AYRIK GİZLİ MARKOV MODELİNİ KULLANARAK SEGMENTASYON YOLUYLA TREND TESPİTİ

Gizli değişkenler kavramından yararlanan ve pratikte sıklıkla kullanılan iki model bulunmaktadır. Bunlar gizli Markov modeli ve Markov rejim değişikliği modelidir. Ekonometri alanında Markov rejim değişikliği modelini uygulayan çok sayıda rejim değişikliği çalışması olmasına rağmen, ekonometrik zaman serilerindeki rejimleri incelemek için gizli Markov modelini kullanan çalışma sayısı çok azdır. Bu çalışmada, dört yüksek frekanslı zaman serisine ayrık gizli Markov modeli uygulanmış ve modellemede dönüştürülmüş ayrık veri kullanılmasına rağmen, gizli Markov modelinin iki ya da daha fazla rejimi oldukça iyi tanımladığı görülmüştür. Sonuç olarak ayrık gizli Markov modelinin rejimleri etkili bir şekilde tanımlayabildiği, böylece trendleri belirleyip seriyi segmentleyebildiği sonucuna varılmıştır.

Anahtar Kelimeler:

Saklı Markov Modeli, rejim değişikliği, trend belirleme, segmentasyon, örüntü_tanıma

TREND DETECTION THROUGH SEGMENTATION USING DISCRETE HIDDEN MARKOV MODEL

There are two models that benefit from the concept of hidden variables and are used frequently in practice. These are hidden Markov model and Markov switching model. Although there are quite a number of regime switching studies which applied the Markov switching model in the field of econometrics, studies that use the hidden Markov model for examining the regimes in econometric time series are sparse. In this study, we apply discrete hidden Markov model to four high-frequency time series and show that although transformed discrete data have to be used in the model structure, the model identifies two or more regimes quite well. It is concluded that the discrete hidden Markov model is defining regimes effectively, thereby, can also identify and segment trends.

Keywords:

Hidden Markov model, regime switching, trend detection, segmentation, pattern_finding,

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Ailliot, P. and V. Monbet (2012), “Markov-switching autoregressive models for wind time series,” Environmental Modelling & Software, 30, 92–101.
Baum, L. E., T. Petrie, G. Soules, and N. Weiss (1970), “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist., 41, 164–171.
Bhar, R. and S. Hamori (2004), Hidden Markov models: applications to financial economics, Advanced studies in theoretical and applied econometrics, volume v. 40, Boston, Mass. and London: Springer US.
Bilmes, J. (1998), “A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models,” Technical Report ICSI-TR-97-21, International Computer Science Institute, Computer Science Division, U.C. Berkeley.
Borman, S. (2009), “The expectation maximization algorithm - a short tutorial,” URL http://www.seanborman.com/publications/.
Cappé, O., E. Moulines, and T. Rydén (2005), Inference in hidden Markov models, Springer series in statistics, New York: Springer Verlag.
Ceppellini, R., M. Siniscalco, and C. A. Smith (1955), “The estimation of gene frequencies in a random-mating population,” Ann. Hum. Genet., 20, 97–115.
Collins, M. (1997), “The EM algorithm,” Technical report, Department of Computer and Information Science, University of Pennsylvania.
Costa, M. and L. De Angelis (2010), “Model selection in hidden Markov models: a simulation study,” Quaderni di Dipartimento 7, Department of Statistics, University of Bologna.
De Angelis, L. and L. J. Paas (2013), “A dynamic analysis of stock markets using a hidden Markov model,” Journal of Applied Statistics, 40, 1682–1700.
Demidenko, E. (2013), Mixed models: theory and applications with R, Wiley series in probability and statistics, John Wiley & Sons, Inc., 2nd edition.
Dempster, A. P., N. M. Laird, and D. B. Rubin (1977), “Maximum likelihood from incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B (Methodological), 39, 1–38.
Do, C. and S. Batzoglou (2008), “What is the expectation maximization algorithm?” Nature Biotechnology, 26, 897–899.
Fahrmeir, L. and G. Tutz (2001), Multivariate statistical modelling based on generalized linear models, Springer series in statistics, New York: Springer, 2nd edition.
Fink, G. A. (2007), Markov models for pattern recognition: from theory to applications, Berlin, Heidelberg: Springer-Verlag.
Ghahramani, Z. and M. I. Jordan (1994), “Supervised learning from incomplete data via an EM approach,” in Advances in Neural Information Processing Systems 6, Morgan Kaufmann, 120–127.
Giudici, P., T. Rydén, and P. Vandekerkhove (2000), “Likelihood-ratio tests for hidden Markov models,” Biometrics, 56, 742–747.
Gupta, M. and Y. Chen (2010), “Theory and use of the EM algorithm,” Foundations and Trends in Signal Processing, 4, 223–296.
Hamilton, J. D. (1989), “A new approach to the economic analysis of nonstationary time series and the business cycle,” Econometrica, 57, 357–384.
Hamilton, J. D. (2010), Regime switching models, London: Palgrave Macmillan UK, 202–209.
Hamilton, J. D. and B. Raj (2002), “New directions in business cycle research and financial analysis,” Empirical Economics, 27, 149–162.
Hartley, H. O. (1958), “Maximum likelihood estimation from incomplete data,” Biometrics, 14, 174–194.
Jamshidian, M. and R. I. Jennrich (1997), “Acceleration of the EM algorithm by using quasi-Newton methods,” Journal of the Royal Statistical Society. Series B (Methodological), 59, 569–587.
Janczura, J. and R. Weron (2010), “An empirical comparison of alternate regimes witching models for electricity spot prices,” Energy Economics, 32, 1059–1073.
Jiang, J. (2007), Linear and generalized linear mixed models and their applications, Springer series in statistics, New York and London: Springer.
Juang, B. H. and L. R. Rabiner (1985a), “Mixture autoregressive hidden Markov models for speech signals,” IEEE Transactions on Acoustics, Speech, and Signal Processing, 33, 1404–1413.
Kneib, T. and G. Tutz (2010), Statistical modelling and regression structures: festschrift in honour of Ludwig Fahrmeir, Heidelberg [u.a.]: Physica-Verl.
Kobayashi, H., B. L. Mark, and W. Turin (2012), Probability, random processes, and statistical analysis, Cambridge and New York: Cambridge University Press.
Koller, D. and N. Friedman (2009), Probabilistic graphical models: principles and techniques - adaptive computation and machine learning, The MIT Press.
Levinson, S. E., L. R. Rabiner, and M. M. Sondhi (1983), “An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition,” The Bell System Technical Journal, 62, 1035–1074.
Lindgren, G. (1978), “Markov regime models for mixed distributions and switching regressions,” Scandinavian Journal of Statistics, 5, 81–91.
Louis, T. A. (1982), “Finding the observed information matrix when using the EM algorithm,” Journal of the Royal Statistical Society. Series B (Methodological), 44, 226–233.
McLachlan, G. J. and T. Krishnan (2008), The EM algorithm and extensions, Wiley series in probability and statistics, Hoboken and NJ: Wiley-Interscience, 2nd edition.
Moon, T. K. (1996), “The expectation-maximization algorithm,” IEEE Signal Processing Magazine, 13, 47–60.
Poritz, A. (1982), “Linear predictive hidden Markov models and the speech signal,” in ICASSP ’82. IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 7, 1291–1294.
Rabiner, L. R. (1989), “A tutorial on hidden Markov models and selected applications in speech recognition,” Proceedings of the IEEE, 77, 257–286.
Rabiner, L. R. and B. H. Juang (1986), “An introduction to hidden Markov models,” IEEE ASSP Mag., 3, 4–16.
Rabiner, L. R., S. E. Levinson, and M. M. Sondhi (1983), “On the application of vector quantization and hidden Markov models to speaker-independent, isolated word recognition,” The Bell System Technical Journal, 62, 1075–1105.
Rydén, T., T. Teräsvirta, and S. Åsbrink (1998), “Stylized facts of daily return series and the hidden Markov model,” Journal of Applied Econometrics, 13, 217–244.
Schlattmann, P. (2009), Medical applications of finite mixture models, Statistics for Biology and Health, Berlin and Heidelberg: Springer-Verlag Berlin Heidelberg.
Tanner, M. A. (1996), Tools for statistical inference: methods for the explorations of posterior distribution and likelihood functions, Springer series in statistics, New York: Springer, 3rd edition.
Vaseghi, S. V. (2007), Multimedia signal processing: theory and applications in speech, music and communications, Chichester and West Sussex and England and Hoboken and NJ: J. Wiley.
Viterbi, A. (1967), “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm,” IEEE Trans. Inf. Theor., 13, 260–269.
Wu, C. F. J. (1983), “On the convergence properties of the EM algorithm,” Ann. Statist., 11, 95–103.
Wu, L. (2009), Mixed effects models for complex data, C & H/CRC Monographs on Statistics & Applied Probability, 113, volume v. 113, Hoboken: Chapman & Hall/CRC.