483612e-03 0.07576829
## u_space_time 7.969247e-03 0.07789457
## u_space_time -8.735144e-03 0.06807214
## u_space_time -2.972962e-03 0.07596177
## u_space_time -1.870698e-02 0.08185468
## u_space_time 1.511540e-02 0.07079883
## u_space_time -2.076539e-02 0.10267890
## u_space_time 7.580744e-03 0.06675050
## u_space_time 2.433233e-02 0.06623922
## u_space_time -1.079847e-03 0.07148389
## u_space_time 1.220971e-02 0.07503116
## u_space_time -8.728292e-03 0.10002587
## u_space_time 1.118320e-02 0.06138782
## u_space_time -3.084259e-02 0.10496312
## u_space_time 1.372844e-02 0.06823996Hyper parameter comparison
summary(sdr9, "fixed")## Estimate Std. Error
## log_prec_space_time 5.574152 1.155823cbind("mean" = inlafit9C$misc$theta.mode,
"se" = sqrt(diag(inlafit9C$misc$cov.intern)))## mean se
## [1,] 5.578561 1.157753Fixed effects (Intercept)
summary(sdr9, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.002680206 0.02933691inlafit9C$summary.fixed[ , 1:2]Random effects mean and standard deviation
plot(inlafit9C$summary.random[[1]][,2], summary(sdr9, "report")[,1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(inlafit9C$summary.random[[1]][,3], summary(sdr9, "report")[,2],
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation")
abline(0, 1, col = "red")
10.2 Model 9b: Explicit sum-to-zero constraint
This one gets slow!
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_MATRIX(L_space_time)
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(LRL_space_time);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_VECTOR(u_raw_space_time);
vector<Type> u_space_time(L_space_time * u_raw_space_time * sigma_space_time);
val += GMRF(LRL_space_time)(u_raw_space_time);
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
qrc <- qr(t(Aconstr))
L_space_time <- qr.Q(qrc, complete=TRUE)[ , (nrow(Aconstr)+1):ncol(Aconstr)]
LRL <- as(t(L_space_time) %*% R_space_time %*% L_space_time, "dgCMatrix")
LRL_adj <- LRL + Matrix::Diagonal(ncol(LRL), diagval)
tmbdata <- list(y = data$Observed,
E = data$Expected,
L_space_time = L_space_time,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
LRL_space_time = LRL_adj)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
u_raw_space_time = numeric(ncol(tmbdata$L_space_time)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr9b <- TMB::sdreport(obj)
summary(sdr9b, "all")## Estimate Std. Error
## log_prec_space_time 5.572942e+00 1.15592828
## beta0 -2.679220e-03 0.02933672
## u_raw_space_time -3.577492e-02 1.27749115
## u_raw_space_time 2.470870e-01 1.02202678
## u_raw_space_time -1.502620e-01 1.12606194
## u_raw_space_time 2.559488e-01 1.34799904
## u_raw_space_time -4.927598e-01 1.05082459
## u_raw_space_time 1.170255e-01 1.56641214
## u_raw_space_time 6.551324e-03 1.03973898
## u_raw_space_time -2.475447e-01 0.93724336
## u_raw_space_time -1.163255e-01 1.05572958
## u_raw_space_time -1.801598e-01 1.10189275
## u_raw_space_time 5.212785e-02 1.35555608
## u_raw_space_time -7.827538e-02 0.98749415
## u_raw_space_time 5.268476e-01 1.53767806
## u_raw_space_time -2.529911e-01 1.03759416
## u_raw_space_time -6.969417e-01 0.88472258
## u_raw_space_time -1.092774e-01 1.14316860
## u_raw_space_time -3.810248e-01 1.09490285
## u_raw_space_time 3.317466e-01 1.26159426
## u_raw_space_time 9.607264e-01 1.54668385
## u_raw_space_time 2.203972e-01 1.08023453
## u_raw_space_time -2.486459e-01 1.26074127
## u_raw_space_time -1.721961e-01 1.34210412
## u_raw_space_time 1.844905e-01 1.37538303
## u_raw_space_time -5.130454e-02 0.99213795
## u_raw_space_time 2.863846e-01 0.91189864
## u_raw_space_time 5.182396e-01 1.83679630
## u_raw_space_time 1.871840e-01 1.29756814
## u_raw_space_time -6.475683e-02 0.84829090
## u_raw_space_time -5.692713e-01 1.37687378
## u_raw_space_time 1.987702e-01 1.25080786
## u_raw_space_time -2.705554e-01 1.43161219
## u_raw_space_time 1.231516e-01 0.98611481
## u_raw_space_time 8.538261e-02 1.21842455
## u_raw_space_time 3.352006e-01 0.92096629
## u_raw_space_time -1.398530e-01 1.07492483
## u_raw_space_time 3.663496e-01 1.20127992
## u_raw_space_time -4.590269e-01 0.98537264
## u_raw_space_time 1.302654e-01 1.45224723
## u_raw_space_time 3.488747e-02 0.92046076
## u_raw_space_time -2.441253e-01 0.85562175
## u_raw_space_time 2.153281e-02 1.00902753
## u_raw_space_time -9.568129e-02 1.01197495
## u_raw_space_time 1.074776e-01 1.32068168
## u_raw_space_time -3.921761e-02 0.81970988
## u_raw_space_time 6.224036e-01 1.48226342
## u_raw_space_time -2.417395e-01 0.89557826
## u_raw_space_time -7.754532e-01 0.96459972
## u_raw_space_time -1.468808e-01 1.13470931
## u_raw_space_time -4.280915e-01 0.99767621
## u_raw_space_time 2.245174e-01 1.14130260
## u_raw_space_time 8.739919e-01 1.43834724
## u_raw_space_time 1.552919e-01 0.97324261
## u_raw_space_time -1.876879e-01 1.10270862
## u_raw_space_time -2.100098e-01 1.15181347
## u_raw_space_time 1.816451e-01 1.31949956
## u_raw_space_time -1.195749e-01 0.99679579
## u_raw_space_time 1.914447e-01 0.88028975
## u_raw_space_time 4.803315e-01 1.66950836
## u_raw_space_time 1.099118e-01 1.12797066
## u_raw_space_time -1.287515e-01 0.88065635
## u_raw_space_time -6.572065e-01 1.25094057
## u_raw_space_time 1.916569e-01 1.16220471
## u_raw_space_time -3.937141e-01 1.34807373
## u_raw_space_time 1.604654e-02 0.97450037
## u_raw_space_time 4.239097e-02 1.02869293
## u_raw_space_time 2.648578e-01 0.86035579
## u_raw_space_time -2.711679e-01 1.03157152
## u_raw_space_time 3.182020e-01 1.06125037
## u_raw_space_time -5.539999e-01 0.94065000
## u_raw_space_time -2.451768e-02 1.37449241
## u_raw_space_time -6.330777e-02 0.90188488
## u_raw_space_time -3.788731e-01 0.88573362
## u_raw_space_time 1.448961e-02 0.90902819
## u_raw_space_time -1.461784e-01 0.97002031
## u_raw_space_time -1.838795e-03 1.21548728
## u_raw_space_time -1.355669e-01 0.83242248
## u_raw_space_time 5.155517e-01 1.27340142
## u_raw_space_time -3.611644e-01 0.93358562
## u_raw_space_time -6.238846e-01 0.81091438
## u_raw_space_time 7.466272e-03 1.03684552
## u_raw_space_time -2.442266e-01 0.88579553
## u_raw_space_time 2.775533e-01 1.11577158
## u_raw_space_time 9.163621e-01 1.37678270
## u_raw_space_time 2.794320e-01 0.93664505
## u_raw_space_time 1.088078e-01 1.03001240
## u_raw_space_time -7.359808e-03 1.14456559
## u_raw_space_time 3.508137e-01 1.13081449
## u_raw_space_time -4.205288e-03 0.88966156
## u_raw_space_time 2.590194e-01 0.87013387
## u_raw_space_time 6.135285e-01 1.49673198
## u_raw_space_time 2.744475e-01 1.12348859
## u_raw_space_time 9.678238e-03 0.77219977
## u_raw_space_time -5.241135e-01 1.23983470
## u_raw_space_time 3.618973e-01 1.09779324
## u_raw_space_time -3.089151e-01 1.18638211
## u_raw_space_time 8.889299e-02 0.92054048
## u_raw_space_time 2.265551e-01 0.98261772
## u_raw_space_time 3.570958e-01 0.80144357
## u_raw_space_time -1.953722e-01 0.91675244
## u_raw_space_time 4.540303e-01 1.03481550
## u_raw_space_time -4.258360e-01 0.87591969
## u_raw_space_time 4.192454e-02 1.19518514
## u_raw_space_time 1.105585e-02 0.82701673
## u_raw_space_time -3.078430e-01 0.81293342
## u_raw_space_time 1.953540e-01 0.85065294
## u_raw_space_time 8.084056e-03 0.87887942
## u_raw_space_time 7.776103e-02 1.13180302
## u_raw_space_time -2.990062e-02 0.78249892
## u_raw_space_time 5.692847e-01 1.24406245
## u_raw_space_time -2.593002e-01 0.80503692
## u_raw_space_time -4.729791e-01 0.80410875
## u_raw_space_time 5.598813e-02 1.05855120
## u_raw_space_time -1.806023e-01 0.83940419
## u_raw_space_time 2.350008e-01 1.08141354
## u_raw_space_time 7.357534e-01 1.25260172
## u_raw_space_time 2.716166e-01 0.81418976
## u_raw_space_time 2.583324e-01 0.96795345
## u_raw_space_time 3.660207e-02 1.06621273
## u_raw_space_time 3.801936e-01 1.08728142
## u_raw_space_time -1.729941e-03 0.83338668
## u_raw_space_time 2.124115e-01 0.86245669
## u_raw_space_time 5.807847e-01 1.39272125
## u_raw_space_time 2.985603e-01 1.07042313
## u_raw_space_time 3.697647e-02 0.72115317
## u_raw_space_time -4.068568e-01 1.15520931
## u_raw_space_time 3.926052e-01 1.05887257
## u_raw_space_time -2.380350e-01 1.16491489
## u_raw_space_time 8.336121e-02 0.87206242
## u_raw_space_time 2.773002e-01 0.85552664
## u_raw_space_time 3.034340e-01 0.78068274
## u_raw_space_time -1.621752e-01 0.88027460
## u_raw_space_time 4.217872e-01 0.91346579
## u_raw_space_time -3.196387e-01 0.85561444
## u_raw_space_time 5.723878e-02 1.21035098
## u_raw_space_time -2.301843e-02 0.76587310
## u_raw_space_time -2.639743e-01 0.79390140
## u_raw_space_time 2.500113e-01 0.79061336
## u_raw_space_time 7.307953e-02 0.90184772
## u_raw_space_time 7.674917e-02 1.07642079
## u_raw_space_time 3.511115e-03 0.73884613
## u_raw_space_time 4.880976e-01 1.14829841
## u_raw_space_time -2.235057e-01 0.75861426
## u_raw_space_time -3.419436e-01 0.75279218
## u_raw_space_time 3.846122e-02 0.96854416
## u_raw_space_time -1.819569e-01 0.84524620
## u_raw_space_time 1.417960e-01 1.02830754
## u_raw_space_time 4.497493e-01 1.16299702
## u_raw_space_time 1.986392e-01 0.79791425
## u_raw_space_time 2.848720e-01 0.96506879
## u_raw_space_time -1.232551e-03 1.04994482
## u_raw_space_time 3.233134e-01 1.00056610
## u_raw_space_time -5.241620e-02 0.78627442
## u_raw_space_time 1.121104e-01 0.82534115
## u_raw_space_time 4.239626e-01 1.32482088
## u_raw_space_time 1.764709e-01 1.05607153
## u_raw_space_time 1.414479e-03 0.69080179
## u_raw_space_time -3.210419e-01 1.10618864
## u_raw_space_time 3.096588e-01 1.02366202
## u_raw_space_time -1.793368e-01 1.12318952
## u_raw_space_time 2.405143e-02 0.81897545
## u_raw_space_time 2.449673e-01 0.87170311
## u_raw_space_time 1.820655e-01 0.70706148
## u_raw_space_time -1.610658e-01 0.84467673
## u_raw_space_time 3.092174e-01 0.86612013
## u_raw_space_time -2.514098e-01 0.79731826
## u_raw_space_time 6.439231e-02 1.10672225
## u_raw_space_time -9.801683e-02 0.71287432
## u_raw_space_time -2.337361e-01 0.75235145
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## u_space_time -3.593679e-03 0.04419474
## u_space_time -1.183696e-02 0.07329946
## u_space_time -8.721986e-03 0.05587389
## u_space_time -2.640932e-03 0.03384431
## u_space_time 1.533734e-02 0.05541283
## u_space_time -4.991093e-03 0.05576850
## u_space_time 1.023853e-02 0.05503748
## u_space_time 1.603682e-03 0.04084551
## u_space_time -3.023762e-03 0.04284509
## u_space_time -7.305651e-03 0.03749114
## u_space_time 9.865896e-03 0.04300459
## u_space_time -4.385223e-03 0.04453944
## u_space_time 1.333113e-02 0.04163091
## u_space_time 1.192284e-02 0.05636482
## u_space_time -3.413500e-03 0.03649982
## u_space_time 4.718843e-03 0.03629479
## u_space_time -1.264435e-04 0.03946172
## u_space_time 1.931061e-03 0.03969638
## u_space_time 2.736542e-04 0.05445033
## u_space_time 1.846384e-03 0.03290221
## u_space_time -6.547093e-03 0.05465095
## u_space_time 3.251335e-03 0.03756517
## u_space_time 1.975612e-02 0.04421475
## u_space_time 4.601553e-03 0.04877597
## u_space_time -2.472186e-03 0.04248047
## u_space_time -5.503803e-03 0.05281485
## u_space_time -3.360992e-02 0.07184154
## u_space_time -9.612507e-03 0.04349857
## u_space_time 1.447909e-02 0.05830448
## u_space_time -6.993210e-03 0.05714106
## u_space_time -2.205337e-03 0.05626133
## u_space_time -3.076998e-03 0.04066409
## u_space_time -7.430032e-03 0.04651319
## u_space_time -1.480218e-02 0.07643431
## u_space_time -1.498984e-02 0.05871682
## u_space_time -1.977039e-03 0.03524625
## u_space_time 2.344178e-02 0.05993957
## u_space_time -3.772161e-03 0.05756964
## u_space_time 1.315217e-02 0.05736021
## u_space_time 6.889186e-04 0.04247922
## u_space_time -2.676144e-03 0.04434822
## u_space_time -1.329294e-02 0.04039923
## u_space_time 1.095698e-02 0.04453540
## u_space_time -1.578677e-02 0.04831159
## u_space_time 1.870527e-02 0.04437659
## u_space_time 1.440826e-02 0.05868903
## u_space_time -2.186236e-03 0.03784635
## u_space_time 1.241191e-02 0.03882486
## u_space_time 1.991091e-03 0.04108620
## u_space_time 6.093108e-03 0.04188782
## u_space_time -6.903135e-04 0.05662493
## u_space_time 5.074548e-03 0.03450898
## u_space_time -1.278551e-02 0.05797314
## u_space_time 8.102307e-03 0.03939533
## u_space_time 3.180190e-02 0.04840252
## u_space_time 6.644158e-03 0.05224887
## u_space_time -1.376640e-03 0.04482180
## u_space_time -8.259556e-03 0.05637723
## u_space_time -4.530939e-02 0.08025463
## u_space_time -1.311965e-02 0.04692182
## u_space_time 2.014132e-02 0.06309490
## u_space_time -6.081584e-03 0.05994280
## u_space_time -2.424385e-03 0.05944455
## u_space_time -2.202162e-03 0.04337439
## u_space_time -1.058309e-02 0.05000740
## u_space_time -1.952438e-02 0.08167670
## u_space_time -1.850825e-02 0.06223531
## u_space_time -8.426427e-04 0.03754563
## u_space_time 2.836937e-02 0.06460291
## u_space_time -3.905651e-03 0.06070068
## u_space_time 1.088960e-02 0.05985071
## u_space_time -6.554087e-04 0.04514729
## u_space_time -1.523767e-03 0.04680947
## u_space_time -1.701924e-02 0.04376948
## u_space_time 1.008090e-02 0.04654298
## u_space_time -2.421190e-02 0.05377323
## u_space_time 2.245045e-02 0.04736827
## u_space_time 1.162506e-02 0.06094875
## u_space_time 7.980558e-04 0.04005605
## u_space_time 1.908868e-02 0.04260864
## u_space_time 2.621343e-03 0.04359120
## u_space_time 9.647158e-03 0.04532981
## u_space_time -2.940618e-03 0.06016539
## u_space_time 8.792124e-03 0.03719689
## u_space_time -1.801572e-02 0.06280747
## u_space_time 1.355392e-02 0.04255560
## u_space_time 3.933118e-02 0.05168348
## u_space_time 5.934564e-03 0.05630418
## u_space_time 6.092549e-04 0.04811139
## u_space_time -9.979737e-03 0.06087227
## u_space_time -5.097868e-02 0.08657138
## u_space_time -1.427248e-02 0.05060824
## u_space_time 1.761540e-02 0.06551229
## u_space_time -5.447616e-03 0.06387171
## u_space_time -6.025739e-03 0.06378348
## u_space_time 4.891998e-04 0.04701947
## u_space_time -1.147724e-02 0.05395408
## u_space_time -2.658797e-02 0.08917336
## u_space_time -2.116938e-02 0.06663222
## u_space_time 2.836707e-04 0.04063296
## u_space_time 3.580364e-02 0.07173961
## u_space_time -7.982246e-03 0.06512353
## u_space_time 8.941156e-03 0.06376415
## u_space_time -1.675471e-04 0.04862385
## u_space_time -1.632244e-03 0.05013095
## u_space_time -1.733157e-02 0.04658504
## u_space_time 1.026028e-02 0.04968177
## u_space_time -2.897528e-02 0.05886311
## u_space_time 2.562685e-02 0.05089963
## u_space_time 7.990448e-03 0.06453246
## u_space_time 5.323149e-03 0.04343738
## u_space_time 2.707017e-02 0.04828790
## u_space_time 1.085887e-03 0.04665368
## u_space_time 1.246103e-02 0.04958482
## u_space_time -3.424800e-03 0.06474711
## u_space_time 1.190283e-02 0.04061360
## u_space_time -2.133580e-02 0.06831424
## u_space_time 1.605963e-02 0.04585522
## u_space_time 4.176327e-02 0.05311921
## u_space_time 3.138415e-03 0.06099800
## u_space_time 7.250900e-03 0.05322941
## u_space_time -1.012632e-02 0.06598932
## u_space_time -5.212164e-02 0.09131040
## u_space_time -1.242231e-02 0.05431991
## u_space_time 9.375241e-03 0.06736865
## u_space_time 8.205060e-04 0.06899028
## u_space_time -9.668495e-03 0.06919936
## u_space_time 3.253548e-03 0.05155598
## u_space_time -1.107713e-02 0.05839482
## u_space_time -3.334624e-02 0.09803191
## u_space_time -1.679148e-02 0.07022849
## u_space_time 1.909487e-03 0.04439921
## u_space_time 4.000983e-02 0.07815763
## u_space_time -1.380714e-02 0.07093723
## u_space_time 5.596428e-03 0.06882877
## u_space_time 5.273230e-04 0.05281060
## u_space_time -2.621215e-03 0.05417285
## u_space_time -1.494158e-02 0.04929453
## u_space_time 9.872586e-03 0.05350887
## u_space_time -2.843271e-02 0.06217841
## u_space_time 2.445531e-02 0.05321907
## u_space_time 1.277338e-03 0.06957657
## u_space_time 7.747985e-03 0.04744493
## u_space_time 3.219195e-02 0.05337901
## u_space_time -2.105646e-03 0.05039015
## u_space_time 1.248761e-02 0.05372536
## u_space_time -1.829109e-03 0.07016021
## u_space_time 1.287597e-02 0.04410769
## u_space_time -2.193732e-02 0.07386330
## u_space_time 1.667461e-02 0.04930627
## u_space_time 5.036430e-02 0.05885835
## u_space_time 1.895867e-03 0.06658142
## u_space_time 1.573667e-02 0.06057527
## u_space_time -1.404199e-02 0.07253692
## u_space_time -5.596665e-02 0.09878532
## u_space_time -1.032320e-02 0.05881552
## u_space_time 7.511398e-03 0.07223273
## u_space_time 1.051555e-02 0.07646944
## u_space_time -1.040307e-02 0.07521514
## u_space_time 5.314065e-03 0.05665689
## u_space_time -1.320984e-02 0.06400430
## u_space_time -3.635225e-02 0.10666088
## u_space_time -1.192326e-02 0.07533714
## u_space_time 4.577319e-03 0.04886202
## u_space_time 3.797829e-02 0.08210272
## u_space_time -1.598264e-02 0.07716531
## u_space_time 3.112273e-03 0.07498866
## u_space_time -1.714244e-03 0.05769718
## u_space_time -5.556100e-05 0.05899743
## u_space_time -1.410139e-02 0.05325292
## u_space_time 6.423547e-03 0.05783122
## u_space_time -2.760003e-02 0.06628048
## u_space_time 2.167869e-02 0.05599888
## u_space_time -7.929535e-03 0.07686349
## u_space_time 9.366850e-03 0.05188930
## u_space_time 3.409186e-02 0.05720215
## u_space_time -2.470377e-03 0.05484130
## u_space_time 1.344680e-02 0.05868856
## u_space_time -4.806529e-03 0.07665048
## u_space_time 1.445801e-02 0.04823826
## u_space_time -2.605792e-02 0.08136664
## u_space_time 1.646698e-02 0.05320788
## u_space_time 5.917795e-02 0.06626645
## u_space_time -1.433653e-03 0.07274472
## u_space_time 2.200972e-02 0.06822140
## u_space_time -1.574475e-02 0.07934086
## u_space_time -5.408552e-02 0.10419956
## u_space_time -8.061815e-03 0.06398715
## u_space_time -1.766607e-03 0.07753503
## u_space_time 1.877214e-02 0.08547409
## u_space_time -1.739629e-02 0.08293846
## u_space_time 7.796404e-03 0.06231026
## u_space_time -1.331329e-02 0.06972494
## u_space_time -4.395153e-02 0.11806418
## u_space_time -7.640183e-03 0.08191627
## u_space_time 5.210514e-03 0.05343530
## u_space_time 3.615002e-02 0.08727276
## u_space_time -2.269776e-02 0.08529296
## u_space_time 6.303142e-03 0.08171808
## u_space_time -2.251014e-03 0.06304912
## u_space_time -1.181355e-04 0.06434011
## u_space_time -1.135823e-02 0.05750573
## u_space_time 4.857947e-03 0.06306640
## u_space_time -2.500844e-02 0.07062503
## u_space_time 2.157848e-02 0.06051861
## u_space_time -1.204179e-02 0.08453001
## u_space_time 1.210060e-02 0.05710646
## u_space_time 3.458904e-02 0.06090087
## u_space_time -5.974578e-03 0.05994239
## u_space_time 1.315891e-02 0.06378000
## u_space_time -4.411937e-03 0.08364511
## u_space_time 1.560178e-02 0.05267158
## u_space_time -2.755692e-02 0.08864534
## u_space_time 1.750580e-02 0.05797667
## u_space_time 5.752002e-02 0.06889865
## u_space_time -2.164711e-03 0.07942301
## u_space_time 2.473938e-02 0.07473032
## u_space_time -1.675424e-02 0.08658062
## u_space_time -5.401752e-02 0.11181174
## u_space_time -6.852677e-03 0.06976224
## u_space_time 6.647957e-04 0.08488873
## u_space_time 2.537633e-02 0.09515017
## u_space_time -1.702417e-02 0.09031941
## u_space_time 7.468300e-03 0.06777543
## u_space_time -1.403047e-02 0.07604174
## u_space_time -4.159313e-02 0.12662793
## u_space_time -3.557880e-03 0.08971896
## u_space_time 6.006264e-03 0.05835528
## u_space_time 3.090040e-02 0.09229959
## u_space_time -1.971214e-02 0.09209272
## u_space_time 6.677741e-03 0.08929887
## u_space_time -4.140381e-03 0.06898973
## u_space_time 3.838306e-03 0.07069111
## u_space_time -9.435817e-03 0.06252244
## u_space_time 1.980120e-03 0.06901805
## u_space_time -2.289779e-02 0.07604527
## u_space_time 1.869550e-02 0.06521239
## u_space_time -1.625840e-02 0.09310215
## u_space_time 1.066604e-02 0.06179534
## u_space_time 3.015646e-02 0.06306303
## u_space_time -4.286034e-03 0.06538781
## u_space_time 1.189630e-02 0.06907864
## u_space_time -4.060845e-03 0.09131530
## u_space_time 1.346412e-02 0.05679605
## u_space_time -2.832881e-02 0.09626963
## u_space_time 1.506495e-02 0.06270431
## u_space_time 5.299803e-02 0.07246614
## u_space_time -1.316146e-03 0.08647950
## u_space_time 2.997566e-02 0.08312906
## u_space_time -2.018571e-02 0.09491529
## u_space_time -5.020519e-02 0.11898629
## u_space_time -4.829981e-03 0.07594847
## u_space_time 5.428417e-03 0.09377145
## u_space_time 3.109455e-02 0.10550687
## u_space_time -1.431296e-02 0.09824074
## u_space_time 6.950248e-03 0.07351152
## u_space_time -1.676376e-02 0.08318996
## u_space_time -3.864179e-02 0.13625540
## u_space_time 2.609564e-03 0.09891339
## u_space_time 7.856517e-03 0.06376346
## u_space_time 2.424154e-02 0.09844541
## u_space_time -1.448472e-02 0.09979255
## u_space_time 8.062476e-03 0.09765269
## u_space_time -8.513663e-03 0.07578189
## u_space_time 7.989998e-03 0.07791561
## u_space_time -8.692681e-03 0.06809249
## u_space_time -3.046173e-03 0.07597689
## u_space_time -1.867091e-02 0.08187311
## u_space_time 1.505267e-02 0.07081224
## u_space_time -2.087957e-02 0.10270878
## u_space_time 7.587768e-03 0.06678320
## u_space_time 2.430836e-02 0.06626139
## u_space_time -1.068907e-03 0.07148799
## u_space_time 1.220395e-02 0.07506821
## u_space_time -8.755110e-03 0.10003521
## u_space_time 1.115584e-02 0.06142880
## u_space_time -3.083016e-02 0.10497269
## u_space_time 1.368183e-02 0.06827142Hyper parameter comparison
summary(sdr9, "fixed")## Estimate Std. Error
## log_prec_space_time 5.574152 1.155823summary(sdr9b, "fixed")## Estimate Std. Error
## log_prec_space_time 5.572942 1.155928cbind("mean" = inlafit9C$misc$theta.mode,
"se" = sqrt(diag(inlafit9C$misc$cov.intern)))## mean se
## [1,] 5.578561 1.157753Fixed effects (Intercept)
summary(sdr9, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.002680206 0.02933691summary(sdr9b, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.00267922 0.02933672inlafit9C$summary.fixed[ , 1:2]Random effects mean and standard deviation
plot(summary(sdr9, "report")[,1], summary(sdr9b, "report")[,1],
xlab = "TMB soft constraint)", ylab = "TMB hard constraint",
main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(summary(sdr9, "report")[,2], summary(sdr9b, "report")[,2],
xlab = "TMB soft constraint", ylab = "TMB hard constraint",
main = "Random effect standard devation")
abline(0, 1, col = "red")
11 Model 10: ICAR space + RW1 time + ICAR space x RW1 time
Model fit with full Type IV constraints
R_space <- diag(rowSums(adj.mat)) - adj.mat
R_space_scaled <- inla.scale.model(R_space, constr = list(A = matrix(1, ncol = ncol(R_space)), e = 0))
R_space_scaled_adj <- R_space_scaled + Matrix::Diagonal(ncol(R_space_scaled), diagval)
D_time <- diff(diag(length(levels(data$Yearf))), differences = 1)
R_time <- Matrix::Matrix(t(D_time) %*% D_time)
R_time_scaled <- inla.scale.model(R_time, constr = list(A = matrix(1, ncol = ncol(R_time)), e = 0))
R_time_scaled_adj <- R_time_scaled + Matrix::Diagonal(ncol(R_time_scaled), diagval)
R_space_time <- kronecker(R_time_scaled, R_space_scaled)
eig <- eigen(R_space_time)
eigval <- zapsmall(eig$values)
eigvec <- eig$vectors
rankdef <- nrow(R_space) + ncol(R_time) - 1
sum(eigval == 0)## [1] 50rankdef## [1] 50Aconstr <- t(eigvec[ , eigval == 0])
inlafit10 <- inla(Observed ~
f(ID, model = "besag", graph = adj.mat, hyper = prec.prior,
diagonal = diagval, scale.model = TRUE) +
f(Year, model = "rw1", hyper = prec.prior,
diagonal = diagval, scale.model = TRUE) +
f(id.area.year, model = "generic0", Cmatrix = R_space_time,
hyper = prec.prior, diagonal = diagval, rankdef = rankdef,
extraconstr = list(A = Aconstr, e = numeric(nrow(Aconstr)))),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep(".*rank.*", inlafit10$logfile, value = TRUE)## [1] " rank-deficiency is *defined* [1]"
## [2] " computed/guessed rank-deficiency = [1]"
## [3] " rank-deficiency is *defined* [50]"The precision parameter estimates match
summary(inlafit10)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.88, Running = 0.729, Post = 0.0331, Total = 3.64
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.046 0.036 -0.118 -0.046 0.025 NA 0
##
## Random effects:
## Name Model
## ID Besags ICAR model
## Year RW1 model
## id.area.year Generic0 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for ID 141.35 143.81 22.45 99.03 521.15 NA
## Precision for Year 280.50 397.47 23.70 162.71 1271.87 NA
## Precision for id.area.year 523.49 517.25 61.10 371.01 1920.75 NA
##
## Marginal log-Likelihood: -1247.95
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Hyper parameters
inlafit10$internal.summary.hyperpar[, 1:2]11.1 TMB
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_space);
DATA_SPARSE_MATRIX(Z_time);
DATA_MATRIX(C_space_time);
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_time);
DATA_SPARSE_MATRIX(R_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space);
val -= dlgamma(log_prec_space, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space(exp(-0.5 * log_prec_space));
PARAMETER_VECTOR(u_raw_space);
vector<Type> u_space(u_raw_space * sigma_space);
val += GMRF(R_space)(u_raw_space);
val -= dnorm(u_raw_space.sum(), Type(0), Type(0.001) * u_raw_space.size(), true); // soft sum-to-zero constraint
PARAMETER(log_prec_time);
val -= dlgamma(log_prec_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_time(exp(-0.5 * log_prec_time));
PARAMETER_VECTOR(u_raw_time);
vector<Type> u_time(u_raw_time * sigma_time);
val += GMRF(R_time)(u_raw_time);
val -= dnorm(u_raw_time.sum(), Type(0), Type(0.001) * u_raw_time.size(), true); // soft sum-to-zero constraint
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_raw_space_time_v(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time * sigma_space_time);
val += SEPARABLE(GMRF(R_time), GMRF(R_space))(u_raw_space_time);
val -= dnorm(C_space_time * u_raw_space_time_v, Type(0), Type(0.001) * u_raw_space_time.cols(), true).sum(); // soft sum-to-zero constraint
vector<Type> mu(beta0 +
Z_space * u_space +
Z_time * u_time +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space);
ADREPORT(u_time);
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
R_space_time_adj <- R_space_time + Matrix::Diagonal(ncol(R_space_time), 1e-6)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_space = Matrix::sparse.model.matrix(~0 + IDf, data),
Z_time = Matrix::sparse.model.matrix(~0 + Yearf, data),
C_space_time = Aconstr,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_space = R_space_scaled_adj,
R_time = R_time_scaled_adj)
tmbpar <- list(beta0 = 0,
log_prec_space = 0,
u_raw_space = numeric(nrow(tmbdata$R_space)),
log_prec_time = 0,
u_raw_time = numeric(nrow(tmbdata$R_time)),
log_prec_space_time = 0,
u_raw_space_time = array(0, c(nrow(tmbdata$R_space), nrow(tmbdata$R_time))))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space", "u_raw_time", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr10 <- TMB::sdreport(obj)
sdr10sum <- summary(sdr10, "all")Hyper parameter comparison
summary(sdr10, "fixed")## Estimate Std. Error
## log_prec_space 4.457034 0.874251
## log_prec_time 5.020457 1.086855
## log_prec_space_time 6.012000 1.152610cbind("mean" = inlafit10$misc$theta.mode,
"se" = sqrt(diag(inlafit10$misc$cov.intern)))## mean se
## [1,] 4.489781 0.9048292
## [2,] 5.014909 1.0368246
## [3,] 6.014394 1.4714114Fixed effects (Intercept)
summary(sdr10, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.04673713 0.03797077inlafit10$summary.fixed[ , 1:2]Random effects mean and standard deviation
par(mfrow = c(1, 2))
plot(inlafit10$summary.random[[1]][,2],
sdr10sum[rownames(sdr10sum) == "u_space", 1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(inlafit10$summary.random[[1]][,3],
sdr10sum[rownames(sdr10sum) == "u_space", 2],
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation")
abline(0, 1, col = "red")
plot(inlafit10$summary.random[[2]][,2],
sdr10sum[rownames(sdr10sum) == "u_time", 1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(inlafit10$summary.random[[2]][,3],
sdr10sum[rownames(sdr10sum) == "u_time", 2],
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation")
abline(0, 1, col = "red")
plot(inlafit10$summary.random[[3]][,2],
sdr10sum[rownames(sdr10sum) == "u_space_time", 1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
plot(inlafit10$summary.random[[3]][,3],
sdr10sum[rownames(sdr10sum) == "u_space_time", 2],
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation")
abline(0, 1, col = "red")
12 Model 11: AR1 time
This has row constraints on the interaction
inlafit11 <- inla(Observed ~
f(Year, model = "ar1", hyper = c(prec.prior, rho.prior)),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep(".*rank.*", inlafit11$logfile, value = TRUE)## [1] " computed/guessed rank-deficiency = [0]"summary(inlafit11)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.97, Running = 0.23, Post = 0.0106, Total = 3.21
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) 0.003 0.065 -0.124 0.003 0.129 NA 0
##
## Random effects:
## Name Model
## Year AR1 model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for Year 220.585 379.718 13.48 112.164 1107.316 NA
## Rho for Year 0.672 0.365 -0.37 0.814 0.993 NA
##
## Marginal log-Likelihood: -799.75
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Hyper parameters
inlafit11$internal.summary.hyperpar[, 1:2]12.1 TMB
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_time);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_time);
val -= dlgamma(log_prec_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_time(exp(-0.5 * log_prec_time));
PARAMETER(logit_rho_time);
val -= dnorm(logit_rho_time, Type(0.0), Type(1.0 / sqrt(0.15)), true);
Type rho_time(2 * exp(logit_rho_time)/(1 + exp(logit_rho_time)) - 1);
PARAMETER_VECTOR(u_raw_time);
vector<Type> u_time(u_raw_time * sigma_time);
val += AR1(rho_time)(u_raw_time);
vector<Type> mu(beta0 +
Z_time * u_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_time = Matrix::sparse.model.matrix(~0 + Yearf, data))
tmbpar <- list(beta0 = 0,
log_prec_time = 0,
logit_rho_time = 0,
u_raw_time = numeric(ncol(tmbdata$Z_time)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr11 <- TMB::sdreport(obj)
sdr11sum <- summary(sdr11, "all")Hyper parameter comparison
summary(sdr11, "fixed")## Estimate Std. Error
## log_prec_time 4.618542 1.016693
## logit_rho_time 2.130039 1.340042cbind("mean" = inlafit11$misc$theta.mode,
"se" = sqrt(diag(inlafit11$misc$cov.intern)))## mean se
## [1,] 4.620572 1.024576
## [2,] 2.130285 1.331142Fixed effects (Intercept)
summary(sdr11, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 0.002596854 0.06547004inlafit11$summary.fixed[ , 1:2]Random effects mean and standard deviation
par(mfrow = c(1, 2))
plot(inlafit11$summary.random[[1]][,2],
sdr11sum[rownames(sdr11sum) == "u_time", 1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
x <- inlafit11$summary.random[[1]][,3]
y <- sdr11sum[rownames(sdr11sum) == "u_time", 2]
plot(x, y,
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation",
xlim = range(c(x, y)), ylim = range(c(x, y)))
abline(0, 1, col = "red")
13 Model 12: IID space x AR1 time
inlafit12 <- inla(Observed ~
f(ID, model = "iid", hyper = prec.prior,
group = ID.Year, control.group = list(model = "ar1", hyper = rho.prior)),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep(".*rank.*", inlafit12$logfile, value = TRUE)## [1] " computed/guessed rank-deficiency = [0]"summary(inlafit12)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 3.11, Running = 0.29, Post = 0.0167, Total = 3.42
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.035 0.04 -0.113 -0.035 0.043 NA 0
##
## Random effects:
## Name Model
## ID IID model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for ID 98.214 121.702 16.553 62.138 404.367 NA
## GroupRho for ID 0.797 0.244 0.075 0.889 0.995 NA
##
## Marginal log-Likelihood: -799.74
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Hyper parameters
inlafit12$internal.summary.hyperpar[, 1:2]13.1 TMB
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER(logit_rho_time);
val -= dnorm(logit_rho_time, Type(0.0), Type(1.0 / sqrt(0.15)), true);
Type rho_time(2 * exp(logit_rho_time)/(1 + exp(logit_rho_time)) - 1);
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time * sigma_space_time);
val += SEPARABLE(AR1(rho_time), GMRF(R_space))(u_raw_space_time);
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
n_space <- length(levels(data$IDf))
n_time <- length(levels(data$Yearf))
R_space <- Matrix::sparseMatrix(1:n_space, 1:n_space, x = rep(1L, n_space))
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_space = R_space)
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
logit_rho_time = 0,
u_raw_space_time = array(0, c(n_space, n_time)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr)
sdr12 <- TMB::sdreport(obj)
sdr12sum <- summary(sdr12, "all")Hyper parameter comparison
summary(sdr12, "fixed")## Estimate Std. Error
## log_prec_space_time 3.785073 0.789077
## logit_rho_time 2.575710 1.455927cbind("mean" = inlafit12$misc$theta.mode,
"se" = sqrt(diag(inlafit12$misc$cov.intern)))## mean se
## [1,] 3.794426 0.7725143
## [2,] 2.556174 1.4372996Fixed effects (Intercept)
summary(sdr12, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.03514655 0.04185747inlafit12$summary.fixed[ , 1:2]Random effects mean and standard deviation
par(mfrow = c(1, 2))
plot(inlafit12$summary.random[[1]][,2],
sdr12sum[rownames(sdr12sum) == "u_space_time", 1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
x <- inlafit12$summary.random[[1]][,3]
y <- sdr12sum[rownames(sdr12sum) == "u_space_time", 2]
plot(x, y,
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation",
xlim = range(c(x, y)), ylim = range(c(x, y)))
abline(0, 1, col = "red")
14 Model 13: ICAR space x AR1 time
This has row constraints on the interaction
inlafit13 <- inla(Observed ~
f(ID, model = "besag", graph = adj.mat,
scale.model = TRUE, hyper = prec.prior,
group = ID.Year, control.group = list(model = "ar1", hyper = rho.prior)),
data = data, E = Expected, family = "poisson",
control.inla = list(strategy = "gaussian", int.strategy = "eb"),
control.compute = list(config = TRUE))
grep(".*rank.*", inlafit13$logfile, value = TRUE)## [1] " rank-deficiency is *defined* [1]"summary(inlafit13)##
## Call:
## c("inla.core(formula = formula, family = family, contrasts = contrasts,
## ", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
## scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
## ", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
## verbose, ", " lincomb = lincomb, selection = selection, control.compute
## = control.compute, ", " control.predictor = control.predictor,
## control.family = control.family, ", " control.inla = control.inla,
## control.fixed = control.fixed, ", " control.mode = control.mode,
## control.expert = control.expert, ", " control.hazard = control.hazard,
## control.lincomb = control.lincomb, ", " control.update =
## control.update, control.lp.scale = control.lp.scale, ", "
## control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
## ", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
## num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
## working.directory = working.directory, ", " silent = silent, inla.mode
## = inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
## .parent.frame)")
## Time used:
## Pre = 2.89, Running = 0.502, Post = 0.0157, Total = 3.4
## Fixed effects:
## mean sd 0.025quant 0.5quant 0.975quant mode kld
## (Intercept) -0.033 0.035 -0.101 -0.033 0.035 NA 0
##
## Random effects:
## Name Model
## ID Besags ICAR model
##
## Model hyperparameters:
## mean sd 0.025quant 0.5quant 0.975quant mode
## Precision for ID 137.692 159.073 21.69 90.361 546.654 NA
## GroupRho for ID 0.823 0.224 0.15 0.907 0.996 NA
##
## Marginal log-Likelihood: -888.62
## is computed
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Hyper parameters
inlafit13$internal.summary.hyperpar[, 1:2]Constraints
nrow(inlafit13$misc$configs$constr$A)## [1] 1914.1 TMB
The precision matrix for the ICAR x AR1 interacation is expressed as \(Q = Q_{time} \otimes Q_{space}\).
SEPARABLE calculates the density assuming both are full rank.
To apply the correct density we need to adjust the log posterior for the rank deficiency of the ICAR precision matrix.
The log determinant of the Kronecker product is \(\log |Q| = n_{time} \cdot \log |Q_{space}| + n_{space} \cdot \log |Q_{time}|\).
The log determinant adjusted for rank deficiency is \(\log |Q| = n_{time} \cdot \log |Q_{space}| + (n_{space} - k_{space}) \cdot \log |Q_{time}|\), where \(k_{space}\) is the rank deficiency \(Q_{space}\). Thus we need to subtract \(0.5 \cdot k_space \cdot \log |Q_{time}|\) to the log posterior.
For and \(AR1(\rho)\) time component, \(|Q^{-1}_{time}| = (1-\rho^2) ^ (n_{time} - 1)\).
mod <- '
#include <TMB.hpp>
template<class Type>
Type objective_function<Type>::operator() ()
{
using namespace density;
DATA_VECTOR(y);
DATA_VECTOR(E);
DATA_SPARSE_MATRIX(Z_space_time);
DATA_SPARSE_MATRIX(R_space);
DATA_SCALAR(rankdef_R_space);
Type val(0);
PARAMETER(beta0);
// beta0 ~ 1
PARAMETER(log_prec_space_time);
val -= dlgamma(log_prec_space_time, Type(0.001), Type(1.0 / 0.001), true);
Type sigma_space_time(exp(-0.5 * log_prec_space_time));
PARAMETER(logit_rho_time);
val -= dnorm(logit_rho_time, Type(0.0), Type(1.0 / sqrt(0.15)), true);
Type rho_time(2 * exp(logit_rho_time)/(1 + exp(logit_rho_time)) - 1);
PARAMETER_ARRAY(u_raw_space_time);
vector<Type> u_space_time(u_raw_space_time * sigma_space_time);
val += SEPARABLE(AR1(rho_time), GMRF(R_space))(u_raw_space_time);
// Adjust normalising constant for rank deficience of R_space
Type log_det_Qar1((u_raw_space_time.cols() - 1) * log(1 - rho_time * rho_time));
val -= rankdef_R_space * 0.5 * (log_det_Qar1 - log(2 * PI));
for (int i = 0; i < u_raw_space_time.cols(); i++) {
val -= dnorm(u_raw_space_time.col(i).sum(), Type(0), Type(0.001) * u_raw_space_time.rows(), true);
}
vector<Type> mu(beta0 +
Z_space_time * u_space_time +
log(E));
val -= dpois(y, exp(mu), true).sum();
ADREPORT(u_space_time);
return val;
}
'dll <- tmb_compile_and_load(mod)
n_space <- length(levels(data$IDf))
n_time <- length(levels(data$Yearf))
R_space <- diag(rowSums(adj.mat)) - adj.mat
R_space_scaled <- inla.scale.model(R_space, constr = list(A = matrix(1, ncol = ncol(R_space)), e = 0))
R_space_scaled_adj <- R_space_scaled + Matrix::Diagonal(ncol(R_space_scaled), diagval)
tmbdata <- list(y = data$Observed,
E = data$Expected,
Z_space_time = Matrix::sparse.model.matrix(~0 + IDf:Yearf, data),
R_space = R_space_scaled_adj,
rankdef_R_space = nrow(R_space_scaled) - as.integer(rankMatrix(R_space_scaled)))
tmbpar <- list(beta0 = 0,
log_prec_space_time = 0,
logit_rho_time = 2,
u_raw_space_time = array(0, c(n_space, n_time)))
obj <- TMB::MakeADFun(data = tmbdata,
parameters = tmbpar,
random = c("beta0", "u_raw_space_time"),
DLL = dll,
silent = TRUE)
tmbfit <- nlminb(obj$par, obj$fn, obj$gr,
control = list(iter.max = 1000,
eval.max = 1000))
sdr13 <- TMB::sdreport(obj)
sdr13sum <- summary(sdr13, "all")Hyper parameter comparison
summary(sdr13, "fixed")## Estimate Std. Error
## log_prec_space_time 4.283482 0.8063502
## logit_rho_time 2.823639 1.4215524cbind("mean" = inlafit13$misc$theta.mode,
"se" = sqrt(diag(inlafit13$misc$cov.intern)))## mean se
## [1,] 4.280829 0.8096733
## [2,] 2.804004 1.3524356Fixed effects (Intercept)
summary(sdr13, "random")[1, , drop = FALSE]## Estimate Std. Error
## beta0 -0.03317819 0.03668285inlafit13$summary.fixed[ , 1:2]Random effects mean and standard deviation
par(mfrow = c(1, 2))
plot(inlafit13$summary.random[[1]][,2],
sdr13sum[rownames(sdr13sum) == "u_space_time", 1],
xlab = "INLA", ylab = "TMB", main = "Random effect point estimates")
abline(0, 1, col = "red")
x <- inlafit13$summary.random[[1]][,3]
y <- sdr13sum[rownames(sdr13sum) == "u_space_time", 2]
plot(x, y,
xlab = "INLA", ylab = "TMB", main = "Random effect standard deviation",
xlim = range(c(x, y)), ylim = range(c(x, y)))
abline(0, 1, col = "red")
15 Original computing environment
sessionInfo()## R version 4.2.0 (2022-04-22)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Big Sur/Monterey 10.16
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
##
## attached base packages:
## [1] parallel stats graphics grDevices utils datasets methods
## [8] base
##
## other attached packages:
## [1] forcats_0.5.1 stringr_1.4.0 dplyr_1.0.9 purrr_0.3.4
## [5] readr_2.1.2 tidyr_1.2.0 tibble_3.1.7 ggplot2_3.3.6
## [9] tidyverse_1.3.1 DClusterm_1.0-1 DCluster_0.2-8 MASS_7.3-56
## [13] spdep_1.2-4 sf_1.0-7 spData_2.0.1 boot_1.3-28
## [17] spacetime_1.2-8 INLA_22.05.07 sp_1.4-7 foreach_1.5.2
## [21] Matrix_1.5-1
##
## loaded via a namespace (and not attached):
## [1] nlme_3.1-157 fs_1.5.2 xts_0.12.2 lubridate_1.8.0
## [5] httr_1.4.3 TMB_1.9.1 tools_4.2.0 backports_1.4.1
## [9] bslib_0.3.1 utf8_1.2.2 R6_2.5.1 KernSmooth_2.23-20
## [13] mgcv_1.8-40 DBI_1.1.2 colorspace_2.0-3 raster_3.5-15
## [17] withr_2.5.0 tidyselect_1.1.2 compiler_4.2.0 cli_3.3.0
## [21] rvest_1.0.2 xml2_1.3.3 bookdown_0.26 sass_0.4.1
## [25] scales_1.2.0 classInt_0.4-3 proxy_0.4-26 digest_0.6.29
## [29] rmarkdown_2.14 pkgconfig_2.0.3 htmltools_0.5.2 highr_0.9
## [33] dbplyr_2.1.1 fastmap_1.1.0 rlang_1.0.2 readxl_1.4.0
## [37] rstudioapi_0.13 jquerylib_0.1.4 generics_0.1.2 zoo_1.8-10
## [41] jsonlite_1.8.0 magrittr_2.0.3 s2_1.0.7 Rcpp_1.0.8.3
## [45] munsell_0.5.0 fansi_1.0.3 lifecycle_1.0.1 terra_1.5-21
## [49] stringi_1.7.6 yaml_2.3.5 grid_4.2.0 crayon_1.5.1
## [53] deldir_1.0-6 lattice_0.20-45 haven_2.5.0 splines_4.2.0
## [57] hms_1.1.1 knitr_1.39 pillar_1.7.0 codetools_0.2-18
## [61] wk_0.6.0 reprex_2.0.1 glue_1.6.2 evaluate_0.15
## [65] modelr_0.1.8 vctrs_0.4.1 tzdb_0.3.0 MatrixModels_0.5-0
## [69] cellranger_1.1.0 gtable_0.3.0 assertthat_0.2.1 xfun_0.31
## [73] broom_0.8.0 e1071_1.7-9 class_7.3-20 intervals_0.15.2
## [77] iterators_1.0.14 units_0.8-0 ellipsis_0.3.2