Skip to main content
Springer Nature Link
Log in

On Finite Mixture Modeling of Change-point Processes

  • Published:
Journal of Classification Aims and scope Submit manuscript

An Erratum to this article was published on 16 September 2021

This article has been updated

Abstract

Change point estimation in standard process observed over time is an important problem in literature with applications in various fields. We study this problem in a heterogeneous population. A model-based clustering procedure relying on skewed matrix-variate mixture is proposed. It is capable of capturing the heterogeneity pattern and estimating change points from all data groups simultaneously. The appeal of such approach also lies in its flexibility to model the skewness and dependence in data with good interpretability. Two novel algorithms called matrix power mixture with abrupt change model and matrix power mixture with gradual change model are developed. The approaches are illustrated by simulation studies across a variety of settings. The models are then tested on the US crime data with promising results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Change history

References

  • Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716–723.

    Article  MathSciNet  MATH  Google Scholar 

  • Atkinson, A.C., Riani, M., Cerioli, A. (2003). Exploring multivariate data with the forward search. Oxford: Clarendon Press.

    MATH  Google Scholar 

  • Bouveyron, C., & Brunet-Saumard, C. (2014). Model-based clustering of high-dimensional data: a review. Computational Statistics and Data Analysis, 71, 52–78.

    Article  MathSciNet  MATH  Google Scholar 

  • Box, G.E., & Cox, D.R. (1964). An analysis of transformations. Journal of the Royal Statistical Society Series B, 26(2), 211–252.

    MathSciNet  MATH  Google Scholar 

  • Chen, J., & Gupta, A. (2007). A Bayesian approach to the statistical analysis of a smooth-abrupt change point model. Advances and Applications in Statistics, 7(1), 115–126.

    MathSciNet  MATH  Google Scholar 

  • Chen, J. (2012). Parametric statistical change point analysis. Basel: Birkhäuser.

    Book  Google Scholar 

  • Chen, J., & Gupta, A.K. (1997). Testing and locating variance change points with application to stock prices. Journal of the American Statistical Association, 92, 739–747.

    Article  MathSciNet  MATH  Google Scholar 

  • Coppin, P., Jonckheere, I., Nackaerts, B., Muys, B., Lambin, E. (2004a). Review articlegigital change detection methods in ecosystem monitoring: a review. International Journal of Remote Sensing, 25, 1565–1596.

    Article  Google Scholar 

  • Chen, J., & Gupta, A. (2004b). Review articlegigital change detection methods in ecosystem monitoring: a review. International Journal of Remote Sensing, 25, 1565–1596.

    Article  Google Scholar 

  • Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum likelihood for incomplete data via the EM algorithm (with discussion). Jounal of the Royal Statistical Society Series B, 39, 1–38.

    MATH  Google Scholar 

  • Doğru, F.Z., Bulut, Y.M., Arslan, O. (2016). Finite mixtures of matrix variate t distributions. Journal of Science, 25, 335–341.

    Google Scholar 

  • Forgy, E. (1965). Cluster analysis of multivariate data: efficiency vs. interpretability of classifications. Biometrics, 21, 768–780.

    Google Scholar 

  • Fraley, C., & Raftery, A.E. (2002). Model-Based Clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association, 97, 611–631.

    Article  MathSciNet  MATH  Google Scholar 

  • Gallaugher, M., & McNicholas, P. (2019). Mixtures of skewed matrix variate bilinear factor analyzers. In Advances in data analysis and classification (pp. 1–20).

  • Gallaugher, M., & McNicholas, P. (2020). Parsimonious mixtures of matrix variate bilinear factor analyzers. In Advanced Studies in Behaviormetrics and Data Science:, Essays in Honor of Akinori Okada (pp. 177–196).

  • Grubesic, T.H., & Murray, A.T. (2001). Detecting hot spots using cluster analysis and GIS. In Fifth annual international crime mapping research conference.

  • Guild, L.S., Cohen, W.B., Kauffman, J.B. (2004). Detection of deforestation and land conversion in Rondania, Brazil using change detection techniques. International Journal of Remote Sensing, 25, 731–750.

    Article  Google Scholar 

  • Gupta, A.K., & Nagar, D.K. (1999). Matrix variate distributions. Boca Raton: Chapman & Hall/CRC.

    MATH  Google Scholar 

  • Hsu, D. (1977). Tests for variance shifts at an unknown time point. Journal of the Royal Statistical Society Series C, 26(3), 279–284.

    Google Scholar 

  • Hubert, L., & Arabie, P. (1985). Comparing partitions. Journal of Classification, 2, 193–218.

    Article  MATH  Google Scholar 

  • Huth, R., Beck, C., Philipp, A., Demuzere, M., Ustrnul, Z., Cahynová, M., Kyselý, J. (2008). Classifications of atmospheric circulation patterns. In ANNALS of the New York Academy of sciences, (Vol. 1146 pp. 105–152).

  • MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. Proceedings of the Fifth berkeley symposium, 1, 281–297.

    MathSciNet  MATH  Google Scholar 

  • Manly, B.F.J. (1976). Exponential data transformations. Biometrics Unit, 25, 37–42.

    Google Scholar 

  • McLachlan, G.J., & Peel, D. (2000). Finite mixture models. New York: Wiley.

    Book  MATH  Google Scholar 

  • McNicholas, P.D. (2016). Model-based clustering. Journal of Classification, 33(3), 331–373.

    Article  MathSciNet  MATH  Google Scholar 

  • Melnykov, V., & Zhu, X. (2018a). Manly transformation in finite mixture modeling. Computational Statistics and Data Analysis, 190–208.

  • Melnykov, V., & Zhu, X. (2018b). On model-based clustering of skewed matrix data. Journal of Multivariate Analysis, 181–194.

  • Melnykov, V., & Zhu, X. (2019). Studying crime trends in the USA over the years 2000–2012. Advances in Data Analysis and Classification, 13(1), 325–341.

    Article  MathSciNet  MATH  Google Scholar 

  • Page, E.S. (1957). On problem in which a change in parameter occurs at an unknown points. Biometrika, 42, 248–252.

    Article  MATH  Google Scholar 

  • Perry, M.B., & Pignatiello, J.J. (2008). A change point model for the location parameter of exponential family densities. IIE Transactions, 40, 947–956.

    Article  Google Scholar 

  • Pettitt, A.N. (1979). A non-parametric approach to the change point problem. Journal of the American Statistical Association, 28, 126–135.

    MATH  Google Scholar 

  • Sarkar, S., Zhu, X., Melnykov, V., & Ingrassia, S. (2020). On parsimonious models for modeling matrix data. Computational Statistics & Data Analysis, 142, 106822.

    Article  MathSciNet  MATH  Google Scholar 

  • Schwarz, G. (1978). Estimating the dimensions of a model. Annals of Statistics, 6, 461–464.

    Article  MathSciNet  MATH  Google Scholar 

  • Sneath, P. (1957). The application of computers to taxonomy. Journal of General Microbiology, 17, 201–226.

    Article  Google Scholar 

  • Sunaga, D.Y., Nievola, J.C., Ramos, M.P. (2007). Statistical and biological validation methods in cluster analysis of gene expression. In Sixth international conference on machine learning and applications (pp. 494–499).

  • Tomarchio, S.D., Punzo, A., Bagnato, L. (2020). Two new matrix-varaite distributions with application in model-based clustering. Computational Statistics and Data Analysis, 152(C).

  • Velilla, S. (1993). A note on the multivariate box-Cox transformation to normality. Statistics & Probability Letters, 17(4), 259–263.

    Article  MathSciNet  MATH  Google Scholar 

  • Vilasuso, J. (1996). Changes in the duration of economic expansions and contractions in the United States. Applied Economics Letters, 3(12), 803–806.

    Article  Google Scholar 

  • Viroli, C. (2011a). Finite mixtures of matrix normal distributions for classifying three-way data. Statistics and Computing, 21, 511–522.

    Article  MathSciNet  MATH  Google Scholar 

  • Viroli, C. (2011b). Model based clustering for three-way data structures. Bayesian Analysis, 6, 573–602.

    Article  MathSciNet  MATH  Google Scholar 

  • Viroli, C. (2011c). On matrix-variate regression analysis. Journal of Multivariate Analysis, 111, 296–309.

    Article  MathSciNet  MATH  Google Scholar 

  • Ward, J.H. (1963). Hierarchical grouping to optimize an objective function. Journal of the American Statistical Association, 58, 236–244.

    Article  MathSciNet  Google Scholar 

  • Worsley, K.J. (1979). On the likelihood ratio test for a shift in location of normal populations. Journal of the American Statistical Association, 74, 365–367.

    MathSciNet  MATH  Google Scholar 

  • Yeo, I., & Johnson, R.A. (2000). A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954–959.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, L.C., Krishnaiah, P.R., Bai, Z.D. (1986a). On detection of the number of signals in presence of white noise. Journal of Multivariate Analysis, 20, 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, L.C., Krishnaiah, P.R., Bai, Z.D. (1986b). On detection of the number of signals when the noise covariance matrix is arbitrary. Journal of Multivariate Analysis, 20, 26–49.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information Systems, Statistics, and Management Science, University of Alabama, Tuscaloosa, AL, 35487, USA

    Xuwen Zhu & Yana Melnykov

Authors
  1. Xuwen Zhu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Yana Melnykov
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Xuwen Zhu.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

(PDF 284 KB )

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, X., Melnykov, Y. On Finite Mixture Modeling of Change-point Processes. J Classif 39, 3–22 (2022). https://doi.org/10.1007/s00357-021-09385-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00357-021-09385-6

Keywords