Chaos expansions
Polynomial chaos, generalized polynomial chaos, arbitrary chaos etc
May 20, 2020 — February 14, 2021
Placeholder, for a topic which has a slightly confusing name. To explore: Connection to/difference from other methods of keeping track of the evolution of uncertainty in dynamical systems. C&C Gaussian process regression as used in Gratiet, Marelli, and Sudret (2016), functional data analysis etc.
This is not the same thing as chaos in the sense of the deterministic chaos made famous by dynamical systems theory and fractal t-shirts. Different term
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 \[ \left\langle \psi, \phi\right\rangle=\int \psi(\xi) \phi(\xi) p_{\xi}(\xi) d \xi \] with respect to the probability density function \(p_{\xi}\) of some random variate \(\Xi\) which we call the germ.
Suppose we have chosen a functional basis comprising polynomials \(\psi_{0}=1, \psi_{1}, \psi_{2}, \ldots,\) where \(\psi_{j}\) is a polynomial of order \(j\) and where they satisfy the orthogonality condition that for all \(j \neq k\) \[ \left\langle\psi_{j}, \psi_{k}\right\rangle=0. \]
We write \[ X=f(\Xi)=\sum_{j=0}^{\infty} x_{j} \psi_{j}(\Xi) \] 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 \[ x_{j}=\langle f, \psi j\rangle /\left\langle\psi_{j}, \psi_{j}\right\rangle \] An expansion of \(X\) in this form is called a polynomial chaos expansion. Since there are many possible functions \(f\) for given \(X\) and \(\Xi\) 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, \[ X_{p}=f_{p}(\Xi)=\sum_{j=0}^{p} x_{j} \psi_{j}(\Xi). \]
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.