🤖 AI Summary
This study addresses the ill-conditioning issue in finite-memory Volterra models under non-Gaussian inputs, which arises from monomial parameterization. To mitigate this, the authors propose performing input-distribution-matched Gram–Schmidt orthogonalization of monomials in the \(L^2(P)\) space, thereby constructing a Volterra–Wiener–Kunchenko (VWK) basis. This approach interprets the VWK basis as a generalized polynomial chaos coordinate system for data-driven Volterra system identification. The work establishes, for the first time, a second-order misspecification penalty theorem and formally verifies—using Lean 4—the Krawtchouk polynomial structure under arbitrary-order binomial distributions. Experimental results demonstrate that the VWK basis substantially improves the condition number of the basis matrix, with the excess risk governed by the input skewness coefficient \(\delta\); notably, this risk vanishes exactly for symmetric inputs and outperforms conventional power bases under finite-sample regimes.
📝 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.