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    458.485    524.649    562.5877    550.723    589.119    682.143   100
#>  n1e3   1231.358   1440.257   1572.6511   1519.610   1658.590   3984.354   100
#>  n1e4  10738.645  12317.551  15327.7901  15091.916  16083.537  93523.184   100
#>  n1e5 162991.509 186056.265 215360.6871 209083.372 251890.306 325993.178   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    725.794    755.380    788.005    781.6485    807.807    899.618   100
#>   p25   2754.770   2794.744   2921.774   2841.7365   2993.209   5718.578   100
#>  p100  21364.299  23960.528  26000.107  24283.1150  28909.611  44203.299   100
#>  p400 238647.328 262425.380 297338.443 284630.0480 312886.745 495070.681   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.