Universal Modelling of Autocovariance Functions via Spline Kernels
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908425421586432 |
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| author | Astfalck, Lachlan |
| author_facet | Astfalck, Lachlan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21953 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universal Modelling of Autocovariance Functions via Spline Kernels Astfalck, Lachlan Methodology Machine Learning 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. |
| title | Universal Modelling of Autocovariance Functions via Spline Kernels |
| topic | Methodology Machine Learning |
| url | https://arxiv.org/abs/2506.21953 |