Restricted Spatial Regression is Reasonable Statistical Practice: Clarifications, Interpretations, and New Developments

Jonathan R. Bradley

Home/Archive/arXiv

arXiv · 2408.05106

Restricted Spatial Regression is Reasonable Statistical Practice: Clarifications, Interpretations, and New Developments

Jonathan R. Bradley

View original source ↗Read source document ↗Export metadata ↓

Abstract

The spatial linear mixed model (SLMM) consists of fixed and spatial random effects that may be linearly dependent. Partially motivated as a means to address potential issues with confounding, the Restricted spatial regression (RSR) model restricts spatial random effects to be in the orthogonal column space of the covariates. Recent articles have shown that the misspecified Bayesian RSR generally performs worse than the SLMM when the data is generated from the SLMM. However, we show that the misspecified Bayesian RSR model's marginal posterior distribution is equivalent up to a reparameterization to that of the SLMM's marginal posterior distribution, under a certain prior assumption on the orthogonalized regression coefficients. This suggests that the RSR models are not sub-optimal as the subsequent Bayesian analysis can be interpreted as a type of SLMM Bayesian analysis. This equivalence relationship is developed further in the context of unmeasured confounders and nonlinearity, where we explore a semi-parametric property of the orthogonalized regression effects. Several results are provided to demonstrate new benefits of an RSR. In particular, we provide new results that show that the RSR can produce clear computational advantages via a direct sampler from the posterior distribution for all hyperparameters, fixed effects, and random effects. Additionally, a transfer learning approach offers a new interpretation to orthogonalized regression coefficients, which we show empirically can improve inference on dependent regression coefficients in the presence of spatial confounding. Simulations and an illustration using COVID-19 mortality data are provided.

Explore related subjects

stat.ME

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jonathan R. Bradley. 2026-03-22. Restricted Spatial Regression is Reasonable Statistical Practice: Clarifications, Interpretations, and New Developments. https://arxiv.org/abs/2408.05106

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Likelihood Based Inference in Fully and Partially Observed Exponential Family Graphical Models with Intractable Normalizing Constants

Probabilistic graphical models that encode an underlying Markov random field are fundamental building blocks of generative modeling to learn latent representations in modern multivariate data sets with complex dependency structures. Among these, the exponential family graphical models are especially popular, given their fairly well-understood statistical properties and computational scalability to high-dimensional data based on pseudo-likelihood methods. These models have been successfully applied in many fields, such as the Ising model in statistical physics and count graphical models in genomics. Another strand of models allows some nodes to be latent, so as to allow the marginal distribution of the observable nodes to depart from exponential family to capture more complex dependence. These approaches form the basis of generative models in artificial intelligence, such as the Boltzmann machines and their restricted versions. A fundamental barrier to likelihood-based (i.e., both maximum likelihood and fully Bayesian) inference in both fully and partially observed cases is the intractability of the likelihood. The usual workaround is via adopting pseudo likelihood-based approaches, following the pioneering work of Besag(1974). The goal of this paper is to demonstrate that full likelihood-based analysis of these models is feasible in a computationally efficient manner under a logarithmically sparse setting. The chief innovation lies in utilizing the tractable independence model underlying an intractable graphical model, to estimate the normalizing constant, as well as its gradient. Extensive numerical results, supporting theory and comparisons with pseudo likelihood-based approaches demonstrate the applicability of the proposed method.

stat.ME↗

arXiv2026

Interpretable Deep Neural Network for Modeling Functional Surrogates

Developing surrogates for computer models has become increasingly important for addressing complex problems in science and engineering. This article introduces an artificial intelligent (AI) surrogate, referred to as the DeepSurrogate, for analyzing functional outputs with vector-valued inputs. The relationship between the functional output and vector-valued input is modeled as an infinite sequence of unknown functions, each representing the relationship at a specific location within the functional domain. These spatially indexed functions are expressed through a combination of basis functions and their corresponding coefficient functions, both of which are modeled using deep neural networks (DNN). The proposed framework accounts for spatial dependencies across locations, while capturing the relationship between the functional output and scalar predictors. It also integrates a Monte Carlo (MC) dropout strategy to quantify prediction uncertainty, enhancing explainability in the deep neural network architecture. The proposed method enables efficient inference on datasets with approximately 50,000 spatial locations and 20 simulations, achieving results in under 10 minutes using standard hardware. The approach is validated on extensive synthetic datasets and a large-scale simulation from the Sea Lake and Overland Surge from Hurricanes (SLOSH) simulator. An open-source Python package implementing the method is made available.

stat.ME↗

arXiv2026

Deep Generative Modeling with Spatial and Network Images: An Explainable AI (XAI) Approach

This article addresses the challenge of modeling the amplitude of spatially indexed low frequency fluctuations (ALFF) in resting state functional MRI as a function of cortical structural features and a multi-task coactivation network in the Adolescent Brain Cognitive Development (ABCD) Study. It proposes a generative model that integrates effects of spatially-varying inputs and a network-valued input using deep neural networks to capture complex non-linear and spatial associations with the output. The method models spatial smoothness, accounts for subject heterogeneity and complex associations between network and spatial images at different scales, enables accurate inference of each images effect on the output image, and allows prediction with uncertainty quantification via Monte Carlo dropout, contributing to one of the first Explainable AI (XAI) frameworks for heterogeneous imaging data. The model is highly scalable to high-resolution data without the heavy pre-processing or summarization often required by Bayesian methods. Empirical results demonstrate its strong performance compared to existing statistical and deep learning methods. We applied the XAI model to the ABCD data which revealed associations between cortical features and ALFF throughout the entire brain. Our model performed comparably to existing methods in predictive accuracy but provided superior uncertainty quantification and faster computation, demonstrating its effectiveness for large-scale neuroimaging analysis. Open-source software in Python for XAI is available.

stat.ME↗