# Chaos expansions

## Polynomial chaos, generalized polynomial chaos, arbitrary chaos etc Placeholder, for a topic which has a slightly confusing name. To explore: Connection to/difference from other methods of keeping track of evolution of uncertainty in dynamical systems. C&C Gaussian process regression as used in , 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

## 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 . For my money, the most direct is 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 propagation of error through a model.

There is a veritable zoo of bases to consider.

Here associated with means orthogonal with respect to.

## “Generalized” chaos expansion

Wikipedia credits Xiu (2010) with the particular generalisation 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.

## Arbitrary chaos expansion

Learnable sparse basis-style chaos expansions. See .

As far as I can tell from brusque mentions about the place, these methods construct polynomial basis expansions weighted by the empirical distribution of observationss. 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.

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