Performance

All the tests were done on an Arch Linux x86_64 machine with an Intel(R) Core(TM) i7 CPU (1.90GHz).

Empirical likelihood computation

We show the performance of computing empirical likelihood with el_mean(). We test the computation speed with simulated data sets in two different settings: 1) the number of observations increases with the number of parameters fixed, and 2) the number of parameters increases with the number of observations fixed.

Increasing the number of observations

We fix the number of parameters at \(p = 10\), and simulate the parameter value and \(n \times p\) matrices using rnorm(). In order to ensure convergence with a large \(n\), we set a large threshold value using el_control().

library(ggplot2)
library(microbenchmark)
set.seed(3175775)
p <- 10
par <- rnorm(p, sd = 0.1)
ctrl <- el_control(th = 1e+10)
result <- microbenchmark(
  n1e2 = el_mean(matrix(rnorm(100 * p), ncol = p), par = par, control = ctrl),
  n1e3 = el_mean(matrix(rnorm(1000 * p), ncol = p), par = par, control = ctrl),
  n1e4 = el_mean(matrix(rnorm(10000 * p), ncol = p), par = par, control = ctrl),
  n1e5 = el_mean(matrix(rnorm(100000 * p), ncol = p), par = par, control = ctrl)
)

Below are the results:

result
#> Unit: microseconds
#>  expr        min         lq       mean      median         uq        max neval
#>  n1e2    403.916    499.812    538.213    530.2205    583.862    644.816   100
#>  n1e3   1209.266   1424.515   1556.652   1513.5910   1644.164   3977.231   100
#>  n1e4  11258.896  12883.701  15995.856  15791.5755  16900.065  89868.220   100
#>  n1e5 202795.874 230583.762 268387.647 259927.2260 316020.897 349137.740   100
#>  cld
#>  a  
#>  a  
#>   b 
#>    c
autoplot(result)
#> Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
#> ℹ Please use tidy evaluation idioms with `aes()`.
#> ℹ See also `vignette("ggplot2-in-packages")` for more information.
#> ℹ The deprecated feature was likely used in the microbenchmark package.
#>   Please report the issue at
#>   <https://github.com/joshuaulrich/microbenchmark/issues/>.
#> This warning is displayed once per session.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.

Increasing the number of parameters

This time we fix the number of observations at \(n = 1000\), and evaluate empirical likelihood at zero vectors of different sizes.

n <- 1000
result2 <- microbenchmark(
  p5 = el_mean(matrix(rnorm(n * 5), ncol = 5),
    par = rep(0, 5),
    control = ctrl
  ),
  p25 = el_mean(matrix(rnorm(n * 25), ncol = 25),
    par = rep(0, 25),
    control = ctrl
  ),
  p100 = el_mean(matrix(rnorm(n * 100), ncol = 100),
    par = rep(0, 100),
    control = ctrl
  ),
  p400 = el_mean(matrix(rnorm(n * 400), ncol = 400),
    par = rep(0, 400),
    control = ctrl
  )
)

result2
#> Unit: microseconds
#>  expr        min          lq        mean      median         uq       max neval
#>    p5    703.745    768.0695    827.7765    793.8415    833.189   3373.58   100
#>   p25   2846.115   2911.2485   3036.9704   2957.9785   3054.601   6536.03   100
#>  p100  24167.098  27019.8250  29315.1103  27564.5970  31929.380  49370.84   100
#>  p400 288952.699 319144.4565 357593.8390 343041.9260 386157.552 503220.76   100
#>  cld
#>  a  
#>  a  
#>   b 
#>    c
autoplot(result2)

On average, evaluating empirical likelihood with a 100000×10 or 1000×400 matrix at a parameter value satisfying the convex hull constraint takes less than a second.