Computational Statistics

, Volume 29, Issue 6, pp 1481–1496 | Cite as

Wavelet improvement in turning point detection using a hidden Markov model: from the aspects of cyclical identification and outlier correction

Original Paper


The hidden Markov model (HMM) has been widely used in regime classification and turning point detection for econometric series after the decisive paper by Hamilton (Econometrica 57(2):357–384, 1989). The present paper will show that when using HMM to detect the turning point in cyclical series, the accuracy of the detection will be influenced when the data are exposed to high volatilities or combine multiple types of cycles that have different frequency bands. Moreover, outliers will be frequently misidentified as turning points. The present paper shows that these issues can be resolved by wavelet multi-resolution analysis based methods. By providing both frequency and time resolutions, the wavelet power spectrum can identify the process dynamics at various resolution levels. We apply a Monte Carlo experiment to show that the detection accuracy of HMMs is highly improved when combined with the wavelet approach. Further simulations demonstrate the excellent accuracy of this improved HMM method relative to another two change point detection algorithms. Two empirical examples illustrate how the wavelet method can be applied to improve turning point detection in practice.


HMM Turning point Wavelet Wavelet power spectrum Outlier 


  1. Anders K (1997) Two methods for improving performance of a HMM and their application for gene finding. In: Proceedings of the 5th international conference on intelligent systems for molecular biology 1997, vol 5, pp 179–186Google Scholar
  2. Barnett V, Lewis T (1994) Outliers in statistical data, 3rd edn. Wiley, ChichesterMATHGoogle Scholar
  3. Baum LE, Eagon JA (1967) An inequality with applications to statistical estimation for probabilistic functions of a Markov process and to a model for ecology. Bull Am Math Soc 73(3):360–363CrossRefMATHMathSciNetGoogle Scholar
  4. Baum LE, Petrie T (1966) Statistical inference for probabilistic functions of finite state Markov chains. Ann Math Stat 37(6):1554–1563CrossRefMATHMathSciNetGoogle Scholar
  5. Baum LE et al (1970) A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann Math Stat 41(1):164–171CrossRefMATHMathSciNetGoogle Scholar
  6. Bellone B, Saint-Martin D (2003) Detecting turning points with many predictors through hidden Markov models. In: Working paper presented in Séminaire Fourgeaud, December, 2003:3, Etudes pour la conjonctureGoogle Scholar
  7. Burns AF, Mitchell WC (1946) Measuring business cycles. NBER, New YorkGoogle Scholar
  8. Canan B, Huzurbazar S (2002) Wavelet-based detection of outliers in time series. J Comput Graph Stat 11(2):311–327CrossRefGoogle Scholar
  9. Chengalvarayan R (1999) Robust energy normalization using speech/non-speech discriminator for German connected digit recognition. In: Proceedings of EUROSPEECH, pp. 61–64 Budapest, Hungary, September 1999, ISCAGoogle Scholar
  10. Dempster A, Laird N, Rubin D (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc B 39:1–38Google Scholar
  11. Edwards AWF, Cavalli-Sforza LL (1965) A method for cluster analysis. Biometrics 21:362–375Google Scholar
  12. Elliott RJ, Aggoun L, Moore JB (1995) Hidden Markov models. Springer, New YorkMATHGoogle Scholar
  13. Gençay R, Selçuk F, Whitcher B (2001) An introduction to wavelets and other filtering methods in finance and economic. Academic Press, San DiegoGoogle Scholar
  14. Grané A, Veiga H (2009) Wavelet-based detection of outliers in volatility models. In: Statistics and econometrics working papers. Universidad Carlos III de Madrid, Calle Madrid, Getafe, SpainGoogle Scholar
  15. Grossman A, Morlet J (1984) Decomposition of Hardy functions into square integrable wavelets of constant shape. Soc Ind Appl Math J Math Anal 15:732–736Google Scholar
  16. Hamilton JD (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2):357–384CrossRefMATHMathSciNetGoogle Scholar
  17. Hamilton J, Perez-Quiros G (1996) What do the leading indicators lead? J Bus 69:27–49CrossRefGoogle Scholar
  18. Huang XD, Ariki Y, Jack MA (1990) Hidden Markov models for speech recognition. Edinburg university press, EdinburgGoogle Scholar
  19. Killick R, Fearnhead P, Eckley IA (2012) Optimal detection of change points with a linear computational cost. J Am Stat Assoc 107(500):1590–1598CrossRefMATHMathSciNetGoogle Scholar
  20. Kim CJ, Nelson CR (1998) Business cycle turning points, a new coincident index, and tests of duration dependence based on a dynamic factor model with regime switching. Rev Econ Stat 80(2):188–201CrossRefGoogle Scholar
  21. Krolzig HM (2003) Constructing turning point chronologies with Markov-switching vector autoregressive models: the Euro-zone business cycle. In: Paper presented at the colloquium on modern tools for business cycle analysis, LuxembourgGoogle Scholar
  22. Layton AP (1996) Dating and predicting phase changes in the US business cycle. Int J Forecast 12:417–428CrossRefMathSciNetGoogle Scholar
  23. Levinson SE, Rabiner LR, Sondhi MM (1983) An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition. Bell Syst Tech J 62:1035–1074CrossRefMATHMathSciNetGoogle Scholar
  24. Leroux BG (1992) Maximum-likelihood estimation for hidden Markov models. Stoch Process Appl 40:127–143CrossRefMATHMathSciNetGoogle Scholar
  25. MacDonald IL, Zucchini W (1997) Hidden Markov and other models for discrete-valued time series. Chapman and Hall, LondonMATHGoogle Scholar
  26. Mallat SG (1989) A theory for multiresolution signal decomposition: the wavelet representation. Pattern Anal Mach Intell IEEE Trans 11(7):674–693CrossRefMATHGoogle Scholar
  27. Percival DB, Walden AT (2000) Wavelet methods for time series analysis. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  28. Rabiner LR (1989) A tutorial on hidden Markov models and selected applications in speech recognition. In: Proceedings of the IEEE 77(2):257–286Google Scholar
  29. Stock JH, Watson MW (1989) New indexes of coincident and leading economic indicators. In: Blanchard O, Fisher S (eds) NBER macroeconomics annual. Mit Press, Cambridge, pp 352–394Google Scholar
  30. Vidakovic B (1999) Statistical modelling by wavelets. Wiley, New YorkCrossRefGoogle Scholar
  31. Wachter M, Demuynck K, Compernolle VD (2007) Outlier correction for local distance measures in example based speech recognition. Proceedings ICASSP 4:433–436Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Business and Management ScienceNorwegian School of EconomicsBergenNorway
  2. 2.Department of EconomicsLund UniversityLundSweden

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