# 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 *mixed 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 mixed system in terms of elementary components 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 mixed 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 dependencyrelations 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 mixed 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 will appear in *Theoretical Computer Science*, and is available [1].

## Keywords

stochastic systems mixed systems belief functions communicating processes simulation estimation## Preview

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## References

- 1.A. Benveniste, B. Levy, E. Fabre, and P. L. Guernic, “A calculus of stochastic systems: specification, simulation, and hidden state estimation,” Tech. Rep. 2465, INRIA, Rocquencourt, France, January 1995.Google Scholar
- 2.N. Viswanadham and Y. Narahari,
*Performance Modeling of Automated Manufacturing Systems*. Englewood Cliffs, NJ: Prentice Hall, 1992.Google Scholar - 3.L. R. Rabiner and B. H. Juang, “An introduction to hidden Markov models,”
*IEEE ASSP Magazine*, vol. 3, pp. 4–16, Jan. 1986.Google Scholar - 4.S. Geman and D. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,”
*IEEE Trans. on Pattern Analysis and Machine Intelligence*, vol. 6, pp. 721–741, Nov. 1984.Google Scholar - 5.R. C. Dubes and A. K. Jain, “Random field models in image analysis,”
*J. Applied Stat.*, vol. 12, pp. 131–164, 1989.Google Scholar - 6.G. W. Hart, “Nonintrusive applicance load monitoring,”
*Proc. IEEE*, vol. 80, pp. 1870–1891, Dec. 1992.Google Scholar - 7.M. Basseville and I. V. Nikiforov,
*Detection of Abrupt Changes: Theory and Applications*. Englewood Cliffs, NJ: Prentice Hall, 1993.Google Scholar - 8.T. Soderstrom and P. Stoica,
*System Identification*. Englewood Cliffs, NJ: Prentice Hall, 1989.Google Scholar - 9.M. Molloy, “Performance analysis using stochastic Petri nets,”
*IEEE Trans. Computers*, vol. 31, pp. 913–917, Sep. 1982.Google Scholar - 10.B. Plateau and K. Atif, “Stochastic automata network for modeling parallel systems,”
*IEEE Trans. on Software Engineering*, vol. 17, pp. 1093–1108, Oct. 1991.Google Scholar - 11.B. Plateau and J. Fourneau, “A methodology for solving Markov models of parallel systems,”
*J. Parallel and Distributed Comput.*, vol. 12, pp. 370–387, 1991.Google Scholar - 12.R. van Glabbeek, S. A. Smolka, B. Steffen, and C. Tofts, “Reactive, generative, and stratified models of probabilistic processes,” in
*Proc. 5th IEEE Int. Symp. on Logic in Computer Science*, (Philadelphia,PA), pp. 130–141, June 1990.Google Scholar - 13.H. Hansson and B. Jonsson, “A calculus for communicating systems with time and probabilities,” in
*Proc. of the 11th IEEE Real-Time Systems Symposium*, (Los Alamitos), pp. 278–287, Dec. 1990.Google Scholar - 14.B. Jonsson and K. Larsen, “Specification and refinement of probabilistic processes,” in
*Proc. 6th IEEE Int. Symp. on Logic in Computer Science*, (Amsterdam), pp. 266–277, July 1991.Google Scholar - 15.A. Giacalone, C. Jou, and S. Smolka, “Algebraic reasoning for probabilistic concurrent systems,” in
*Proc. IFIP TC2 Working Conference on Programming Concepts and Methods*, (), p., 1989.Google Scholar - 16.S. Hart and M. Sharir, “Probabilistic propositional temporal logic,”
*Information and Control*, vol. 70, pp. 97–155, 1986.Google Scholar - 17.R. Alur, C. Courcoubetis, and D. Dill, “Model checking for probabilistic real-time systems,” in
*Proc. 18th Int. Coll. on Automata Languages and Programming (ICALP)*, (), p., 1991.Google Scholar - 18.A. Benveniste, “Constructive probability and the Signalea language: building and handling random processes with programming,” Tech. Rep. 1532, Institut National de Recherche en Informatique et Automatique, Rocquencourt, France, Oct. 1991.Google Scholar
- 19.B. C. Levy, A. Benveniste, and R. Nikoukhah, “High-level primitives for recursive maximum likelihood estimation,” Tech. Rep. 767, IRISA, Rennes, France, Oct. 1993.Google Scholar
- 20.G. D. Forney, “The Viterbi algorithm,”
*Proc. IEEE*, vol. 61, pp. 268–278, March 1973.Google Scholar - 21.A. P. Dempster, “Upper and lower probabilities induced by a multivalued mapping,”
*Annals Math. Statistics*, vol. 38, pp. 325–339, 1967.Google Scholar - 22.A. P. Dempster, “A generalization of Bayesian inference (with discussion),”
*Royal Stat. Soc., Series B*, vol. 30, pp. 205–247, 1968.Google Scholar - 23.G. Shafer,
*A Mathematical Theory of Evidence*. Princeton, NJ: Princeton Univ. Press, 1976.Google Scholar - 24.P. P. Shenoi and G. Shafer, “Axioms for probability and belief function propagation,” in
*Uncertainty in Artificial Intelligence*, (R. D. Shachter, T. S. Levitt, L. N. Kanal, and J. F. Lemmer, eds.), pp. 169–198, Amsterdam: North-Holland, 1990.Google Scholar - 25.J. Pearl, “Fusion, propagation, and structuring in belief networks,”
*Artificial Intelligence*, vol. 29, pp. 241–288, Sep. 1986.Google Scholar - 26.M. A. Peot and R. D. Shachter, “Fusion and propagation with multiple observations in belief networks,”
*Artificial Intelligence*, vol. 48, pp. 299–318, 1991.Google Scholar - 27.S. L. Lauritzen and D. J. Spiegelhalter, “Local computations with probabilities on graphical structures and their application to expert systems (with discussion),”
*J. Royal Stat. Soc., Series B*, vol. 50, pp. 157–224, 1988.Google Scholar - 28.B. Jonsson, C. Ho-Stuart, and Y. Wang, “Testing and refinement for nondeterministic and probabilistic processes,” in
*Lecture Notes in Computer Science*, pp. 418–430, Berlin: Springer Verlag, 1994.Google Scholar - 29.R. Kindermann and J. L. Snell,
*Markov Random Fields and their Applications*. Providence, RI: American Mathematical Society, 1980.Google Scholar - 30.C. Robert,
*Modèles Statistiques pour l'Intelligence Artificielle*. Paris: Masson, 1991.Google Scholar - 31.B. Prum and J. Fort,
*Stochastic Processes on a Lattice and Gibbs Measure*. Boston, MA: Kluwer Acad. Publ., 1991.Google Scholar - 32.C. Dellacherie and P. Meyer,
*Probabilités et Potentiels*. Paris: Hermann, 1976.Google Scholar - 33.A. P. Dempster, “Construction and local computation aspects of network belief functions,” in
*Influence Diagrams, Belief Nets, and Decision analysis*, (R. M. Oliver and J. Q. Smith, eds.), ch. 6, pp. 121–141, Chichester, England: J. Wiley, 1990.Google Scholar - 34.P. Le Guernic, T. Gauthier, M. Le Borgne, and C. Le Maire, “Programming real-time applications with Signal,”
*Proc. IEEE*, vol. 79, pp. 1321–1336, Sep. 1991.Google Scholar - 35.A. Benveniste and P. Le Guernic, “Hybrid dynamical systems theory and the Signal language,”
*IEEE Trans. Automat. Contr.*, vol. 35, pp. 535–546, May 1990.Google Scholar - 36.A. Benveniste, M. Le Borgne, and P. Le Guernic, “Hybrid systems: the Signal approach,” in
*Lecture Notes in Computer Science*, pp. 230–254, Berlin: Springer Verlag, 1993.Google Scholar