Example 1

Given a random vector and a model , a PCE approximation has already been derived as

Consider the coefficients as given. The polynomials are also given as

Compute the first order and total Sobol indices.

Solution

First Order Sobol Indices

  1. Total Variance for PCE
  1. Solo-Effect Term: Only corresponds to the solo effect of the first input.
  2. Partial Variance Square of the the contributing coefficient .

Total Effect

  1. Polynomials involving (the solo effect term) & (the interaction term).
  2. Partial Variance:
  3. Total Effect Index