Universal Modelling of Autocovariance Functions via Spline Kernels

TL;DR

This paper introduces a new non-parametric class of autocovariance functions (ACFs) using spline kernels, enabling flexible, closed-form modeling of stationary processes with proven density and convergence properties.

Contribution

It develops a novel, closed-form non-parametric ACF class based on inverse Fourier transforms of B-spline spectral bases, supporting multivariate and non-separable structures.

Findings

Achieves dense approximation in the space of weakly stationary ACFs.

Supports multivariate and multidimensional processes.

Demonstrates accurate process recovery on simulated and real data.

Abstract

Flexible modelling of the autocovariance function (ACF) is central to time-series, spatial, and spatio-temporal analysis. Modern applications often demand flexibility beyond classical parametric models, motivating non-parametric descriptions of the ACF. Bochner's Theorem guarantees that any positive spectral measure yields a valid ACF via the inverse Fourier transform; however, existing non-parametric approaches in the spectral domain rarely return closed-form expressions for the ACF itself. We develop a flexible, closed-form class of non-parametric ACFs by deriving the inverse Fourier transform of B-spline spectral bases with arbitrary degree and knot placement. This yields a general class of ACF with three key features: (i) it is provably dense, under an $L^1$ metric, in the space of weakly stationary, mean-square continuous ACFs with mild regularity conditions; (ii) it accommodates univariate, multivariate, and multidimensional processes; and (iii) it naturally supports non-separable structure without requiring explicit imposition. Jackson-type approximation bounds establish convergence rates, and empirical results on simulated and real-world data demonstrate accurate process recovery. The method provides a practical and theoretically grounded approach for constructing a non-parametric class of ACF.