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Deep Probabilistic Spatial Modeling for Multivariate Mixed-Type Responses

Yeseul Jeon, Kyeong Eun Lee, Joon Jin Song

TL;DR

MultiDeepGP is developed, a scalable and statistically principled framework for joint modeling of multivariate mixed outcomes in spatial settings that introduces a shared latent spatial component that governs cross-outcome dependence while allowing outcome-specific distributions.

Abstract

Many scientific applications involve mixed spatially indexed outcomes of heterogeneous types that are driven by shared latent mechanisms. Modeling such data is challenging due to complex, nonlinear, and potentially nonstationary spatial dependence, as well as the need for coherent joint inference across mixed outcome distributions. Existing multivariate mixed outcome models often rely on restrictive linear assumptions, while recent deep learning approaches emphasize predictive flexibility but typically lack coherent joint modeling and uncertainty quantification for spatial data. We develop MultiDeepGP, a scalable and statistically principled framework for joint modeling of multivariate mixed outcomes in spatial settings. The proposed approach introduces a shared latent spatial component that governs cross-outcome dependence while allowing outcome-specific distributions. Spatial dependence and nonlinear structure are captured through a deep latent representation, and uncertainty quantification is enabled via an efficient Monte Carlo-based inference strategy. This construction balances modeling flexibility with probabilistic interpretability and computational feasibility. The proposed method is evaluated through simulation studies designed to reflect key challenges in mixed outcome spatial modeling, as well as an application to georeferenced environmental and public health data from the African Great Lakes region. The results demonstrate that the proposed framework provides accurate joint prediction and reliable uncertainty quantification in complex spatial settings.

Deep Probabilistic Spatial Modeling for Multivariate Mixed-Type Responses

TL;DR

MultiDeepGP is developed, a scalable and statistically principled framework for joint modeling of multivariate mixed outcomes in spatial settings that introduces a shared latent spatial component that governs cross-outcome dependence while allowing outcome-specific distributions.

Abstract

Many scientific applications involve mixed spatially indexed outcomes of heterogeneous types that are driven by shared latent mechanisms. Modeling such data is challenging due to complex, nonlinear, and potentially nonstationary spatial dependence, as well as the need for coherent joint inference across mixed outcome distributions. Existing multivariate mixed outcome models often rely on restrictive linear assumptions, while recent deep learning approaches emphasize predictive flexibility but typically lack coherent joint modeling and uncertainty quantification for spatial data. We develop MultiDeepGP, a scalable and statistically principled framework for joint modeling of multivariate mixed outcomes in spatial settings. The proposed approach introduces a shared latent spatial component that governs cross-outcome dependence while allowing outcome-specific distributions. Spatial dependence and nonlinear structure are captured through a deep latent representation, and uncertainty quantification is enabled via an efficient Monte Carlo-based inference strategy. This construction balances modeling flexibility with probabilistic interpretability and computational feasibility. The proposed method is evaluated through simulation studies designed to reflect key challenges in mixed outcome spatial modeling, as well as an application to georeferenced environmental and public health data from the African Great Lakes region. The results demonstrate that the proposed framework provides accurate joint prediction and reliable uncertainty quantification in complex spatial settings.
Paper Structure (14 sections, 27 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 27 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Scientific Motivation and Data Description
  3. Environmental--health coupling in the African Great Lakes region
  4. Model
  5. Nonlinear shared representation using neural networks
  6. Deep Gaussian Process Models with Mixed Likelihoods via Neural Networks
  7. Inference
  8. Predictive distribution
  9. Simulation Study
  10. Competitors and Metric of Comparison
  11. Simulation Case 1: One Dimensional Stationary Gaussian Process with Mixed Outcomes
  12. Simulation Case 2: Two Dimensional Nonstationary Surface with Mixed Outcomes
  13. Coupled Environmental–Health Prediction in the African Great Lakes Region
  14. Discussion

Figures (3)

  • Figure 1: Illustration of the proposed MultiDeepGP framework. Conditional on the shared latent representation, outcomes are assumed independent, while dependence across responses is induced through the common deep latent spatial process.
  • Figure 2: Shared spatial surface summarizing common latent effects across outcomes.The figure displays a proxy for the shared latent spatial structure learned by MultiDeepGP. Colors indicate the relative strength and direction of the shared spatial influence, with higher values corresponding to locations where all outcomes are jointly elevated relative to their respective means. The surface is obtained by interpolating pointwise predictions at cross-validation locations onto a regular grid for visualization. Black dots denote observation locations.
  • Figure 3: Pointwise predictive diagnostics for the African Great Lakes real data. Each panel displays MultiDeepGP predictions evaluated at cross-validation locations. From top to bottom: binary vegetation indicator, malaria incidence (count), and water availability (continuous). The plots enable a direct spatial comparison between observed outcomes, predictive means, and local prediction errors across heterogeneous outcome types.