Status: Very shallow exploration of suggested quick heuristic.

Example on simulated exponential data

base <- 2
time_period <- 14

y <- base^{1:time_period}
plot(y)

log_y <- log(y)
plot(log_y)

y_prop <- y / 10^6
plot(log(y_prop))

diff(log_y)
##  [1] 0.6931472 0.6931472 0.6931472 0.6931472 0.6931472 0.6931472 0.6931472 0.6931472 0.6931472 0.6931472 0.6931472 0.6931472 0.6931472

The increments at each time-step are \(\log(b)\) since \[ \log(b^t) - \log(b^{t + 1}) = t\log(b) - (t + 1)\log(b) = \log(b). \]

Example on real data

ts <- c(
  3/308259878, 6/428248420, 1/419143935, 4/382434053, 25/409326241, 8/336993186, 
  6/339782463, 10/393575246, 3/404306425, 2/331453869, 8/443570791, 2/358994390, 
  6/545294572, 15/352576173, 6/298166012, 19/493418258, 18/233816053
)

plot(ts)

plot(log(ts))

log_diff <- diff(log(ts))
log_diff
##  [1]  0.3643869 -1.7702704  1.4779525  1.7646252 -0.9449845 -0.2959250  0.3638589 -1.2308736 -0.2067807  1.0949256 -1.1747437  0.6805929  1.3523502 -0.7486745
## [15]  0.6489727  0.6927553
mean(log_diff)
## [1] 0.1292605
sqrt(sum((log_diff - mean(log_diff))^2))
## [1] 4.159429