In this vignette we compare two different implementations of the cumulative median. The cumstats package provides a naive method, which uses the standard median function in a for loop. Each call to the standard median function is log-linear, so the total expected complexity is log-quadratic. The binsegRcpp package provides a different implementation that uses a log-linear algorithm, previously described in the 2017 NeurIPS research paper Maximum Margin Interval Trees by Alexandre Drouin, Toby Hocking, Francois Laviolette.

expr.list <- c(
  if(requireNamespace("cumstats"))atime::atime_grid(
    "cumstats::cummedian"=cumstats::cummedian(data.vec)),
  if(requireNamespace("binsegRcpp"))atime::atime_grid(
    "binsegRcpp::cum_median"=binsegRcpp::cum_median(data.vec)),
  atime::atime_grid(cumsum=cumsum(data.vec)))
#> Loading required namespace: cumstats
atime.list <- atime::atime(
  N=2^seq(1, 20),
  setup={
    set.seed(1)
    data.vec <- rnorm(N)
  },
  result=TRUE,
  expr.list=expr.list,
  times=5)
plot(atime.list)
#> Warning in ggplot2::scale_y_log10("median line, min/max band"): log-10 transformation introduced infinite values.
#> log-10 transformation introduced infinite values.
#> log-10 transformation introduced infinite values.

(best.list <- atime::references_best(atime.list))
#> references_best list with 86 measurements, best fit complexity:
#> cumstats::cummedian (N^2 kilobytes, N log N seconds)
#> binsegRcpp::cum_median (N kilobytes, N log N seconds)
#> cumsum (N kilobytes, N log N seconds)
## try() to avoid CRAN error 'from' must be a finite number, on
## https://www.stats.ox.ac.uk/pub/bdr/Rblas/README.txt, due to
## https://github.com/r-lib/scales/issues/307
plot(best.list)
#> Warning in ggplot2::scale_y_log10(""): log-10 transformation introduced
#> infinite values.

Exercise for the reader: increase seconds.limit and max N until you can clearly show that binsegRcpp::cum_median should be the preferred method for computing the cumulative median.