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,

center