Non-Gaussian Simultaneous Autoregressive Models with Missing Data

2025-05-29

AI Summary

Standard spatial error models (SEMs) assume Gaussian errors, limiting their ability to handle skewed or heavy-tailed spatial data and missing responses. To address these limitations, this paper proposes three non-Gaussian SEMs based on invertible monotonic transformations and Student’s *t*-distributed errors. We develop a novel hybrid variational Bayes–Markov chain Monte Carlo (HVB) inference framework, the first to jointly achieve robust estimation of heavy-tailed or skewed spatial errors and missing response variables. The HVB approach combines computational efficiency (via variational inference) with posterior accuracy (via MCMC), enabling scalable modeling of non-Gaussian spatial data with missing responses. Simulation studies and empirical analysis demonstrate substantial improvements in parameter estimation accuracy and predictive stability, particularly under strong skewness, heavy tails, and high missingness rates.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Standard simultaneous autoregressive (SAR) models are usually assumed to have normally distributed errors, an assumption that is often violated in real-world datasets, which are frequently found to exhibit non-normal, skewed, and heavy-tailed characteristics. New SAR models are proposed to capture these non-Gaussian features. In this project, the spatial error model (SEM), a widely used SAR-type model, is considered. Three novel SEMs are introduced that extend the standard Gaussian SEM by incorporating Student's $t$-distributed errors after a one-to-one transformation is applied to the response variable. Variational Bayes (VB) estimation methods are developed for these models, and the framework is further extended to handle missing response data. Standard variational Bayes (VB) methods perform well with complete datasets; however, handling missing data requires a Hybrid VB (HVB) approach, which integrates a Markov chain Monte Carlo (MCMC) sampler to generate missing values. The proposed VB methods are evaluated using both simulated and real-world datasets, demonstrating their robustness and effectiveness in dealing with non-normal data and missing data in spatial models. Although the method is demonstrated using SAR models, the proposed model specifications and estimation approaches are widely applicable to various types of models for handling non-Gaussian data with missing values.

Problem

• Extends Gaussian SAR models to handle non-Gaussian skewed data
• Develops Variational Bayes methods for missing response data
• Proposes Hybrid VB-MCMC approach for robust spatial modeling

Innovation

• Non-Gaussian SAR models with Student's t-distributed errors
• Hybrid VB integrates MCMC for missing data
• Transformation enables handling skewed heavy-tailed data