Time-Varying Transition Matrices with Multi-task Gaussian Processes

TL;DR

This paper introduces a multi-task Gaussian Process model for estimating time-varying transition probabilities in a two-state Markov process, capturing correlations and exogenous influences on mobility states.

Contribution

The paper presents a novel kernel-based multi-task GP approach that models dynamic transition probabilities with constraints, improving mobility state analysis.

Findings

Successfully enforces stochasticity and non-negativity constraints.

Learns functional form of transition probabilities over time.

Demonstrates efficiency gains using Toeplitz and Kronecker structures.

Abstract

In this paper, we present a kernel-based, multi-task Gaussian Process (GP) model for approximating the underlying function of an individual's mobility state using a time-inhomogeneous Markov Process with two states: moves and pauses. Our approach accounts for the correlations between the transition probabilities by creating a covariance matrix over the tasks. We also introduce time-variability by assuming that an individual's transition probabilities vary over time in response to exogenous variables. We enforce the stochasticity and non-negativity constraints of probabilities in a Markov process through the incorporation of a set of constraint points in the GP. We also discuss opportunities to speed up GP estimation and inference in this context by exploiting Toeplitz and Kronecker product structures. Our numerical experiments demonstrate the ability of our formulation to enforce the desired constraints while learning the functional form of transition probabilities.