Polynomial Chaos Expansion

Definition

It is a way to represent Random Variables with a general probability distribution through polynomials over random variables with well known distributions like Gaussian, Uniform etc.

Let $X$ be a random variable with arbitrary $\mathrm{PDF}{f}_X$, for which the mean value and variance exist $(\mathbb{E}[X],\mathbb{V}[X]<\infty )$. Using the PC Expansion

$$X(\theta )=\sum _{i=0}^{\infty }{q}_i{\Phi }_i(\xi (\theta ))$$ $$\boxed{X=\sum _{i=0}^{\infty }{q}_i{\Phi }_i(\xi )}$$

where,

• $\xi \sim \mathcal{N}(0,1)$
• ${\Phi }_i\to$ Hermite polynomials.

Hermit Polynomial	Value
$H_0(\xi )$	1
$H_1(\xi )$	$\xi$
$H_2(\xi )$	${\xi }^2-1$
$H_3(\xi )$	${\xi }^3-3\xi$
$H_4(\xi )$	${\xi }^4-6{\xi }^2+3$
…	…

Note

• Each $\xi$ term is called a germ.
• The Hermite Polynomials are orthogonal to the the germ.

Example

Random Variable

$$X\sim \mathcal{N}(2,1)$$

Polynomial Chaos expansion

$$X=2+\xi$$

Where, $\xi \sim \mathcal{N}(0,1)$