Learning Network-Structured Dependence From Non-Stationary Multivariate Point Process Data

Learning the sparse network-structured dependence among nodes from multivariate point process data $\{{ {T}}_{i}\}_{i\in \mathcal {V} }$ has wide applications in information transmission, social science, and computational neuroscience. This paper develops new continuous-time stochastic models of the conditional intensity functions $\{\lambda _{i}(t | \mathscr {F} _{t}): t\ge 0\}_{i\in \mathcal {V} }$ , dependent on past event counts of parent nodes, to uncover the network structure within an array of non-stationary multivariate counting processes $\{\boldsymbol {N}(t): t\ge 0\}$ for $\{{ {T}}_{i}\}_{i\in \mathcal {V} }$ . The stochastic mechanism is crucial for statistical inference of graph parameters relevant to structure recovery but does not satisfy the key assumptions of commonly used processes like the Poisson process, Cox process, Hawkes process, queuing model, and piecewise deterministic Markov process. We introduce a new marked point process for intensity discontinuities, derive compact representations of their conditional distributions, and demonstrate the cyclicity property of ${N}(t)$ driven by recurrence time points. These new theoretical properties enable us to establish statistical consistency and convergence properties of the proposed penalized M-estimators for graph parameters under mild regularity conditions. Simulation evaluations demonstrate computational simplicity and increased estimation accuracy compared to existing methods. Real multiple neuron spike train recordings are analyzed to infer connectivity in neuronal networks.