A calculus of stochastic systems: For the specification, simulation, and hidden state estimation of hybrid stochastic/non-stochastic systems

Abstract

In this paper, we consider hybrid systems containing both stochastic and non-stochastic 3 components. To compose such systems, we introduce a general combinator which allows the specification of an arbitrary hybrid system in terms of elementary primitives of only two types. Thus, systems are obtained hierarchically, by composing subsystems, where each subsystem can be viewed as an “increment” in the decomposition of the full system. The resulting hybrid stochastic system specifications are generally not “executable”, since they do not necessarily permit the incremental simulation of the system variables. Such a simulation requires compiling the dependency relations existing between the system variables. Another issue involves finding the most likely internal states of a stochastic system from a set of observations. We provide a small set of primitives for transforming hybrid systems, which allows the solution of the two problems of incremental simulation and estimation of stochastic systems within a common framework. The complete model is called CSS (a Calculus of Stochastic Systems), and is implemented by the SIG language, derived from the SIGNAL synchronous language. Our results are applicable to pattern recognition problems formulated in terms of Markov random fields or hidden Markov models (HMMs), and to the automatic generation of diagnostic systems for industrial plants starting from their risk analysis. A full version of this paper is available [1], omitted proofs can be found in this reference.