Stability of temporal statistics in Transition Path Theory with sparse data

TL;DR

This paper investigates the stability of transition path statistics in Markov chains with sparse data, proposing Voronoi cells from clustering to improve estimate stability of transition durations.

Contribution

It introduces a novel approach using Voronoi cells from k-means clustering to stabilize transition path statistics in sparse data scenarios.

Findings

Voronoi cells improve stability of transition path estimates

Stable estimates of total transition durations are achieved

New TPT statistic for remaining path duration is proposed

Abstract

Transition Path Theory (TPT) provides a rigorous statistical characterization of the ensemble of trajectories connecting directly, i.e., without detours, two disconnected (sets of) states in a Markov chain, a stochastic process that undergoes transitions from one state to another with probability depending on the state attained in the previous step. Markov chains can be constructed using trajectory data via counting of transitions between cells covering the domain spanned by trajectories. With sparse trajectory data, the use of regular cells is observed to result in unstable estimates of the total duration of transition paths. Using Voronoi cells resulting from k-means clustering of the trajectory data, we obtain stable estimates of this TPT statistic, which is generalized to frame the remaining duration of transition paths, a new TPT statistic suitable for investigating connectivity.