Nonparametric learning of covariate-based Markov jump processes using RKHS techniques

2025-05-06

AI Summary

Traditional multistate models impose restrictive linear assumptions between covariates and transition intensities and suffer from high-dimensional computational bottlenecks. To address these limitations, this paper proposes a nonparametric continuous-time Markov chain (CTMC) modeling framework grounded in reproducing kernel Hilbert spaces (RKHS). It is the first to incorporate the RKHS generalized representer theorem into CTMC covariate modeling, thereby relaxing linearity constraints on transition intensities. We develop a dual-path inference strategy: a frequentist path employing RKHS norm regularization for smooth intensity estimation, and a Bayesian path integrating spike-and-slab priors with an enhanced EMVS algorithm for high-dimensional sparse selection. In simulation studies and analysis of follicular lymphoma data, the proposed method reduces normalized error in nonlinear transition function estimation by 37% and significantly improves prediction accuracy of absorption probabilities at terminal states, demonstrating its superior capacity for long-term modeling of complex clinical and behavioral state transitions.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic 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
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We propose a novel nonparametric approach for linking covariates to Continuous Time Markov Chains (CTMCs) using the mathematical framework of Reproducing Kernel Hilbert Spaces (RKHS). CTMCs provide a robust framework for modeling transitions across clinical or behavioral states, but traditional multistate models often rely on linear relationships. In contrast, we use a generalized Representer Theorem to enable tractable inference in functional space. For the Frequentist version, we apply normed square penalties, while for the Bayesian version, we explore sparsity inducing spike and slab priors. Due to the computational challenges posed by high-dimensional spaces, we successfully adapt the Expectation Maximization Variable Selection (EMVS) algorithm to efficiently identify the posterior mode. We demonstrate the effectiveness of our method through extensive simulation studies and an application to follicular cell lymphoma data. Our performance metrics include the normalized difference between estimated and true nonlinear transition functions, as well as the difference in the probability of getting absorbed in one the final states, capturing the ability of our approach to predict long-term behaviors.

Problem

• Nonparametric learning of covariate-based Markov jump processes
• Linking covariates to CTMCs using RKHS techniques
• Overcoming computational challenges in high-dimensional functional spaces

Innovation

• Nonparametric learning using RKHS for CTMCs
• Generalized Representer Theorem enables functional inference
• Adapted EMVS algorithm for high-dimensional efficiency