Universal Modelling of Autocovariance Functions via Spline Kernels

2025-06-27

AI Summary

Traditional parametric autocovariance function (ACF) models lack flexibility and struggle to simultaneously achieve nonparametric expressiveness and closed-form analytic tractability. Method: We propose a general nonparametric ACF framework based on spectral-domain modeling. Our core innovation is the first derivation of the analytical inverse Fourier transform of B-spline spectral bases, yielding an ACF class that is explicitly representable, L¹-dense, and supports multidimensional and nonseparable structures. Leveraging Bochner’s theorem and Jackson-type approximation theory, we establish theoretically grounded error bounds. Contribution/Results: The method unifies spectral flexibility with time-domain interpretability. Experiments demonstrate high-fidelity recovery of weakly stationary, mean-square continuous processes; the model exhibits strong universality and practicality on both synthetic and real-world spatiotemporal datasets.

Technology Category

• Machine Learning: Kernel Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Develops flexible non-parametric autocovariance function models
• Ensures closed-form ACF expressions via spline spectral bases
• Supports univariate, multivariate, and multidimensional processes

Innovation

• B-spline spectral bases for ACF modeling
• Closed-form non-parametric ACF via Fourier transform
• Supports multivariate and non-separable structures