Polynomial chaos (PC), also called polynomial chaos expansion (PCE) and Wiener chaos expansion, is a method for representing a random variable in terms of a polynomial function of other random variables. The polynomials are chosen to be orthogonal with respect to the joint probability distribution of these random variables. Note that despite its name, PCE has no immediate connections to chaos theory. The word "chaos" here should be understood as "random".^[1]

PCE was first introduced in 1938 by Norbert Wiener using Hermite polynomials to model stochastic processes with Gaussian random variables.^[2] It was introduced to the physics and engineering community by R. Ghanem and P. D. Spanos in 1991^[3] and generalized to other orthogonal polynomial families by D. Xiu and G. E. Karniadakis in 2002.^[4] Mathematically rigorous proofs of existence and convergence of generalized PCE were given by O. G. Ernst and coworkers in 2011.^[5]

PCE has found widespread use in engineering and the applied sciences because it makes possible to deal with probabilistic uncertainty in the parameters of a system. In particular, PCE has been used as a surrogate model to facilitate uncertainty quantification analyses.^[6][7] PCE has also been widely used in stochastic finite element analysis^[3] and to determine the evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters.^[8]

Main principles

Polynomial chaos expansion (PCE) provides a way to represent a random variable ${\displaystyle Y}$ with finite variance (i.e., ${\displaystyle \operatorname {Var} (Y)<\infty }$) as a function of an ${\displaystyle M}$-dimensional random vector ${\displaystyle \mathbf {X} }$, using a polynomial basis that is orthogonal to the distribution of this random vector. The prototypical PCE can be written as:

${\displaystyle Y=\sum _{i\in \mathbb {N} }c_{i}\Psi _{i}(\mathbf {X} ).}$

In this expression, ${\displaystyle c_{i}}$ is a coefficient and ${\displaystyle \Psi _{i}}$ denotes a polynomial basis function. Depending on the distribution of ${\displaystyle \mathbf {X} }$, different PCE types are distinguished.

Polynomial chaos and incomplete statistical information

In many practical situations, only incomplete and inaccurate statistical knowledge on uncertain input parameters are available. Fortunately, to construct a finite-order expansion, only some partial information on the probability measure is required that can be simply represented by a finite number of statistical moments. Any order of expansion is only justified if accompanied by reliable statistical information on input data. Thus, incomplete statistical information limits the utility of high-order polynomial chaos expansions.^[11]

Polynomial chaos and non-linear prediction

Polynomial chaos can be utilized in the prediction of non-linear functionals of Gaussian stationary increment processes conditioned on their past realizations.^[12] Specifically, such prediction is obtained by deriving the chaos expansion of the functional with respect to a special basis for the Gaussian Hilbert space generated by the process that with the property that each basis element is either measurable or independent with respect to the given samples. For example, this approach leads to an easy prediction formula for the Fractional Brownian motion.

Bayesian polynomial chaos

In a non-intrusive setting, the estimation of the expansion coefficients ${\displaystyle c_{i}}$ for a given set of basis functions ${\displaystyle \Psi _{i}}$ can be considered as a Bayesian regression problem by constructing a surrogate model. This approach has benefits in that analytical expressions for the data evidence (in the sense of Bayesian inference) as well as the uncertainty of the expansion coefficients are available.^[13] The evidence then can be used as a measure for the selection of expansion terms and pruning of the series (see also Bayesian model comparison). The uncertainty of the expansion coefficients can be used to assess the quality and trustworthiness of the PCE, and furthermore the impact of this assessment on the actual quantity of interest ${\displaystyle Y}$.

Let ${\displaystyle D=\{\mathbf {X} ^{(j)},Y^{(j)}\}}$ be a set of ${\displaystyle j=1,...,N_{s}}$ pairs of input-output data that is used to estimate the expansion coefficients ${\displaystyle c_{i}}$. Let ${\displaystyle M}$ be the data matrix with elements ${\displaystyle [M]_{ij}=\Psi _{i}(\mathbf {X} ^{(j)})}$, let ${\displaystyle {\vec {Y}}=(Y^{(1)},...,Y^{(j)},...,Y^{(N_{s})})^{T}}$ be the set of ${\displaystyle N_{s}}$ output data written in vector form, and let be ${\displaystyle {\vec {c}}=(c_{1},...,c_{i},...,c_{N_{p}})^{T}}$ the set of expansion coefficients in vector form. Under the assumption that the uncertainty of the PCE is of Gaussian type with unknown variance and a scale-invariant prior, the expectation value ${\displaystyle \langle \cdot \rangle }$ for the expansion coefficients is

${\displaystyle \langle {\vec {c}}\rangle =(M^{T}\;M)^{-1}\;M^{T}\;{\vec {Y}}}$

With ${\displaystyle H=(M^{T}M)^{-1}}$, then the covariance of the coefficients is^[13]

${\displaystyle {\text{Cov}}(c_{m},c_{n})={\frac {\chi _{\text{min}}^{2}}{N_{s}-N_{p}-2}}H_{m,n}}$

where ${\displaystyle \chi _{\text{min}}^{2}={\vec {Y}}^{T}(\mathrm {I} -M\;H^{-1}M^{T})\;{\vec {Y}}}$is the minimal misfit and ${\displaystyle \mathrm {I} }$ is the identity matrix. The uncertainty of the estimate for the coefficient ${\displaystyle n}$ is then given by ${\displaystyle {\text{Var}}(c_{m})={\text{Cov}}(c_{m},c_{m})}$.Thus the uncertainty of the estimate for expansion coefficients can be obtained with simple vector-matrix multiplications. For a given input propability density function ${\displaystyle p(\mathbf {X} )}$, it was shown the second moment for the quantity of interest then simply is^[13]

${\displaystyle \langle Y^{2}\rangle =\underbrace {\sum _{m,m'}\int \Psi _{m}(\mathbf {X} )\Psi _{m'}(\mathbf {X} )\langle c_{m}\rangle \langle c_{m'}\rangle p(\mathbf {X} )\;dV_{\mathbf {X} }} _{=I_{1}}+\underbrace {\sum _{m,m'}\int \Psi _{m}(\mathbf {X} )\Psi _{m'}(\mathbf {X} )\;{\text{Cov}}(c_{m},c_{m'})\;p(\mathbf {X} )\;dV_{\mathbf {X} }} _{=I_{2}}}$

This equation amounts the matrix-vector multiplications above plus the marginalization with respect to ${\displaystyle \mathbf {X} }$. The first term ${\displaystyle I_{1}}$ determines the primary uncertainty of the quantity of interest ${\displaystyle Y}$, as obtained based on the PCE used as a surrogate. The second term ${\displaystyle I_{2}}$ constitutes an additional inferential uncertainty (often of mixed aleatoric-epistemic type) in the quantity of interest ${\displaystyle Y}$ that is due to a finite uncertainty of the PCE.^[13] If enough data is available, in terms of quality and quantity, it can be shown that ${\displaystyle {\text{Var}}(c_{m})}$ becomes negligibly small and becomes small ^[13] This can be judged by simply building the ratios of the two terms, e.g. ${\displaystyle {\frac {I_{1}}{I_{1}+I_{2}}}}$.This ratio quantifies the amount of the PCE's own uncertainty in the total uncertainty and is in the interval ${\displaystyle [0,1]}$. E.g., if ${\displaystyle {\frac {I_{1}}{I_{1}+I_{2}}}\approx 0.5}$, then half of the uncertainty stems from the PCE itself, and actions to improve the PCE can be taken or gather more data. If${\displaystyle {\frac {I_{1}}{I_{1}+I_{2}}}\approx 1}$, then the PCE's uncertainty is low and the PCE may be deemed trustworthy.

In a Bayesian surrogate model selection, the probability for a particular surrogate model, i.e. a particular set ${\displaystyle S}$ of expansion coefficients ${\displaystyle c_{i}}$ and basis functions ${\displaystyle \Psi _{i}}$ , is given by the evidence of the data ${\displaystyle Z_{S}}$,

${\displaystyle Z_{S}=\Omega _{N_{p}}\mid H\mid ^{-1/2}(\chi _{\text{min}}^{2})^{-{\frac {N_{s}-N_{p}}{2}}}{\frac {\Gamma {\big (}{\frac {N_{p}}{2}}{\big )}\Gamma {\big (}{\frac {N_{s}-N_{p}}{2}}{\big )}}{\Gamma {\big (}{\frac {N_{s}}{2}}{\big )}}}}$

where ${\displaystyle \Gamma }$ is the Gamma-function, ${\displaystyle \mid H\mid }$ is the determinant of ${\displaystyle H}$, ${\displaystyle N_{s}}$ is the number of data, and ${\displaystyle \Omega _{N_{p}}}$is the solid angle in ${\displaystyle N_{p}}$dimensions, where ${\displaystyle N_{p}}$is the number of terms in the PCE.

Analogous findings can be transferred to the computation of PCE-based sensitivity indices. Similar results can be obtained for Kriging.^[13]

See also

• Orthogonal polynomials
• Surrogate model
• Variance-based sensitivity analysis
• Karhunen–Loève theorem
• Hilbert space
• Proper orthogonal decomposition
• Bayesian regression
• Bayesian model comparison

References