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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.15146 |
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| _version_ | 1866915800548376576 |
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| author | Godoy, Lucas da Cunha Prates, Marcos Oliveira Quintana, Fernando Andrés Yan, Jun |
| author_facet | Godoy, Lucas da Cunha Prates, Marcos Oliveira Quintana, Fernando Andrés Yan, Jun |
| contents | Distances between sets arise naturally when modeling stochastic dependence on collections of spatial supports, including settings with point-referenced and areal observations. However, commonly used constructions of distances on sets, including those derived from the Hausdorff distance, generally fail to be conditionally negative definite, precluding their use in isotropic covariance models. We propose the ball--Hausdorff distance, defined as the Hausdorff distance between the minimum enclosing balls of bounded sets in a metric space. For length spaces, we derive an explicit representation of this distance in terms of the associated centers and radii. We show that the ball--Hausdorff distance is conditionally negative definite whenever the underlying metric is conditionally negative definite. By Schoenberg's theorem, this implies an isometric embedding into a Hilbert space and guarantees the validity of broad classes of isotropic covariance functions, including the Matérn and powered exponential families, for set-indexed random fields. The construction reduces dependence between sets to low-dimensional geometric summaries, leading to substantial simplifications in covariance evaluation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Validity of Isotropic Covariance Functions for Set-indexed Random Fields Godoy, Lucas da Cunha Prates, Marcos Oliveira Quintana, Fernando Andrés Yan, Jun Methodology 62M99 (Primary), 62M30 (Primary), 60K99 (Secondary) Distances between sets arise naturally when modeling stochastic dependence on collections of spatial supports, including settings with point-referenced and areal observations. However, commonly used constructions of distances on sets, including those derived from the Hausdorff distance, generally fail to be conditionally negative definite, precluding their use in isotropic covariance models. We propose the ball--Hausdorff distance, defined as the Hausdorff distance between the minimum enclosing balls of bounded sets in a metric space. For length spaces, we derive an explicit representation of this distance in terms of the associated centers and radii. We show that the ball--Hausdorff distance is conditionally negative definite whenever the underlying metric is conditionally negative definite. By Schoenberg's theorem, this implies an isometric embedding into a Hilbert space and guarantees the validity of broad classes of isotropic covariance functions, including the Matérn and powered exponential families, for set-indexed random fields. The construction reduces dependence between sets to low-dimensional geometric summaries, leading to substantial simplifications in covariance evaluation. |
| title | On the Validity of Isotropic Covariance Functions for Set-indexed Random Fields |
| topic | Methodology 62M99 (Primary), 62M30 (Primary), 60K99 (Secondary) |
| url | https://arxiv.org/abs/2502.15146 |