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Observable Operator Models for Discrete Stochastic Time Series

German National Research Center for Information Technology, Institute for Intelligent Autonomous Systems, D-53754 Sankt Augustin, Germany

A widely used class of models for stochastic systems is hidden Markov models. Systems that can be modeled by hidden Markov models are a proper subclass of linearly dependent processes, a class of stochastic systems known from mathematical investigations carried out over the past four decades. This article provides a novel, simple characterization of linearly dependent processes, called observable operator models. The mathematical properties of observable operator models lead to a constructive learning algorithm for the identification of linearly dependent processes. The core of the algorithm has a time complexity of O(N+nm³), where N is the size of training data, n is the number of distinguishable outcomes of observations, and m is model state-space dimension.