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 × 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
#>  n1e2    445.591    482.520    515.6633    499.3265    549.9805    665.060
#>  n1e3   1198.074   1405.321   1545.9846   1488.6355   1597.3335   5499.694
#>  n1e4  10783.957  12937.993  14705.6650  15065.8505  15934.4450  19893.191
#>  n1e5 171542.793 189765.480 231912.0660 224741.4230 256666.6580 380635.069
#>  neval cld
#>    100 a  
#>    100 a  
#>    100  b 
#>    100   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    732.375    772.21    838.1332    797.1365    851.1125   3755.312   100
#>   p25   2882.514   2936.23   3067.7938   2966.6315   3032.0030   7546.050   100
#>  p100  23318.308  25866.79  29510.7933  26971.0830  30854.6690 165750.865   100
#>  p400 266235.449 292424.40 324949.4690 313093.4115 341615.5010 447584.186   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.