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A Refined Generalization Analysis for Extreme Multi-class Supervised Contrastive Representation Learning Ensemble Distributionally Robust Bayesian Optimisation The Proxy Presumption: From Semantic Embeddings to Valid Social Measures Modulated learning for private and distributed regression with just a single sample per client device Query-efficient model evaluation using cached responses Functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks Optimal Experiments for Partial Causal Effect Identification Order-Agnostic Autoregressive Modelling with Missing Data Grokking or Glitching? How Low-Precision Drives Slingshot Loss Spikes Tuning Derivatives for Causal Fairness in Machine Learning Spherical Flows for Sampling Categorical Data Bayesian Rain Field Reconstruction using Commercial Microwave Links and Diffusion Model Priors GRALIS: A Unified Canonical Framework for Linear Attribution Methods via Riesz Representation Sharp Capacity Thresholds in Linear Associative Memory: From Winner-Take-All to Listwise Retrieval Unified Framework of Distributional Regret in Multi-Armed Bandits and Reinforcement Learning Jacobian-Velocity Bounds for Deployment Risk Under Covariate Drift Self-Attention as Transport: Limits of Symmetric Spectral Diagnostics Perturbation is All You Need for Extrapolating Language Models Adapt or Forget: Provable Tradeoffs Between Adam and SGD in Nonstationary Optimization Realizable Bayes-Consistency for General Metric Losses Graph Convolutional Support Vector Regression for Robust Spatiotemporal Forecasting of Urban Air Pollution Segmenting Human-LLM Co-authored Text via Change Point Detection Stochastic Schrödinger Diffusion Models for Pure-State Ensemble Generation Understanding Self-Supervised Learning via Latent Distribution Matching The Geometric Mechanics of Contrastive Representation Learning: Alignment Potentials, Entropic Dispersion, and Cross-modal Divergence Imbalanced Classification under Capacity Constraints On the Spectral Structure and Objective Equivalence of Orthogonal Multilabel Fisher Discriminants Partially Observed Structural Causal Models First-Order Efficiency for Probabilistic Value Estimation via A Statistical Viewpoint Robust and Fast Training via Per-Sample Clipping
Non-stationary Spatial Modeling Using Fractional SPDEs
[Submitted on 21 Dec 2025 (v1), last revised 24 Jun 2026 (this v · 2026-06-26 · via stat updates on arXiv.org

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Abstract:We construct a Gaussian random field (GRF) that combines fractional smoothness with spatially varying anisotropy. The GRF is defined through a stochastic partial differential equation (SPDE), where the range, marginal variance, and anisotropy vary spatially according to a spectral parametrization of the SPDE coefficients. Priors are constructed to reduce overfitting in this flexible covariance model, and we estimate parameters using a gradient-based optimization approach based on automatic differentiation. In a simulation study, we investigate how many observations are required to reliably estimate fractional smoothness and non-stationarity, and find that one realization containing 500 observations or more is needed in the scenario considered. We also find that the proposed penalization prevents overfitting across varying numbers of observation locations. Two case studies demonstrate that the relative importance of fractional smoothness and non-stationarity is application dependent. Non-stationarity improves predictions in an application to ocean salinity, whereas fractional smoothness improves predictions in an application to precipitation. Predictive ability is assessed using mean squared error and the continuous ranked probability score. In addition to prediction, the proposed approach can be used as a tool to explore the presence of fractional smoothness and non-stationarity.

Submission history

From: Elling Svee [view email]
[v1] Sun, 21 Dec 2025 15:15:30 UTC (3,322 KB)
[v2] Wed, 24 Jun 2026 18:38:15 UTC (6,568 KB)