spatial basis function modeling

Existing spatial functional regression models neglect spatial dependence, while functional kriging—though capable of predicting unobserved location curves—lacks parametric modeling and Bayesian inference capabilities. To address this, we propose the first hierarchical Bayesian function-on-function regression framework that jointly accounts for intra-curve, inter-curve, and spatial dependencies. We introduce a novel synchronized band scoring method to identify statistically significant regions of influence for the regression functions. Our approach integrates basis expansion for dimension reduction, an extension of functional kriging that avoids specifying variogram functions, and a tailored MCMC algorithm to mitigate slow mixing induced by high-dimensional parameters. Experiments on areal and point-referenced data demonstrate that our method is computationally efficient, yields accurate parameter estimates, and achieves significantly higher predictive accuracy than state-of-the-art alternatives.

Traditional spatial modeling relies on stationarity and isotropy assumptions, which increasingly fail in real-world applications; meanwhile, existing nonstationary approaches often sacrifice either flexibility or computational efficiency. To address this, we propose a covariate-driven, modular covariance function—built upon the Matérn kernel—that enables parameterized, decoupled modeling of multiple sources of nonstationarity, including marginal standard deviation, geometric anisotropy, and smoothness. Our formulation ensures interpretability, computational scalability, and intuitive visualizability. By incorporating a covariate-adaptive construction mechanism and a large-scale adaptation algorithm, the method achieves superior predictive performance over state-of-the-art nonstationary models in both simulation studies and real-world analysis of Swiss precipitation data. It simultaneously attains high model fidelity and low computational overhead, demonstrating robust scalability to large spatial datasets.

To address the challenges of basis function selection and modeling observation error correlations in functional data smoothing, this paper proposes a fast Bayesian basis selection method that supports both univariate and multivariate functional data jointly. It is the first to explicitly model the correlation structure of observational errors within a Bayesian framework. The method employs variational inference combined with an EM algorithm, replacing conventional MCMC sampling to achieve substantial computational speedup without compromising estimation accuracy. Extensive evaluations on simulated data and real-world applications—including motorcycle acceleration and Canadian weather datasets—demonstrate significant improvements in adjusted R², accurate recovery of basis coefficients, optimal basis sets, and within-curve correlation patterns. An open-source R implementation is publicly available.

Existing spatial spectral analysis methods are constrained by data types (e.g., point processes, lattice fields, irregularly sampled processes) and domain structures (limited to regular grids). To address these limitations, this paper proposes a unified multitaper spectral estimation framework. Methodologically, it introduces, for the first time, a theoretical framework coupling discrete and continuous taper windows, thereby relaxing classical Fourier-based assumptions of Cartesian domains and uniform sampling. It establishes rigorous asymptotic and finite-sample statistical foundations for partial spectral coherence estimation and significance testing. The framework integrates multitaper windowing, tapered discrete Fourier transforms, and efficient computational algorithms. Empirical validation on large-scale ecological datasets demonstrates robust estimation of cross-spectral associations among heterogeneous spatial processes—spanning point patterns, gridded fields, and irregular samples—while delivering interpretable, statistically principled inference.

In spatial prediction tasks—such as weather forecasting and pollution modeling—the validation and prediction locations are fixed and non-overlapping, violating the i.i.d. assumption underlying conventional validation methods (including those correcting for covariate shift), which presume stochastic sampling rather than deterministic spatial sampling. This work formally introduces the notion of *validation consistency*: as the density of validation locations tends to infinity, the validation error must converge arbitrarily closely to the true prediction error. Building upon this principle, we propose the first theoretically guaranteed consistent spatial validation framework, integrating spatial sampling theory with weighted density estimation to accommodate both gridded and irregularly spaced observational structures. We prove its consistency under mild regularity conditions. Empirical evaluation on meteorological and air pollution datasets demonstrates that our method significantly outperforms standard cross-validation and importance-weighting baselines, achieving an average 37% reduction in estimation error.

This work proposes a multiresolution cokriging model to address the challenge of sparse or missing observations in multivariate spatial prediction. By jointly modeling latent effects across multiple correlated spatial processes, the method enables information sharing and collaborative prediction. It integrates spatial basis functions with Gaussian Markov random field coefficients, explicitly capturing cross-process dependence without assuming global stationarity, while preserving the computational efficiency of fixed-rank kriging. Parameter estimation is carried out via an expectation–maximization algorithm. Simulation studies demonstrate that the approach effectively leverages densely observed variables to improve prediction accuracy in regions with sparse or no observations. The practical utility of the method is further validated through an application to PM10 concentration prediction in northern Italy.

This study addresses the lack of systematic comparisons among effective approaches for handling spatial autocorrelation in random forests. Integrating machine learning with geostatistical principles, it presents the first unified evaluation of four strategies—Gaussian process augmentation, observation-driven correlation structures, spatial basis functions, and local geographical fitting—to enhance the spatial predictive performance of random forests. Through both simulation experiments and an empirical analysis of air pollution in Blantyre, Malawi, the research demonstrates that while no single method consistently outperforms others across all scenarios, spatial basis functions exhibit robust and superior performance throughout. These findings offer practical and reliable guidance for spatial modeling of environmental processes using random forests.

This study addresses the limitations of traditional spatial prediction methods on spherical domains, which often suffer from distance distortion due to reliance on Euclidean metrics or planar projections and struggle to scale to large datasets. To overcome these challenges, the authors propose Spherical DeepKriging—a novel framework that integrates intrinsic spherical thin-plate spline basis functions with deep learning to flexibly capture complex spatial structures on the sphere. This approach represents the first fusion of native spherical basis functions and neural networks for scalable geostatistical modeling. It effectively circumvents the constraints of classical kriging under global-scale and big-data scenarios, demonstrating superior predictive accuracy and computational scalability over existing methods in both synthetic and real-world global datasets.

This study addresses the challenges of high computational complexity and spurious dependencies among irrelevant variables in estimating covariance structures for high-dimensional multivariate spatial Gaussian random fields. The authors propose a sparse estimation framework based on LASSO penalization, which, for the first time, incorporates sparse regularization into the Cholesky factor of the multivariate Matérn covariance matrix. This approach simultaneously ensures positive semi-definiteness and automatically identifies conditionally independent variable pairs. A composite likelihood-based objective function is formulated and efficiently optimized via a projected block coordinate descent algorithm. Empirical results demonstrate that the method accurately recovers the underlying sparse dependence structure, substantially reduces estimation error, and successfully enables spatial prediction on a geochemical dataset comprising 36 variables observed at 3,998 spatial locations—a task infeasible for conventional methods.