  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.
