Determining a purely symbolic transfer function from symbol streams: Theory and algorithms

Transfer function modeling is a standard technique in classical linear time invariant and statistical process control. The work of Box and Jenkins was seminal in developing methods for identifying parameters associated with classical (r, s, k) transfer functions. Discrete event systems are often used for modeling hybrid control structures and high-level decision problems. Examples include discrete time, discrete strategy repeated games. For these games, a discrete transfer function in the form of an accurate hidden Markov model of input-output relations could be used to derive optimal response strategies. In this paper, we develop an algorithm for creating probabilistic Mealy machines that act as transfer function models for discrete event dynamic systems (DEDS). Our models are defined by three parameters, (l₁,l₂,k) just as the Box-Jenkins transfer function models. Here h is the maximal input history lengths to consider, l₂ is the maximal output history lengths to consider and k is the response lag. Using related results, We show that our Mealy machine transfer functions are optimal in the sense that they maximize the mutual information between the current known state of the DEDS and the next observed input/output pair.