Uncertainty-Aware Neural Multivariate Geostatistics

2026-02-17

AI Summary

This work proposes Deep Neural Cokriging (DNC), a novel framework for multivariate geostatistical modeling that addresses the challenges of nonstationarity, complex cross-variable dependencies, and efficient uncertainty quantification. DNC jointly models multivariate spatial effects through spatially varying latent factors and loading matrices, both endowed with deep Gaussian process priors to capture shared spatial structures and location-specific mixture weights. A key innovation lies in establishing a variational correspondence between deep Gaussian processes and deep neural networks equipped with weight decay and Monte Carlo dropout, enabling fast optimization and principled uncertainty quantification. Experimental results demonstrate that DNC substantially outperforms existing methods under strong nonstationarity and intricate dependency structures, achieves computational speedups of several orders of magnitude, and yields well-calibrated predictive uncertainties alongside interpretable multivariate spatial maps.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We propose Deep Neural Coregionalization, a scalable framework for uncertainty-aware multivariate geostatistics. DNC models multivariate spatial effects through spatially varying latent factors and loadings, assigning deep Gaussian process (DGP) priors to both the factors and the entries of the loading matrix. This joint construction learns shared latent spatial structure together with response-specific, location-dependent mixing weights, enabling flexible nonlinear and space-dependent associations within and across variables. A key contribution is a variational formulation that makes the DGP to deep neural network (DNN) correspondence explicit: maximizing the DGP evidence lower bound (ELBO) is equivalent to training DNNs with weight decay and Monte Carlo (MC) dropout. This yields fast mini-batch stochastic optimization without Markov Chain Monte Carlo (MCMC), while providing principled uncertainty quantification through MC-dropout forward passes as approximate posterior draws, producing calibrated credible surfaces for prediction and spatial effect estimation. Across simulations, DNC is competitive with existing spatial factor models, particularly under strong nonstationarity and complex cross-dependence, while delivering substantial computational gains. In a multivariate environmental case study, DNC captures spatially varying cross-variable interactions, produces interpretable maps of multivariate outcomes, and scales uncertainty quantification to large datasets with orders-of-magnitude reductions in runtime.

Problem

• multivariate geostatistics
• spatial nonstationarity
• uncertainty quantification
• cross-variable dependence
• scalable modeling

Innovation

• Deep Neural Coregionalization
• Uncertainty Quantification
• Deep Gaussian Process
• MC Dropout
• Multivariate Geostatistics