- #1

- 163

- 1

My problem is as follows: The material says that the points which should be used and their weight can be described the following way:

(LaTex notation)

$A(l, d)= \sum_{l+1 \leq |i| \leq l+1} (-1)^{l+d-|i|} {d-1 \choose l+d-|i|}(U^{i_1} \otimes ... \otimes U^{i_d})

(Regular notation)

∑

_{(l+1≤|i|≤l+d)}(-1)

^{(l+d-|i|)}(d-1 l+d-|i|)(U

^{i1}⊗...⊗U

^{id})

Where d is the number of random variables, 3, l is the level, 1, |i| is the graded lexicographic order, (U

^{i1}⊗...⊗U

^{id}) are the tensor product of the nodes for each dimension and (d-1 l+d-|i|) is a binomial coefficient.

Also, μ=f(ξ

^{(1)}ξ

^{(2)})°(ω), which means that the mean of the process is equal to the sum of all the realizations multiplied by their weight.

Then, if l+1=2 and l+d =4, I should add all the terms which are listed in the lexicographic matrix with |i|=2, 3 and 4. However, since the terms which have a zero as one of the index don't have nodes, these are discarted, leaving me with (1 1 1), (2 1 1) and (1 1 2), which in total produce 7 realizations. However, if I use these realizations, my mean looks signifficantly different from the realizations as can be seen in the figure.

For what I have read, the amount of realizations that should be used is way higher than 7; however, I cannot find where my mistake is.

Any answer is appreciaded.

Thanks.