Volterra--Wiener--Kunchenko Orthogonalization: From Wiener--Hermite to Distribution-Matched Volterra Bases

Abstract

The monomial parameterization of finite-memory Volterra identification is ill-conditioned under non-Gaussian input, and the Wiener--Hermite expansion removes this ill-conditioning only for Gaussian white-noise input. We construct the distribution-matched Volterra--Wiener--Kunchenko (VWK) basis by oriented Gram--Schmidt orthogonalization of monomials in $L^2(P)$ and use it as an arbitrary-polynomial-chaos coordinate system for finite-memory Volterra identification from data, following the generalized polynomial chaos of Xiu and Karniadakis (2002) and the data-driven arbitrary polynomial chaos of Oladyshkin and Nowak (2012). The basis itself is classical; the contribution is the Volterra-estimation reading. First, an order-2 misspecification-penalty theorem shows that a self-normalized diagonal estimator in the variance-matched Gaussian basis incurs an excess $L^2(P)$ risk governed by the skew coefficient $δ=μ_3/σ^2$, vanishing exactly for symmetric inputs. Second, conditioning experiments separate the constructional fact that the population matched Gram is the identity from the finite-sample design Gram: at $n=2000$, the centered-exponential empirical VWK Gram remains far better conditioned than the power Gram, although it degrades with degree. Third, a machine-checked Lean 4 proof establishes the Binomial$(N,p)$ Krawtchouk row for arbitrary $N$. Full least squares over a fixed span is basis-invariant, so VWK stabilizes diagonal cross-correlation and regularized coordinate fits rather than claiming universal prediction superiority. The analysis is moment-based, finite-memory, and restricted to product input laws.