Quadrature to Approximate Expected Value

The general formula is an approximation based on a weighted sum of the model evaluated at specific points:

$$\mathbb{E}[\mathcal{M}(X)]={\int }_{{\Xi }_X}\mathcal{M}(x)f_X(x)\mathrm{d}x\approx \sum _{i=1}^{n_{\text{qu }}}{w}^{(i)}\mathcal{M}\left(x^{(i)}\right)$$

• $x^{(i)}$ → the integration points or nodes
• $w^{(i)}$ → the integration weights

• ${\int }_{{\Xi }_x}$ → represents integral over the support
• $\mathcal{M}$ → model function

Key Principle

The key principle is to achieve the highest possible accuracy for integrating polynomials. A Gauss rule with $n_{qu}$ points can exactly integrate polynomials of degree $2n_{qu}-1$ or less.

The nodes are chosen as the roots of polynomials that are orthogonal with respect to the input variable’s PDF ($f_X(x)$) as the weighting function. Note:

• Points ${\left(x^{(i)}\right)}_{i=1}^{n_{qu}}$ are chosen as roots of polynomials ${\Phi }_i$ satisfying:

  $${\int }_{{\Xi }_{\chi }}{\Phi }_i(x){\Phi }_j(x)f_{\chi }(x)\mathrm{d}x\propto {\delta }_{ij}$$