non-Intrusive Surrogate Modeling

From Orthogonal Projection the following analogies are drawn

Linear algebra	Probability
Dot product	Expectation
$\text{a}$	Model $Y=\mathcal{M}(\mathbf{\zeta })$
${\text{c}}^{\text{a}}$	Surrogate $\tilde{\mathcal{M}}(\mathbf{\zeta })$
plane $\mathcal{P}$	gPC space
basis vectors ${\text{b}}_i$	gPC basis ${\Phi }_i$

$\therefore$ gPC Coefficient is given by

$$q_i=\frac{\mathbb{E}\left[\mathcal{M}(\xi ){\Phi }_i(\xi )\right]}{\mathbb{E}\left[{\Phi }_i(\xi {)}^2\right]}$$

Projected gPC expansion

$$\mathcal{Y}\approx \tilde{\mathcal{M}}(\xi )=\sum _{i=0}^n\frac{\mathbb{E}\left[\mathcal{M}(\xi ){\Phi }_i(\xi )\right]}{\mathbb{E}\left[{\Phi }_i(\xi {)}^2\right]}{\Phi }_i(\xi )$$

where ${\text{c}}^{\text{a}}$ is the projected point given by $\left[c^a=\frac{\left(a\cdot b_1\right)}{b_1\cdot b_1}{b}_1+\frac{\left(a\cdot b_2\right)}{b_2\cdot b_2}{b}_2\right]$