4d monte carlo integration R - r

Question

How to integrate function using the Monte Carlo method
F(X[1],X[2],X[3],X[4])
Which depends on 4 variable by 4 dimensions?
I mean int_{0}^{1} int_{0}^{1} int_{0}^{1} int_{0}^{1} X[1]X[2]X[3]X[4] dX[1] dX[2] dX[3] dX[4]
UPD
function is
data1 = rnorm(100, 0, 1)
data2 = rnorm(100, 0.1, 0.5)
data3 = rnorm(100, 0.2, 0.8)
data4 = rnorm(100, 0.3, 0.9)
kernel1 = kdensity(data1,kernel = 'gaussian')
kernel2 = kdensity(data2,kernel = 'gaussian')
kernel3 = kdensity(data3,kernel = 'gaussian')
kernel4 = kdensity(data4,kernel = 'gaussian')
f <- function(X) {
return(X[1]*kernel1(X[1])*kernel2(X[2])*kernel3(X[3])*kernel4(X[4]))
}
and i want to integrate it
int_{0}^{1} int_{0}^{1} int_{0}^{1} int_{0}^{1} f dX[1] dX[2] dX[3] dX[4]

Answer 1

Due to your particular structure of integral expression, you can rewrite your nested integral into a product of integrals.
Thus, the solution below may be a example (with random seed set.seed(1)) for you:
g <- Vectorize(function(ker) {integrate(ker,0,1)}$value)
gE <- function(ker) {integrate(function(x) x*ker(x),0,1)}$value
res1 <- prod(c(gE(kernel1),g(c(kernel2,kernel3,kernel4))))
such that
> res1
[1] 0.01559343
Nested Integral: you need to rewrite your function f first, i.e.,
f <- function(x1,x2,x3,x4) {
return(x1*kernel1(x1)*kernel2(x2)*kernel3(x3)*kernel4(x4))
}
res2 <- integrate(Vectorize(function(x4)
integrate(Vectorize(function(x3,x4)
integrate(Vectorize(function(x2,x3,x4)
integrate(f,0,1,x2,x3,x4)$value),
0,
1,
x3,
x4)$value),
0,
1,
x4)$value),
0,
1)$value
such that
> res2
[1] 0.01559343