Time-Varying Transition Matrices with Multi-task Gaussian Processes
This is an incremental improvement for modeling individual mobility patterns, with potential applications in fields like transportation or urban planning.
The paper tackles the problem of modeling individual mobility states as a time-inhomogeneous Markov process with moves and pauses, using a multi-task Gaussian Process to approximate transition probabilities that vary with exogenous variables while enforcing stochasticity and non-negativity constraints. The results demonstrate the formulation's ability to learn functional forms of transition probabilities under these constraints.
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.