auto.arima runtime stepwise increase

[Someone894]

Hello,

some time ago the runtime of the forecast we are using doubled from one (monthly) execution to the next. So I've wrote a small test script to analyze how the execution time of some of the forecasting method performs with different length timeseries:

start <- 1
end <- 100

# FROM: https://github.com/robjhyndman/M4metalearning/blob/61ddc7101680e9df7219c359587d0b509d2b50d6/R/forec_methods_list.R#L40
auto_arima_forec <- function(x, h) {
  model <- forecast::auto.arima(x, stepwise = FALSE, approximation = FALSE)
  forecast::forecast(model, h = h)$mean
}

generate_random_ts <- function(a = 1:1000, b = 0, c = 7) {
  sin(a * rnorm(1, b, c)) +
	sin(a * rnorm(1, b^2, c^2)) +
	sin(a * rnorm(1, b^3, c^3)) +
	sin(a * rnorm(1, b^4, c^4)) +
	seq(a) / runif(a, 1, 10)
}

set.seed(4711)
truex <- generate_random_ts()
#plot(truex, type = "l")

timeseries <- list()
for (i in start:end) {
  x <- ts(head(truex, i), frequency = 12)
  timeseries <- rlist::list.append(timeseries, x)
}

jobs <- lapply(
  timeseries[1:length(timeseries)],
  function(x) {
	local({
	  x <- x
	  bquote(auto_arima_forec(.(x), 24))
	})
  }
)

names(jobs) <- paste0(
  "Length ts: ",
  stringr::str_pad(start:end, 3, pad = "0")
)

mbm <- microbenchmark::microbenchmark(list = jobs)

save(mbm, file = "mbm.RData")

options(microbenchmark.unit = "eps")
print(mbm)

options(microbenchmark.unit = "relative")
print(mbm)

plot <- ggplot2::autoplot(mbm)
print(plot)

Most of them just (more or less) linearly increase in runtime for longer timeseries, which is to be expected. But the auto.arima method seems to be exploding (4-5 times more execution time) when the timeseries becomes longer then 71 month (=6 Years).

Do you know why this is the case and is it possible to decrease the magnitude of this sudden jump?

In order to keep this Issue short I only posted code and image for the auto.arima case.

[ncooder]

@Someone894 If you look at these lines:

forecast/R/newarima2.R

Line 173 in 0583d2f

max.p <- min(max.p, floor(serieslength / 3))

For monthly data if you have exactly 71 months, then floor(71 / (3 * 12)) = floor(71 / 36) = 1.
So max.P and max.Q become 1.
If you have 72 months, then floor(72 / (3 * 12)) = floor(72 / 36) = 2.
So max.P and/or max.Q become 2.
This jump from 1 to 2 in the seasonal orders causes that significant increase the search space. The same is valid for 36 months.