# Performance

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

library(melt)
library(microbenchmark)
library(ggplot2)

## 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().

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    496.721    600.000    751.9864    661.144    818.4215   1671.521   100
#>  n1e3   1431.147   1764.893   2199.2031   1978.675   2381.7740   3931.695   100
#>  n1e4  14558.944  19130.122  22514.8889  22234.505  24996.2615  35005.867   100
#>  n1e5 300308.387 377692.838 434658.5113 403606.535 497906.4695 792190.949   100
#>  cld
#>  a
#>  a
#>   b
#>    c
autoplot(result) ### 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    745.675    887.169   1067.269    936.594   1077.803   2160.35   100
#>   p25   2956.290   3315.738   4075.956   3656.311   4361.149  13528.18   100
#>  p100  24379.049  29547.458  34768.130  33255.675  38310.068  60739.20   100
#>  p400 300339.926 369978.383 425060.884 393062.214 433277.034 863053.36   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.