Abstract: The apparatus, systems, and methods described herein may operate to receive information identifying and describing at least one of a set of events, an initial distribution of a plurality of states, an initial transition matrix, or an initial event matrix; generate, based at least in part on the information, at least one intermediate transition matrix and at least one intermediate event matrix describing a sparse Baum-Welch training that allows no event to occur at one or more time steps; and transform the at least one intermediate transition matrix and the at least one intermediate event matrix into a transition matrix and an event matrix describing a continuous-time Baum-Welch training, the continuous-time Baum-Welch training allowing events to occur simultaneously or at sporadic time intervals in a Markov model including a hidden Markov Model (HMM) having more than two hidden states.

Abstract: The apparatus, systems, and methods described herein may operate to receive information identifying and describing at least one of a set of events, an initial distribution of a plurality of states, an initial transition matrix, or an initial event matrix; generate, based at least in part on the information, at least one intermediate transition matrix and at least one intermediate event matrix describing a sparse Baum-Welch training that allows no event to occur at one or more time steps; and transform the at least one intermediate transition matrix and the at least one intermediate event matrix into a transition matrix and an event matrix describing a continuous-time Baum-Welch training, the continuous-time Baum-Welch training allowing events to occur simultaneously or at sporadic time intervals in a Markov model including a hidden Markov Model (HMM) having more than two hidden states.