Calculating Sobol Indices

Example 1

Given a random vector $\mathbf{X}=\left(X_1,X_2\right)$ and a model $\mathcal{M}(\mathbf{X})$, a PCE approximation has already been derived as

$$\tilde{\mathcal{M}}(\mathbf{X}(\mathbf{\xi }))=\sum _{i=0}^3{q}_i\Phi (\mathbf{X}(\mathbf{\xi })).$$

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

$$\begin{array}{rl}& {\Phi }_0=1\\ & {\Phi }_1={\xi }_1\\ & {\Phi }_2={\xi }_2\\ & {\Phi }_3={\xi }_1{\xi }_2\end{array}$$

Compute the first order and total Sobol indices.

Solution

First Order Sobol Indices

$$S_i=\frac{D_i}{V[M(X)]}$$

1. Total Variance for PCE

   $$\mathbb{V}[\tilde{M}]=\sum _{i=1}^3{q}_i^2=q_1^2+q_2^2+q_3^2$$

2. Solo-Effect Term: Only ${\Phi }_1={\xi }_1$ corresponds to the solo effect of the first input.

3. Partial Variance Square of the the contributing coefficient ⇒$q_1^2$.

   $$S_1=\frac{D_1}{V[M(X)]}=\frac{q_1^2}{q_1^2+q_2^2+q_3^2}$$

Total Effect

1. Polynomials involving ${\xi }_1$ → ${\Phi }_1={\xi }_1$ (the solo effect term) & ${\Phi }_3={\xi }_1{\xi }_2$ (the interaction term).

2. Partial Variance: $q_1^2+q_3^2$

3. Total Effect Index

   $$S_{T,1}=\frac{q_1^2+q_3^2}{{\sum }_{i=1}^3{q}_i^2}$$