Why are my estimated and theoretical results for sobol sensitivity analysis different? - r

Question

I am working on the sobol sensitivity analysis. I am trying to compute the first order effect and total effect indices in both an estimated and theoretical way.
Firstly, I computed the estimated values by following the steps in Wikipedia "Variance-based sensitivity analysis".
Here is the code:
set.seed(123)
x_1<-ceiling(1000*runif(1000)) ## generate x1 from range [1,1000]
x_2<-ceiling(100*runif(1000)) ## generate x2 from range [1,100]
x_3<-ceiling(10*runif(1000)) ## generate x3 from range [1,10]
x<-cbind(x_1,x_2,x_3) ## combine as one matrix
A<-matrix(x[1:500,],ncol=3) ## divide this one matrix into two
B<-matrix(x[501:1000,],ncol=3)
AB1<-cbind(B[,1],A[,-1]) ## replace the first column of sample A by the first column of sample B
AB2<-cbind(A[,1],B[,2],A[,3]) ## replace the second column of sample A by the second column of sample B
AB3<-cbind(A[,-3],B[,3]) ## replace the third column of sample A by the third column of sample B
trial<-function(x){ ## define the trial function:
#x1+x2*x3^2
x[,1]+(x[,2])*(x[,3])^2
}
Y_A<-trial(A) ## the output of A
Y_B<-trial(B) ## the output of B
Y_AB1<-trial(AB1) ## the output of AB1
Y_AB2<-trial(AB2) ## the output of AB2
Y_AB3<-trial(AB3) ## the output of AB3
Y<-matrix(cbind(Y_A,Y_B),ncol=1) ## the matrix of total outputs
S1<-mean(Y_B*(Y_AB1-Y_A))/var(Y) ## first order effect of x1
St1<-(sum((Y_A-Y_AB1)^2)/(2*500))/var(Y) ## total order effect of x1
S2<-mean(Y_B*(Y_AB2-Y_A))/var(Y) ## first order effect of x2
St2<-(sum((Y_A-Y_AB2)^2)/(2*500))/var(Y) ## total order effect of x2
S3<-mean(Y_B*(Y_AB3-Y_A))/var(Y) ## first order effect of x3
St3<-(sum((Y_A-Y_AB3)^2)/(2*500))/var(Y) ## total order effect of x3
S<-matrix(c(S1,St1,S2,St2,S3,St3),nrow=2,ncol=3) ## define the results
matrix
rownames(S)<-list("first order","total")
colnames(S)<-list("X1","X2","X3")
S ## print result
The results are:
X1 X2 X3
first order 0.01734781 0.2337758 0.5261082
total 0.01523861 0.4078107 0.6471387
And then, I wanted to compute the theoretical results to validate the above estimated results. I follow the equations in the picture below:
In order to compute the partial variance, I approached it in two ways: monte carlo integration and doing the integration by hand ( I just don't trust computer...)
Here is the code:
pimc=function(n){ ## monte carlo integration
a=runif(n)
g=(a+mean(x_2)*(mean(x_3)^2))^2
pimc=mean(g)
pimc
}
pimc(10000)/var(Y)
##
# doing integration by hand
(1/3+mean(x_2)*(mean(x_3))^2+((mean(x_2))^2)*((mean(x_3))^4))/var(Y)
## first order effect x1
(mean(x_1)^2+mean(x_1)*mean(x_3)^2+(1/3)*mean(x_3)^4)/var(Y)
## first order effect x2
(mean(x_1)^2+(2/3)*(mean(x_1)*mean(x_2))+(1/5)*(mean(x_2)^2))/var(Y)
## first order effect x3
The results are the following:
[,1]
[1,] 0.4414232 ## first order indice of x1 computed by monte carlo integration
[,1]
[1,] 0.4414238 ## first order indice of x1 computed by hand
[,1]
[1,] 0.0505044 ## first order indice of x2 computed by hand
[,1]
[1,] 0.05086958 ## first order indice of x3 computed by hand
The results are constant by using monte-carlo integration and hand solving. But they are quite different from the estimated results. Why is this?
When we do partial integral, we only look at one parameter, and we treat other parameters as constant numbers. So I chose to take the mean of each parameter as the constant number. Is it wrong?