Chaos expansions

1 Polynomial chaos expansion

Ground Zero, the original famous one. Wikipedia has an inscrutable introduction which I can make no sense of.

Popular introductions to the concept seem to be (R. Ghanem and Spanos 1990; R. G. Ghanem and Spanos 2003; Witteveen and Bijl 2006; Xiu and Karniadakis 2002; Kim et al. 2013). For my money, the most direct is O’Hagan (2013) which runs thusly: We define an inner product $$⟨\psi ,\varphi ⟩=\int \psi (\xi )\varphi (\xi )p_{\xi }(\xi )d\xi$$ with respect to the probability density function $p_{\xi }$ of some random variate $\mathrm{\Xi }$ which we call the germ.

Suppose we have chosen a functional basis comprising polynomials ${\psi }_0=1,{\psi }_1,{\psi }_2,\dots ,$ where ${\psi }_j$ is a polynomial of order $j$ and where they satisfy the orthogonality condition that for all $j\ne k$ $$⟨{\psi }_j,{\psi }_k⟩=0.$$

We write $$X=f(\mathrm{\Xi })=\sum _{j=0}^{\mathrm{\infty }}{x}_j{\psi }_j(\mathrm{\Xi })$$ The combination of $x_j$ (the mode strength) and ${\psi }_j$ (the mode function) is called the $j$-th mode. By orthogonality, given $f$ and the ${\psi }_j$s there is a unique expansion in which the mode strengths are given by $$x_j=⟨f,\psi j⟩/⟨{\psi }_j,{\psi }_j⟩$$ An expansion of $X$ in this form is called a polynomial chaos expansion. Since there are many possible functions $f$ for given $X$ and $\mathrm{\Xi }$ distributions, there are many possible expansions of a given $X$ using a given germ. They will differ in the mode strengths.

In practice we truncate the expansions to a finite number $p$ of terms, $$X_p=f_p(\mathrm{\Xi })=\sum _{j=0}^p{x}_j{\psi }_j(\mathrm{\Xi }).$$

C&C Karhunen—Loève expansion.

It is not clear immediately, but this gives us a tool to track the propagation of error through a model.

There is a veritable zoo of bases to consider.

Here associated with means orthogonal with respect to.

2 “Generalized” chaos expansion

Wikipedia credits Xiu (2010) with the particular generalization which apparently got naming rights for generalized chaos expansion, in the teeth of my private campaign for a moratorium on naming anything generalized [whatever]. I think it is about expanding the list of acceptable polynomial bases? TBC.

3 Arbitrary chaos expansion

Learnable sparse basis-style chaos expansions. See (Witteveen and Bijl 2006; Zhang et al. 2019; Lei et al. 2018; Oladyshkin and Nowak 2012; Wan and Karniadakis 2006; Zheng, Wan, and Karniadakis 2015).

As far as I can tell from brusque mentions about the place, these methods construct polynomial basis expansions weighted by the empirical distribution of observations. That is, instead of taking a given germ, we use an empirical estimate of a germ and calculate a basis over it by the Gram-Schmidt procedure.

4 References