Observable operator models for discrete stochastic time series

A widely used class of models for stochastic systems is hidden Markov model s. Systems that can be modeled by hidden Markov models are a proper subclas s of linearly dependent processes, a class of stochastic systems known from mathematical investigations carried out over the past four decades. This a rticle provides a novel, simple characterization of linearly dependent proc esses, called observable operator models. The mathematical properties of ob servable operator models lead to a constructive learning algorithm for the identification of linearly dependent processes. The core of the algorithm h as a time complexity of O(N + nm(3)), where N is the size of training data, n is the number of distinguishable outcomes of observations, and m is mode l state-space dimension.