Realization problem for hidden Markov models

Mathematics of Control, Signals, and Systems

Volumn 12, Issue 1, 1999, Pages 80-120

  Andersen, Brian D O ^a  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

GEOMETRY; LINEAR SYSTEMS; MATHEMATICAL MODELS; MATRIX ALGEBRA; PROBABILITY;

FINITE STATE MARKOV PROCESS; HANKEL MATRICES; HIDDEN MARKOV MODELS;

MARKOV PROCESSES;

EID: 0032668386     PISSN: 09324194     EISSN: None     Source Type: Journal    
DOI: 10.1007/pl00009846     Document Type: Article

References (30)

1. [A] B. D. O. Anderson, New developments in the theory of positive systems, in Systems and Control in the 21st Century, C. I. Byrnes, B. N. Dalton, D. S. Gilliam and C. F. Martin (Eds.) (ISBN: 0-8176-3881-4/3-7643-3881-4), Birkhäuser, Boston, 1996, pp. 17-36.
2. [ADFB] B. D. O. Anderson, M. Deistler, L. Farina and L. Benvenuti, Nonnegative realization of a linear system with nonnegative impulse response, IEEE Transactions on Circuits and Systems, 1996, vol. 4, pp. 134-142.
3. L. E. Baum and T. Petrie, Statistical inference for probabilistic functions of finite set Markov chains, Annals of Mathematical Statistics, 1966, vol. 37 pp. 1554-1563.
4. [BPSW] L. E. Baum, T. Petrie, G. Soules and N. Weiss, A maximization technique occurring in the statistical analysis of probabilistic function of Markov chains, Annals of the Institute of Statistical Mathematics, 1970, vol. 41, pp. 164-171.
5. A. Berman and R. J. Plemmons, Nonneaative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
6. [BK] D. Blackwell and L. Koopmans, On the identifiability problem for functions of finite Markov chains, Annals of Mathematical Statistics, 1957, vol. 28, pp. 1011-1015.
7. J. W. Carlyle, Stochastic finite-state system theory, in Systems Theory, L. Zadeh and L. Polak (Eds.), McGraw-Hill, New York, 1969, Chapter 10.
8. S. W. Dharmadhikari, Functions of finite Markov chains. Annals of Mathematical Statistics, 1963, vol. 34, pp. 1022-1032.
9. S. W. Dharmadhikari, Sufficient conditions for a stationary process to be a function of a finite Markov chain, Annals of Mathematical Statistics, 1963, vol. 34, pp. 1033-1041.
10. S. W. Dharmadhikari, A characterization of a class of functions of finite Markov chains, Annals of Mathematical Statistics, 1965, vol. 36, pp. 524-528.
11. S. W. Dharmadhikari, Splitting a single state of a stationary process into Markovian states, Annals of Mathematical Statistics, 1968, vol. 38, pp. 1059-1077.
12. S. W. Dharmadhikari, A note on exchangeable processes with states of finite rank, Annals of Mathematical Statistics, 1969, vol. 40, pp. 2207-2208.
13. [DN] S. W. Dharmadhikari and M. G. Nadkarmi, Some regular and non-regular functions of finite Markov chains, Annals of Mathematical Statistics, 1970, vol. 41, pp. 207-213.
14. [E] R. V. Erickson, Functions of Markov chains, Annals of Mathematical Statistics, 1970, vol. 41, pp. 843-850.
15. L. Farina, On the existence of a positive realization, Systems and Control Letters, 1996, vol. 28, pp. 219-226.
16. M. Fliess, Series rationnelles positives et processus stochastiques, Annales de l'Institut Henri Poincaré, Section B, 1975, vol. XI, pp. 1-21.
17. [FR] M. Fox and H. Rubin, Functions of processes with Markovian states, Annals of Mathematical Statistics, 1968, vol. 39, pp. 938-946.
18. [G] E. J. Gilbert, On the identifiability problem for functions of finite Markov chains, Annals of Mathematical Statistics, 1959, vol. 30, pp. 688-697.
19. A. Heller, On stochastic processes derived from Markov chains, Annals of Mathematical Statistics, 1965, vol. 36, pp. 1286-1291.
20. J. M. Van den Hof, Realization of continuous-time positive linear systems, Systems and Control Letters, 1997, vol. 31, pp. 243-253.
21. [HS] J. M. van den Hof and J. H. van Schuppen Realization of positive linear systems using polyhedral cones, Proceedings of the 33rd IEEE Conference on Decision and Control, 1994, pp. 3889-3893.
22. [IAK] H. Ito, S.-I. Amani and K. Kobayoshi, Identifiability of hidden Markov information sources and their minimum degrees of freedom, IEEE Transactions on Information Theory, 1992, vol. 38, pp. 324-333.
23. [KM] T. Kitano and H. Maeda, Positive realization of discrete-time systems by geometric approach, IEEE Transactions on Circuits and Systems, 1998, vol. 45, pp. 308-311.
24. [OMK] Y. Ohta, H. Maeda and S. Kodama, Reachability, observability and realizability of continuous-time positive systems, SIAM Journal of Control and Optimization, 1984, vol. 22, pp. 171-180.
25. A. Paz, Introduction to Probabilistic Automata, Academic Press, New York, 1971.
26. G. Picci, On the internal structure of finite state stochastic processes, in Recent Developments in Variable Structure Systems, Economics and Biology, R. R. Mohler and A. Ruberti (Eds.), Lecture notes in Economics and Mathematical Systems, vol. 162, Springer-Verlag, Berlin, 1978, pp. 288-304.
27. [R] M. Rosenblatt, Markov Processes; Structure and Asymptotic Behaviour, Springer-Verlag, Berlin, 1971, pp. 81-92.
28. E. Seneta, Nonnegative Matrices, George Allen and Unwin, London, 1973.
29. [SAD] L. Shue, B. D. O. Anderson and S. Dey, Exponential stability of filters and smoothers for hidden Markov models. Proceedings of the 1997 European Control Conference [TU-E-1-6272].
30. E. D. Sontag, On certain questions of rationality and decidability, Journal of Computer and Systems Sciences, 1975, vol. 11, pp. 375-381.