regTPS-KLE: A Novel Approach To Approximate A Gaussian Random Field for Bayesian Spatial Modeling
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912839699005440 |
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| author | Cavieres, Joaquin Krumscheid, Sebastian |
| author_facet | Cavieres, Joaquin Krumscheid, Sebastian |
| contents | Gaussian random field is a ubiquitous model for spatial phenomena in diverse scientific disciplines. Its approximation is often crucial for computational feasibility in simulation, inference, and uncertainty quantification. The Karhunen-Loève Expansion provides a theoretically optimal basis for representing a Gaussian random field as a sum of deterministic orthonormal functions weighted by uncorrelated random variables. While this is a well-established method for dimension reduction and approximation of (spatial) stochastic processes, its practical application depends on the explicit or implicit definition of the covariance structure. In this work, we propose a novel approach, referred to as regTPS-KLE, for approximating a Gaussian random field by explicitly constructing its covariance via a regularized thin plate spline (TPS) kernel. Because TPS kernels are conditionally positive definite and lack a direct spectral decomposition, we formulate the covariance as the inverse of a regularized elliptic operator. To evaluate its statistical performance, we compare its predictive accuracy and computational efficiency with a Gaussian random field approximation constructed using the stochastic partial differential equations (SPDE) method and implemented within an MCMC algorithm. In simulation studies, the predictive differences between the SPDE and regTPS-KLE models were minimal when the spatial field was generated using Matèrn and exponential covariance functions, while regTPS-KLE models consistently outperformed the SPDE approach in terms of computational efficiency. In a real data application, regTPS-KLE exhibits superior predictive accuracy compared with SPDE models based on leave-one-out cross-validation while also achieving improved computational efficiency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | regTPS-KLE: A Novel Approach To Approximate A Gaussian Random Field for Bayesian Spatial Modeling Cavieres, Joaquin Krumscheid, Sebastian Computation Gaussian random field is a ubiquitous model for spatial phenomena in diverse scientific disciplines. Its approximation is often crucial for computational feasibility in simulation, inference, and uncertainty quantification. The Karhunen-Loève Expansion provides a theoretically optimal basis for representing a Gaussian random field as a sum of deterministic orthonormal functions weighted by uncorrelated random variables. While this is a well-established method for dimension reduction and approximation of (spatial) stochastic processes, its practical application depends on the explicit or implicit definition of the covariance structure. In this work, we propose a novel approach, referred to as regTPS-KLE, for approximating a Gaussian random field by explicitly constructing its covariance via a regularized thin plate spline (TPS) kernel. Because TPS kernels are conditionally positive definite and lack a direct spectral decomposition, we formulate the covariance as the inverse of a regularized elliptic operator. To evaluate its statistical performance, we compare its predictive accuracy and computational efficiency with a Gaussian random field approximation constructed using the stochastic partial differential equations (SPDE) method and implemented within an MCMC algorithm. In simulation studies, the predictive differences between the SPDE and regTPS-KLE models were minimal when the spatial field was generated using Matèrn and exponential covariance functions, while regTPS-KLE models consistently outperformed the SPDE approach in terms of computational efficiency. In a real data application, regTPS-KLE exhibits superior predictive accuracy compared with SPDE models based on leave-one-out cross-validation while also achieving improved computational efficiency. |
| title | regTPS-KLE: A Novel Approach To Approximate A Gaussian Random Field for Bayesian Spatial Modeling |
| topic | Computation |
| url | https://arxiv.org/abs/2510.04256 |