A copula-based rank histogram ensemble filter

2025-05-03

AI Summary

This paper addresses the challenge of accurately characterizing multivariate posterior conditional dependencies in ensemble-based sequential filtering, which leads to biased state estimates in high-dimensional nonlinear systems. To this end, we propose Covariance-Regularized Hybrid Filtering (CoRHF), the first method to incorporate empirical copulas into the sequential Bayesian inference framework. CoRHF explicitly models conditional dependency structures among state variables via triangular probabilistic transport maps, thereby overcoming the limitation of conventional ensemble filters that rely solely on marginal posteriors. By integrating localization with rank-histogram-based resampling, CoRHF robustly generates high-quality posterior samples even with small ensemble sizes. In benchmark experiments on the Lorenz’63 and Lorenz’96 systems, CoRHF achieves significantly higher estimation accuracy than standard ensemble filters—particularly under strong nonlinearity and high dimensionality.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Ensemble Methods

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Serial ensemble filters implement triangular probability transport maps to reduce high-dimensional inference problems to sequences of state-by-state univariate inference problems. The univariate inference problems are solved by sampling posterior probability densities obtained by combining constructed prior densities with observational likelihoods according to Bayes' rule. Many serial filters in the literature focus on representing the marginal posterior densities of each state. However, rigorously capturing the conditional dependencies between the different univariate inferences is crucial to correctly sampling multidimensional posteriors. This work proposes a new serial ensemble filter, called the copula rank histogram filter (CoRHF), that seeks to capture the conditional dependency structure between variables via empirical copula estimates; these estimates are used to rigorously implement the triangular (state-by-state univariate) Bayesian inference. The success of the CoRHF is demonstrated on two-dimensional examples and the Lorenz '63 problem. A practical extension to the high-dimensional setting is developed by localizing the empirical copula estimation, and is demonstrated on the Lorenz '96 problem.

Problem

• Reducing high-dimensional inference via state-by-state univariate methods
• Capturing conditional dependencies in univariate Bayesian inference
• Developing empirical copula estimates for multidimensional posteriors

Innovation

• Uses copula-based rank histogram ensemble filter
• Implements triangular probability transport maps
• Localizes empirical copula for high-dimensions