# Universal Modelling of Autocovariance Functions via Spline Kernels

arXiv: https://arxiv.org/abs/2506.21953 · HTML: https://mlsanity.com/paper/2506.21953/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 29 references that are in the map.
- First posted on arXiv 27 Jun 2025.

## Builds on (most cited first, 5 of 5)

- [Spectral Mixture Kernels for Multi-Output Gaussian Processes](https://mlsanity.com/paper/1709.01298/) (2017)
- [Band-Limited Gaussian Processes: The Sinc Kernel](https://mlsanity.com/paper/1909.07279/) (2019)
- [The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain](https://mlsanity.com/paper/2103.06950/) (2021)
- [Debiasing Welch's Method for Spectral Density Estimation](https://mlsanity.com/paper/2312.13643/) (2023)
- [Bias correction of quadratic spectral estimators](https://mlsanity.com/paper/2410.12386/) (2024)

## Abstract (arXiv)

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 L1 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.