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Local Statistical Modeling via a Cluster-Weighted Approach with Elliptical Distributions

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Abstract

Cluster-weighted modeling (CWM) is a mixture approach to modeling the joint probability of data coming from a heterogeneous population. Under Gaussian assumptions, we investigate statistical properties of CWM from both theoretical and numerical point of view; in particular, we show that Gaussian CWM includes mixtures of distributions and mixtures of regressions as special cases. Further, we introduce CWM based on Student-t distributions, which provides a more robust fit for groups of observations with longer than normal tails or noise data. Theoretical results are illustrated using some empirical studies, considering both simulated and real data. Some generalizations of such models are also outlined.

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References

  1. ANDERSON, J.A. (1972), “Separate Sample Logistic Discrimination”, Biometrika, 59, 19–35.

  2. ANDREWS, R.L., ANSARI, A., and CURRIM, I.S. (2002), “Hierarchical Bayes Versus Finite Mixture Conjoint Analysis Models: A Comparison of Fit, Prediction, and Partworth Recovery”, Journal of Marketing, 39, 87–98.

  3. ANDREWS, J.L., and McNICHOLAS, P.D. (2011), “Extending Mixtures of Multivariate T-Factor Analyzers”, Statistics and Computing 21(3), 361–373.

  4. BAEK, J., and McLACHLAN, G.J. (2011), “Mixtures of Common T-Factor Analyzers for Clustering High-Dimensional Microarray Data”, Bioinformatics, 27, 1269–1276.

  5. BERNARDO, J.M., and GIRÓN, F.J. (1992), “Robust Sequential Prediction from Non- Random Samples: The Election Night Forecasting Case” in Bayesian Statistics 5, eds. J.M. Bernardo, J.O. Berger, A.P. Dawid, and A.F.M. Smith, Oxford: Oxford University Press, pp. 61–77.

  6. CAMPBELL, N.A., and MAHON, R.J. (1974), “A Multivariate Study of Variation in Two Species of Rock Crab of Genus Letpograspus”, Australian Journal of Zoology, 22, 417–455.

  7. CERIOLI, A. (2010), “Multivariate Outlier Detection with High-Breakdown Estimators”, Journal of the American Statistical Society, 105(489), 147–156.

  8. CUESTA-ALBERTOS, J.A., MATRÁN, C., and MAYO-ISCAR, A. (2008), “Trimming and Likelihood: Robust Location and Dispersion Estimation in the Elliptical Model”, The Annals of Statistics, 36(5), 2284–2318.

  9. DAYTON, C.M., and MACREADY, G.B. (1988), “Concomitant-Variable Latent-Class Models”, Journal of the American Statistical Association, 83, 173–178.

  10. DESARBO, W.S., and CRON, W.L. (1988), “A Maximum Likelihood Methodology for Cluster wise Linear Regression”, Journal of Classification, 5(2), 249–282.

  11. DICKEY, J.T. (1967), “Matricvariate Generalizations of the Multivariate t Distribution and the Inverted Multivariate t Distribution”, The Annals of Mathematical Statistics, 38, 511–518.

  12. EVERITT, B.S., and HAND, D.J. (1981), Finite Mixture Distributions, London: Chapman & Hall.

  13. FARIA, S., and SOROMENHO, G. (2010), “Fitting Mixtures of Linear Regressions, Journal of Statistical Computation and Simulation, 80, 201–225.

  14. FONSECA, J.R.S. (2008), “Mixture Modeling and Information Criteria for Discovering Patterns in Continuous Data”, Eighth International Conference on Hybrid Intelligent Systems, IEEE Computer Society.

  15. FRÜWIRTH-SCHNATTER, S. (2005), Finite Mixture and Markov Switching Models, Heidelberg: Springer.

  16. GALLEGOS, M.T., and RITTER, G. (2009a), “Trimming- Algorithms for Clustering Contaminated Grouped Data and Their Robustness”, Advances in Data Analysis and Classification, 3, 135-167.

  17. GALLEGOS, M.T., and RITTER, G. (2009b), “Trimmed ML Estimation of Contaminated Mixtures”, Sankhya, 71-A, Part 2, 164–220.

  18. GERSHENFELD, N. (1997), “Non Linear Inference and Cluster-Weighted Modeling”, Annals of the New York Academy of Sciences, 808, 18-24.

  19. GERSHENFELD, N. (1999), The Nature of Mathematical Modelling, Cambridge: Cambridge University Press, pp. 101–130.

  20. GERSHENFELD, N., SCHÖNER, B., and METOIS, E. (1999), “Cluster-Weighted Modelling for Time-Series Analysis”, Nature, 397, 329-332.

  21. GRESELIN, F., and INGRASSIA, S. (2010), “Constrained Monotone EM Algorithms of Multivariate t distributions”, Statistics & Computing, 20, 9–22.

  22. INGRASSIA, S. (2004), “A Likelihood-Based Constrained Algorithm for Multivariate Normal Mixture Models”, Statistical Methods & Applications, 13, 151–166.

  23. INGRASSIA, S., and ROCCI, R. (2007), “Constrained Monotone EM Algorithms for Finite Mixture of Multivariate Gaussians”, Computational Statistics & Data Analysis, 51, 5339–5351.

  24. JANSEN, R.C. (1993), “Maximum Likelihood in a Generalized Linear Finite Mixture Model by Using the EM Algorithm”, Biometrics, 49, 227–231.

  25. JORDAN, M.I. (1995), “Why the Logistic Function? A Tutorial Discussion on Probabilities and Neural Networks”, MIT Computational Cognitive Science Report 9503.

  26. JORDAN, M.I., and JACOBS, R.A. (1994), “Hierarchical Mixtures of Experts and the EM Algorithm”, Neural Computation, 6, 181–224.

  27. KAN, R., and ZHOU, G. (2006), “Modelling Non-Normality Using Multivariate t: Implications for Asset Pricing”, Working paper, Washington University, St. Louis.

  28. LANGE, K.L., LITTLE, R.J.A., and TAYLOR, J.M.G. (1989), “Robust StatisticalModeling Using the t Distribution”, Journal of the American Statistical Society, 84(408), 881–896.

  29. LEISCH, F. (2004), “Flexmix: A General Framework for Finite Mixture Models and Latent Class Regression in R”, Journal of Statistical Software, 11(8), 1–18.

  30. LIU, C., and RUBIN, D.M. (1995), “ML Estimation of the t Distribution using EM and its Extensions, ECM and ECME”, Statistica Sinica, 5, 19–39.

  31. MARDIA, K.V., KENT, J.T., and BIBBY, J.M. (1979), Multivariate Analysis, London: Academic Press.

  32. McLACHLAN, G.J., and BASFORD, K.E. (1988), Mixture Models: Inference and Applications to Clustering, New York: Marcel Dekker.

  33. McLACHLAN, G.J., and PEEL, D. (1998), “Robust Cluster Analysis via Mixtures of Multivariate t-distributions”’ in Lecture Notes in Computer Science, Vol. 1451, eds. A. Amin, D. Dori, P. Pudil, and H. Freeman, Berlin: Springer-Verlag, pp. 658–666.

  34. McLACHLAN, G.J., and PEEL, D. (2000), Finite Mixture Models, New York: Wiley.

  35. NADARAJAH, S., and KOTZ, S. (2005), “Mathematical Properties of the Multivariate t Distributions”, Acta Applicandae Mathematicae, 89, 53–84.

  36. NEWCOMB, S. (1886), “A Generalized Theory of the Combination of Observations so as to Obtain the Best Result”, American Journal of Mathematics, 8, 343–366.

  37. NG, S.K., and McLACHLAN, G.J. (2007), “Extension of Mixture-of-Experts Networks for Binary Classification of Hierarchical Data”, Artificial Intelligence in Medicine, 41, 57–67.

  38. NG, S.K., and McLACHLAN, G.J. (2008), “Expert Networks with Mixed Continuous and Categorical Feature Variables: A Location Modeling Approach, in: Machine Learning Research Progress, eds. H. Peters and M. Vogel, New York: Hauppauge, pp. 355–368.

  39. NIERENBERG, D.W., STUKEL, T.A., BARON, J., DAIN, B.J., and GREENBERG, R. (1989), “Determinants of Plasma Levels of Beta-carotene and Retinol”, American Journal of Epidemiology, 130(3), 511–521.

  40. PEARSON, K. (1894), “Contributions to the Mathematical Theory of Evolution”, Philosophical Transactions of the Royal Society of London A, 185, 71–110.

  41. PEEL, D., and McLACHLAN, G.J. (2000), “Robust Mixture Modelling Using the t Distribution”, Statistics & Computing, 10, 339–348.

  42. PENG, F., JACOBS, R.A., and TANNER, M.A. (1996), “Bayesian Inference in Mixtures of- Experts and Hierarchical Mixtures-of-Experts Models with an Application to Speech Recognition”, Journal of the American Statistical Association, 91, 953–960.

  43. PINHEIRO, J.C., LIU, C., and WU, Y.N. (2001), “Efficient Algorithms for Robust Estimation in Linear Mixed-Effects Models Using the Multivariate t Distribution”, Journal of Computational and Graphical Statistics, 10, 249–276.

  44. QUANDT, R.E. (1972), “A New Approach to Estimating Switching Regressions”, Journal of the American Statistical Society, 67, 306–310.

  45. RIANI,M., CERIOLI, A., ATKINSON, A.C., PERROTTA, D., and TORTI, F. (2008), “Fitting Mixtures of Regression Lines with the Forward Search”, in Mining Massive Data Sets for Security, eds. F. Fogelman-Soulié, D. Perrotta, J. Piskorki and R. Steinberg, Amsterdam: IOS Press, pp. 271–286.

  46. RIANI, M., ATKINSON, A.C., and CERIOLI, A. (2009), “Finding an Unknown Number of Multivariate Outliers”, Journal of the Royal Statistical Society B, 71(2), 447–466.

  47. SCHLATTMANN, P. (2009), Medical Applications of Finite Mixture Models, Berlin-Heidelberg: Springer-Verlag.

  48. SCHÖNER, B. (2000), Probabilistic Characterization and Synthesis of Complex Data Driven Systems, Ph.D. Thesis, MIT.

  49. SCHÖNER, B., and GERSHENFELD, N. (2001), “Cluster Weighted Modeling: Probabilistic Time Series Prediction, Characterization, and Synthesis” in Nonlinear Dynamics and Statistics, ed. A.I. Mees, Boston: Birkhauser, pp. 365–385.

  50. TITTERINGTON, D.M., SMITH, A.F.M., and MAKOV, U.E. (1985), Statistical Analysis of Finite Mixture Distributions, New York: Wiley.

  51. WANG, P., PUTERMAN, M.L., COCKBURN, I., and LE, N. (1996), “Mixed Poisson Regression Models with Covariate Dependent Rates”, Biometrics, 52, 381–400.

  52. WEDEL, M. (2002), “Concomitant Variables in Finite Mixture Models”, Statistica Nederlandica, 56(3), 362–375.

  53. WEDEL, M., and DESARBO, W. (1995), “A Mixture Likelihood Approach for Generalized Linear Models”, Journal of Classification, 12, 21–55.

  54. WEDEL, M., and DESARBO, W. (2002), “Market Segment Derivation and Profiling via a Finite Mixture Model Framework”, Marketing Letters, 13, 17–25.

  55. WEDEL, M., and KAMAMURA, W.A. (2000), Market Segmentation. Conceptual and Methodological Foundations, Boston: Kluwer Academic Publishers.

  56. ZELLNER, A. (1976), “Bayesian and Non-Bayesian Analysis of the RegressionModel with Multivariate Student-t Error Terms”, Journal of the American Statistical Society, 71, 400–405.

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Author information

Correspondence to Salvatore Ingrassia.

Additional information

The authors sincerely thank the referees for their interesting comments and valuable suggestions. We also thank Antonio Punzo for helpful discussions.

An erratum to this article is available at http://dx.doi.org/10.1007/s00357-015-9190-2.

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Ingrassia, S., Minotti, S.C. & Vittadini, G. Local Statistical Modeling via a Cluster-Weighted Approach with Elliptical Distributions. J Classif 29, 363–401 (2012). https://doi.org/10.1007/s00357-012-9114-3

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Keywords

  • Cluster-weighted modeling
  • Mixture models
  • Model-based clustering