Abstract
In this paper, we consider hybrid systems containing both stochastic and non-stochastic 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.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
A. Benveniste, B. Levy, E. Fabre, and P. L. Guernic, “A calculus of stochastic systems for the specification, simulation, and hidden state estimation of hybrid stochastic/non-stochastic systems,” Tech. Rep. to appear, Institut National de Recherche en Informatique et Automatique, Rocquencourt, France.
N. Viswanadham and Y. Narahari, Performance Modeling of Automated Manufacturing Systems. Englewood Cliffs, NJ: Prentice Hall, 1992.
L. R. Rabiner and B. H. Juang, “An introduction to hidden Markov models,” IEEE ASSP Magazine, vol. 3, pp. 4–16, Jan. 1986.
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.
R. C. Dubes and A. K. Jain, “Random field models in image analysis,” J. Applied Stat., vol. 12, pp. 131–164, 1989.
G. W. Hart, “Nonintrusive applicance load monitoring,” Proc. IEEE, vol. 80, pp. 1870–1891, Dec. 1992.
M. Basseville and I. V. Nikiforov, Detection of Abrupt Changes: Theory and Applications. Englewood Cliffs, NJ: Prentice Hall, 1993.
T. Soderstrom and P. Stoica, System Identification. Englewood Cliffs, NJ: Prentice Hall, 1989.
M. Molloy, “Performance analysis using stochastic Petri nets,” IEEE Trans. Computers, vol. 31, pp. 913–917, Sept. 1982.
B. Plateau and K. Atif, “Stochastic automata network for modeling parallel systems,” IEEE Trans. on Software Engineering, vol. 17, pp. 1093–1108, Oct. 1991.
B. Plateau and J.-M. Fourneau, “A methodology for solving Markov models of parallel systems,” J. Parallel and Distributed Comput., vol. 12, pp. 370–387, 1991.
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.
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.
A. Giacalone, C. Jou, and S. Smolka, “Algebraic reasoning for probabilistic concurrent systems,” in Proc. IFIP TC2 Working Conference on Programming Concepts and Methods, 1989.
S. Hart and M. Sharir, “Probabilistic propositional temporal logic,” Information and Control, vol. 70, pp. 97–155, 1986.
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), 1991.
A. Benveniste, “Constructive probability and the Sign alea language: Building and handling random processes with programming,” Tech. Rep. 1532, Institut National de Recherche en Informatique et Automatique, Rocquencourt, France, Oct. 1991.
B. C. Levy, A. Benveniste, and R. Nikoukhah, “High-level primitives for recursive maximum likelihood estimation,” Tech. Rep. 767, IRISA, Rennes, France, Oct. 1993.
G. D. Forney, “The Viterbi algorithm,” Proc. IEEE, vol. 61, pp. 268–278, Mar. 1973.
A. P. Dempster, “Upper and lower probabilities induced by a multivalued mapping,” Annals Math. Statistics, vol. 38, pp. 325–339, 1967.
A. P. Dempster, “A generalization of Bayesian inference (with discussion),” Royal Stat. Soc., Series B, vol. 30, pp. 205–247, 1968.
G. Shafer, A Mathematical Theory of Evidence. Princeton, NJ: Princeton Univ. Press, 1976.
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.), vol. 4, pp. 169–198, Amsterdam: North-Holland, 1990.
J. Pearl, “Fusion, propagation, and structuring in belief networks,” Artificial Intelligence, vol. 29, pp. 241–288, Sept. 1986.
M. A. Peot and R. D. Shachter, “Fusion and propagation with multiple observations in belief networks,” Artificial Intelligence, vol. 48, pp. 299–318, 1991.
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.
R. Kindermann and J. L. Snell, Markov Random Fields and their Applications. Providence, RI: American Mathematical Society, 1980.
C. Robert, Modèles Statistiques pour l'Intelligence Artificielle. Paris: Masson, 1991.
B. Prum and J. Fort, Stochastic Processes on a Lattice and Gibbs Measure. Boston, MA: Kluwer Acad. Publ., 1991.
C. Dellacherie and P. Meyer, Probabilités et Potentiels. Paris: Hermann, 1976.
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.
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, Sept. 1991.
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.
A. Benveniste, M. Le Borgne, and P. Le Guernic, “Hybrid systems: the Signal approach,” in Lecture Notes in Computer Science, vol. 736, pp. 230–254, Berlin: Springer Verlag, 1993.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Benveniste, A., Levy, B.C., Fabre, E., Le Guernic, P. (1994). A calculus of stochastic systems. In: Langmaack, H., de Roever, WP., Vytopil, J. (eds) Formal Techniques in Real-Time and Fault-Tolerant Systems. FTRTFT ProCoS 1994 1994. Lecture Notes in Computer Science, vol 863. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58468-4_164
Download citation
DOI: https://doi.org/10.1007/3-540-58468-4_164
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58468-1
Online ISBN: 978-3-540-48984-9
eBook Packages: Springer Book Archive