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

TL;DR

This paper introduces a nonparametric RKHS-based method for modeling covariate-dependent Markov jump processes, enabling flexible, nonlinear transition function estimation with efficient computation and demonstrated effectiveness on simulated and real clinical data.

Contribution

It develops a novel RKHS framework for nonparametric covariate-dependent CTMCs, incorporating both frequentist and Bayesian approaches with efficient variable selection.

Findings

Accurate estimation of nonlinear transition functions.

Effective long-term behavior prediction in clinical data.

Robust performance demonstrated through simulations and lymphoma data.

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.