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Abstract

Markov regime switching models remain enormously popular in speech recognition, economics, finance, etc. Nonparametric segmentation in switching models without probability assignment of jump moments is in many papers by Brodsky and Darkhovsky. We model all regimes as long SCOT strings. Stochastic COntext Tree (abbreviated as SCOT) is $m$ -Markov Chain ( $m$ -MC) with every state of a string independent of the symbols in its more remote past than the context of length determined by the preceding symbols of this state. A parallel super-fast fitting and asymptotically optimal inference in a sparse SCOT model including the nonparametric homogeneity test are described in our previous papers. Our segmentation method is a combination of preliminary online change point detection with its subsequent offline Maximal Likelihood update.

Keywords

Strong mixing SCOT emissions Markov switching model change point detection maximum likelihood HMM

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References

Aminikhanghahi

, & Cook

D.A.

(2017). Survey of methods for time series change point detection. Knowl Inf Syst, 51(2), 339-367.

Billingsley

(1961). Statistical Inference for Markov Chains. University of Chicago Press.

Biscup

(2011). Recent progress on the random conductance model. Probability Surveys, 8, 294-373.

Borovkov

A.A.

(1998). Ergodicity and Stability Of Stochastic Processes. Wiley.

Bradley

R.C.

(2005). Basic properties of strong mixing conditions. A survey and some open questions. Probability Surveys, 2, 107-144.

Brodsky

B.E.

, & Darkhovsky

B.S.

(1993). Nonparametric Methods in Change-Point Problems. Kluwer, Dodrecht.

Chibisov

D.M.

(2009). Lectures on the asymptotic theory of rank tests. Lecture Notes NOTs 14. M: Matematicheskiy Institut im. V. A. Steklova, RAN, In Russian.

Cover

T.M.

, & Thomas

J.A.

(2006). Elements of Information Theory, Second Edition. Wiley, Hoboken.

Cappe

Moulines

, & Rydyen

(2005). Inference in Hidden Markov Models, Springer.

10.

Dumont

(2014). Context tree estimation in variable length hidden markov models. IEEE Trans Inform Theory.

11.

Durbin

Eddy

Krogh

, & Mitchison

(1998). Biological Sequence Analysis. Cambridge University Press.

12.

Feller

(1967). An Introduction to Probability Theory and its Applications, Volume 1, Third edition, Wiley, NY.

13.

Gallager

(2013). Stochastic Processes: Theory for Applications, Cambridge Uni.

14.

Galves, A. & Loecherbach

(2008). Stochastic chains with memory of variable length. Festschrift in Honor of Jorma Rissanen on the Occasion of his 75th Birthday, Tampere, TICSP series No. 38, Tampere Tech Uni, 117-134.

15.

Grinstead

, & Snell

(2006). Introduction to Probability, AMS.

16.

Hamiltons

(2008). Regime switching models. The New Palgrave Dictionary of Economics. Second Edition Durlauf

S.N.

, & Blume

L.E.

, Palgrave Macmillan.

17.

Kharin

, & Maltsau

(2014). Markov chain of conditional order: Properties and statistical analysis. Austrian Journal of Statistics, 4(3-4), 205-216.

18.

Korostelev

, & Korosteleva

(2011). Mathematical Statistics: Asymptotic Minimax Theory, AMS, Providence, RI.

19.

Mächler

, & Bühlmann

(2004). Variable length Markov Chains: Methodology, computing, and software. Journal of Computational and Graphical Statistics, 13(2), 435-455.

20.

Malyutov

, & Protassov

(2002). LAN and LAM, convergence of iterative estimates and optimal design in Gaussian one-way mixed model. Journal of Statistical Planning and Inference, 100(2), 249-279.

21.

Malyutov

, & Grosu

(2017). SCOT approximation, modeling and training. Proceedings of Machine Learning Research, 60, 241-265.

22.

Malyutov

M.B.

Zhang

, & Li

(2013). Time series homogeneity tests via VLMC training. Information Processes, 13(4), 401-414.

23.

Malyutov

M.B.

Zhang

, & Grosu

(2014). SCOT stationary distribution evaluation for some examples. Information Processes, 14(3), 275-283.

24.

Markov

A.A.

(1906). Extension of the limit theorems of probability theory to a sum of variables connected in a chain (in Russian). Appendix B of: Howard R (1971) Dynamic Probabilistic Systems, Volume 1: Markov Chains. John Wiley and Sons.

25.

Malyutov

(2017). Retrospective training slow HMM-SCOT emissions model. Information Processes, 17(3), 199-205.

26.

Meyn

S.P.

, & Tweedy

R.L.

(1993). Markov Chains and Stochastic Stability. Springer.

27.

Rabiner

(1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of IEEE, 77(2), 257-286.

28.

Rissanen

(1983). A universal data compression system. IEEE Trans Inform Theory, 29(5), 656-664.

29.

Roussas

(1972). Contiguity of Probability Measures: Some Applications in Statistics. Cambridge University Press.

30.

Ryabko

Astola

, & Malyutov

(2016). Compression-Based Methods of Prediction and Statistical Analysis of Time Series: Theory and Appllications. Springer International.

31.

Shiryaev

A.N.

(1978). Optimal stopping rules. Applications of Mathematics, 8, Springer, New York.

32.

Tutubalin

V.N.

(1992). Probability and Random Processes Theory. Mathematical Foundations and Applications. Moscow State University Press (In Russian).

33.

Veretennikov

A.Yu.

(2000). Parametric and Nonparametric Estimation for Markov Chains. Moscow State University Press (In Russian).

34.

Veretennikov

A.Yu.

(2017). Ergodic Markov Chains and Poisson equations (lecture notes). Arxiv:1610.09661v3 [math PR].

35.

Volkonskii

V.A.

, & Rozanov

Yu.A.

(1959). Some limit theorems for random functions I. Theor Probab Appl, 4, 178-197.

36.

Yoon

B.J.

(2009). Hidden Markov models and their applications in biological sequence analysis. Current Genomics, 10(9), 402-415.

37.

Zhang

(2017). Perfect memory context trees in time series modeling. Information Processes, 17(1), 70-81.

Offline fitting Markov switching model

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