spatial dependence diagnostics

This study addresses the limitations of existing spatial dependence diagnostics—such as Moran’s I and APLE—in effectively assessing residual spatial autocorrelation after adjusting for large-scale trends and covariates. The authors propose RESAPLE, a novel first-order approximation estimator based on restricted maximum likelihood (REML) residuals, to accurately and efficiently estimate the spatial autocorrelation parameter ρ in spatial error models. By innovatively integrating the Rayleigh quotient formulation with REML residuals, RESAPLE combines the interpretability of exploratory spatial indicators with the rigor of statistical estimation, while also offering a diagnostic tool for selecting spatial weight matrices. Under correctly specified trend models and with moderate to small sample sizes, RESAPLE demonstrates superior estimation accuracy and testing power compared to both Moran’s I and APLE, and is applicable to both regular and irregular lattice data.

This study addresses the susceptibility of the conventional Moran’s I statistic to distributional outliers, which can introduce bias in both global and local spatial autocorrelation estimates. The authors systematically evaluate and compare three classes of robust estimation methods—plug-in robust estimators, trimmed least squares (TLS), and Theil–Sen–type estimators—for constructing robust Moran indices and LISA maps, accompanied by tailored visualization strategies. This work presents the first comprehensive comparison of multiple robust spatial association measures and proposes the Theil–Sen Moran estimator as a preferred default for exploratory spatial data analysis. Empirical results demonstrate its superior performance across diverse scenarios, while plug-in methods also exhibit strong scalability and reliability in large datasets, collectively enhancing the robustness of spatial outlier detection.

This study addresses the limitations of traditional parametric approaches in spatial regression, which are prone to inferential bias due to model misspecification and lack robust procedures for assessing covariate significance. The authors propose a fully nonparametric testing framework based on a Monte Carlo random shift scheme, integrating partial correlation concepts with variance correction. By first removing the influence of nuisance covariates, the method constructs a test statistic from the correlation between the target covariate and residuals. Crucially, it requires no assumptions about the functional form, spatial structure, or sampling distribution of the statistic, and is shown—under sample covariance—to achieve asymptotic exactness, a result established here for the first time. Numerical experiments demonstrate that the procedure accurately controls type I error rates, maintains competitive power when the parametric model is correctly specified, and exhibits superior computational stability.

Traditional geostatistical methods rely on second-order moments and Gaussian assumptions, which are inadequate for capturing non-Gaussian spatial dependence. This work addresses this limitation by leveraging Sklar’s theorem to introduce a copula-based framework that decouples marginal distributions from the spatial dependence structure. The authors systematically develop spatial copula models applicable at both fixed point sets and process levels, emphasizing Kolmogorov consistency to clarify distinctions between these two modeling paradigms. The framework is further extended to spatio-temporal settings and supports flexible marginal specifications. By integrating spatial statistics, copula theory, and stochastic processes, this study establishes a unified approach for modeling non-Gaussian spatial dependence, elucidating the relationships, strengths, and limitations of existing methodologies, thereby advancing both theoretical understanding and practical applications in the field.

This paper addresses the nonparametric testing of spatial dependence in two- and three-dimensional random fields. The proposed method maps spatial grid data onto a one-dimensional sequence via space-filling curves—specifically Hilbert and generalized Gilbert curves—and subsequently applies ordinal pattern-based statistical tests to detect dependence structures. Its key contribution lies in the first integration of locality-preserving space-filling curve mappings with nonparametric ordinal pattern analysis, thereby overcoming dimensional and grid-shape constraints inherent in conventional approaches. The framework supports arbitrary-size regular or irregular grids and extends naturally to higher dimensions. Experimental results demonstrate that the method is robust, computationally efficient, and achieves superior detection accuracy compared to existing spatial ordinal-pattern-based techniques, while maintaining conceptual simplicity and ease of implementation.

This study addresses the challenge that traditional spatial prediction models struggle to simultaneously maintain interpretability of covariates and flexibility in modeling complex spatial dependence structures. To overcome this limitation, the authors propose a semiparametric spatial autoregressive model that integrates linear covariate effects with nonparametrically estimated spatial components. This approach preserves model interpretability while flexibly capturing intricate spatial dependencies, thereby relaxing the strong assumptions on covariance structures commonly imposed by conventional models. The proposed method achieves both high predictive accuracy and strong interpretability, supported by a rigorous asymptotic theory. Empirical evaluations on both simulated and real-world datasets demonstrate that its predictive performance is comparable to that of geostatistical methods, while substantially outperforming classical spatial econometric models in terms of explanatory power.

This study addresses the limitations of traditional residual plot diagnostics—namely, their reliance on subjective human interpretation, low efficiency, and poor scalability—by introducing computer vision techniques for the first time to automate the assessment of residual plots in linear models. The authors develop the R package autovi and an accompanying Shiny-based interactive web application, autovi.web. Their approach leverages deep visual models to quantify the strength of structural signals in residual plots and produces interpretable diagnostic metrics. This methodology significantly enhances the consistency and efficiency of model fit evaluation, offering statisticians and data analysts a robust, objective, and scalable tool for automated diagnostic assessment in statistical modeling.

This study addresses the lack of effective nonparametric tests for spatial independence in irregularly sampled point clouds by extending the ordinal pattern test framework—originally developed for regular lattices—to arbitrary irregular spatial supports. The proposed method constructs symbolic representations of ordinal patterns from neighboring observations and, combined with an additive log-ratio (ALR) transformation, yields an asymptotically pivotal test statistic that requires no assumptions on marginal distributions and involves no redundant parameters. Leveraging the invariance of ordinal patterns under monotonic transformations and their robustness to outliers, the theoretical validity of the test is established via a central limit theorem for graph-dependent processes under α-mixing conditions. Monte Carlo experiments demonstrate that the test accurately controls Type I error rates even at moderate sample sizes and exhibits stable, monotonically increasing power against both linear and nonlinear spatial dependencies.

This study addresses the underestimation of regression standard errors caused by spatial autocorrelation and the lack of systematic guidance for bandwidth selection in existing spatial HAC (heteroskedasticity and autocorrelation consistent) methods. The authors propose a nonparametric, data-driven bandwidth selection procedure based on the empirical covariogram of residuals for spatial HAC standard error estimation. Their analysis reveals an inverted U-shaped relationship between bandwidth and standard errors, challenging the conventional wisdom that larger bandwidths are inherently more conservative, and provides the first formalized bandwidth selection framework. Monte Carlo simulations across diverse spatial structures and sample configurations demonstrate that the method—particularly with Bartlett or Epanechnikov kernels—maintains empirical size close to the nominal 5% level. The approach is further validated using U.S. county-level data, and the accompanying R package SpatialInference is publicly available.