Bayesian Analysis

A Bayesian Nonparametric Approach for Time Series Clustering

Luis E. Nieto-Barajas and Alberto Contreras-Cristán

Full-text: Open access

Abstract

In this work we propose a model-based clustering method for time series. The model uses an almost surely discrete Bayesian nonparametric prior to induce clustering of the series. Specifically we propose a general Poisson-Dirichlet process mixture model, which includes the Dirichlet process mixture model as a particular case. The model accounts for typical features present in a time series like trends, seasonal and temporal components. All or only part of these features can be used for clustering according to the user. Posterior inference is obtained via an easy to implement Markov chain Monte Carlo (MCMC) scheme. The best cluster is chosen according to a heterogeneity measure as well as the model selection criterion LPML (logarithm of the pseudo marginal likelihood). We illustrate our approach with a dataset of time series of share prices in the Mexican stock exchange.

Article information

Source
Bayesian Anal., Volume 9, Number 1 (2014), 147-170.

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.ba/1393251774

Digital Object Identifier
doi:10.1214/13-BA852

Mathematical Reviews number (MathSciNet)
MR3188303

Zentralblatt MATH identifier
1327.62473

Keywords
Bayes nonparametrics dynamic linear model model-based clustering Pitman-Yor process time series analysis

Citation

Nieto-Barajas, Luis E.; Contreras-Cristán, Alberto. A Bayesian Nonparametric Approach for Time Series Clustering. Bayesian Anal. 9 (2014), no. 1, 147--170. doi:10.1214/13-BA852. https://projecteuclid.org/euclid.ba/1393251774


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References

  • Argiento, R., Cremaschi, A. and Guglielmi, A. (2013). A Bayesian nonparametric mixture model for cluster analysis. Technical report Quaderno Imati CNR, 2012 3-MI, Milano. ISSN 1722–8964.
  • Barrios, E., Lijoi, A., Nieto-Barajas, L.E. and Prünster, I. (2013). Modeling with normalized random measure mixture models. Statistical Science. To appear.
  • Campbell, J.Y., Lo A.W. and MacKinlay, A.C. (1997). The econometrics of financial markets. Princeton University Press, Princeton, New Jersey.
  • Carlin, B.P., Polson N.G. and Stoffer, D.S. (1992).A Monte Carlo Approach to Nonnormal and Nonlinear State-Space Modeling.Journal of the American Statistical Association 87, 493–500.
  • Caron, F., Davy, M., Doucet, A., Duflos, E. and Vanheeghe, P. (2008).Bayesian Inference for Linear Dynamic Models with Dirichlet Process Mixtures. IEEE Transactions on Signal Processing 56, 71–84.
  • Carter, C.K. and Kohn, R. (1994). On Gibbs Sampling for State-Space Models. Biometrika 81, 541–553.
  • Carter, C.K. and Kohn, R. (1996). Markov Chain Monte Carlo in Conditionally Gaussian State-Space Models. Biometrika 83, 589–601.
  • Chatfield, C. (1989). The analysis of time series: an introduction.Chapman and Hall, London.
  • Chib, S. and Greenberg, E. (1996). Markov Chain Monte Carlo Simulation Methods in Econometrics. Econometric Theory 12, 409–431.
  • Dahl, D.B. (2006). Model based clustering for expression data via a Dirichlet process mixture model. In Bayesian Inference for Gene Expression and Proteomics, Eds. M. Vanucci, K.-A. Do and P. Müller. Cambridge University Press, Cambridge.
  • Escobar, M.D. and West, M. (1998). Computing nonparametric hierarchical models. In Practical Nonparametric and Semiparametric Bayesian Statistics, Eds. D. Dey, P. Müller and Sinha, D. Springer, New-York.
  • Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics 1, 209–230.
  • Fox, E., Sudderth, E.B., Jordan, M.I. and Willsky, A.S. (2011).Bayesian Nonparametric Inference of Switching Dynamic Linear Models.IEEE Transactions on Signal Processing 59, 1569–1585.
  • Geisser, S. and Eddy, W.F. (1979). A predictive approach to model selection. Journal of the American Statistical Association 74, 153–160.
  • Gelman, A. (2006). Prior distributions for variance parameters inhierarchical models. Bayesian Analysis 1, 515–533.
  • Ghosh, A., Mukhopadhyay., S., Roy, S. and Bhattacharya, S. (2012).Bayesian Inference in Nonparametric Dynamic State-Space Models.arXiv:1108.3262[stat.ME].
  • Granger, C.W.J. and Newbold, P. (1974). Spurious regressions in econometrics. Journal of Econometrics 2, 111–120.
  • Harrison, P.J. and Stevens, P.F. (1976). Bayesian forecasting. Journal of the Royal Statistical Society, Series B 38, 205–247.
  • Heard, N.A., Holmes, C.C. and Stephens, D.A. (2006). A quantitative study of gene regulation involved in the immune response of anopheline mosquitoes: An application of Bayesian hierarchical clustering of curves. Journal of the American Statistical Association 101, 18–29.
  • Ishwaran, H. and James, L.F. (2001). Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association 96, 161–173.
  • Jara, A., Lesaffre, E., De Iorio, M. and Quintana, F. (2010). Bayesian semiparametric inference for multivariate doubly-interval-censored data. The Annals of Applied Statistics 4, 2126–2149.
  • Lijoi, A., Mena, R.H. and Prünster, I. (2007). Controlling the reinforcement in Bayesian nonparametric mixture models. Journal of the Royal Statistical Society, Series B 69, 715–740.
  • MacEachern, S.N. (1994). Estimating normal means with a conjugate style Dirichlet process prior. Communications in Statistics B 23, 727–741.
  • Markowitz, H.M. (1952). Portfolio selection. The Journal of Finance 7, 77–91.
  • Medvedovic, M. and Sivaganesan, S. (2002). Bayesian infinite mixture model based clustering of gene expression profiles. Bioinformatics 18, 1194–1206.
  • Mendoza, M. and Nieto-Barajas, L.E. (2006). Bayesian solvency analysis with autocorrelated observations. Applied Stochastic Models in Business and Industry 22, 169–180.
  • Mukhopadhyay, S. and Gelfand, A.E. (1997). Dirichlet process mixed generalized linear models. Journal of the American Statistical Association 92, 633–639.
  • Navarrete, C., Quintana, F.A. and Müller, P. (2008). Some issues in nonparametric Bayesian modeling using species sampling models. Statistical Modelling 8, 3–21.
  • Neal, R.M. (2000). Markov chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics 9, 249–265.
  • Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probability Theory and Related Fields 102, 145–158.
  • Pitman, J. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. The Annals of Probability 25, 855–900.
  • Ross, S.M. (2000). Introduction to probability models. 7th edition. Harcourt Academic Press, San Diego.
  • Smith, A. and Roberts, G. (1993). Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B 55, 3–23.
  • Stephens, M. (2000). Dealing with label switching in mixture models. Journal of the Royal Statistical Society, Series B 62, 795–809.
  • Tierney, L. (1994). Markov chains for exploring posterior distributions. Annals of Statistics 22, 1701–1722.
  • West, M. and Harrison, J. (1999). Bayesian forecasting and dynamic models. 2nd edition. Springer, New York.
  • Zhou, C. and Wakefield, J. (2006). A Bayesian mixture model for partitioning gene expression data. Biometrics 62, 515–525.