Definition
It is a way to express Random Variables as a series expansion using Orthogonal polynomials of simpler random variables (called the germ)
Formula
Let be a random variable with arbitrary , for which the mean value and variance exist . Using the generalized Polynomial Chaos (gPC) Expansion
Where,
- are deterministic coefficients (also called PC coefficients). They encode important information about distribution and act as weights.
- are Orthogonal Polynomials (e.g.Hermite polynomials)
- is the germ
- Each term is called a germ.
- The Polynomials are orthogonal to the the germ.
Note: The choice of the Orthogonal Polynomials (), depends on the distribution of the germ . According to the Askey Scheme
Example
Random Variable
Polynomial Chaos expansion
Where,