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
#> n1e2 480.702 579.4605 683.9713 651.999 744.1055 1397.514
#> n1e3 1322.316 1639.4090 2089.1867 1879.296 2298.9635 5032.287
#> n1e4 12140.334 17463.6515 20520.6089 19397.234 21753.5280 37504.643
#> n1e5 267149.481 307769.8205 370535.4057 358940.570 414239.7385 560010.771
#> 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 784.742 843.6425 987.3792 880.003 940.296 3610.111 100
#> p25 2971.194 3048.8325 3320.6049 3105.648 3163.769 7695.062 100
#> p100 23538.566 26370.0610 30707.9708 29054.183 32858.164 66575.924 100
#> p400 264627.376 297454.1460 354283.8642 321911.429 407738.503 680908.163 100
#> 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.