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 478.486 567.466 655.2995 610.193 664.700 1283.147 100
#> n1e3 1431.359 1686.219 2018.6906 1892.880 2133.096 3783.401 100
#> n1e4 13955.585 18157.292 20678.7358 20452.990 22623.294 34640.493 100
#> n1e5 288719.574 357686.361 435098.9835 445522.969 490486.745 720364.080 100
#> 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 747.099 828.274 965.7782 884.664 990.1295 1807.282 100
#> p25 2927.671 3153.000 3558.2610 3357.575 3552.0815 7679.986 100
#> p100 24171.973 27815.401 31560.4120 30185.583 34235.3110 49560.778 100
#> p400 301981.597 332774.619 374275.9353 356880.586 382856.2170 564443.300 100
#> 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.