[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Finite mixture biclustering of discrete type multivariate data

  • Regular Article
  • Published:
Advances in Data Analysis and Classification Aims and scope Submit manuscript

Abstract

Many of the methods which deal with clustering in matrices of data are based on mathematical techniques such as distance-based algorithms or matrix decomposition and eigenvalues. In general, it is not possible to use statistical inferences or select the appropriateness of a model via information criteria with these techniques because there is no underlying probability model. This article summarizes some recent model-based methodologies for matrices of binary, count, and ordinal data, which are modelled under a unified statistical framework using finite mixtures to group the rows and/or columns. The model parameter can be constructed from a linear predictor of parameters and covariates through link functions. This likelihood-based one-mode and two-mode fuzzy clustering provides maximum likelihood estimation of parameters and the options of using likelihood information criteria for model comparison. Additionally, a Bayesian approach is presented in which the parameters and the number of clusters are estimated simultaneously from their joint posterior distribution. Visualization tools focused on ordinal data, the fuzziness of the clustering structures, and analogies of various standard plots used in the multivariate analysis are presented. Finally, a set of future extensions is enumerated.

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
£29.99 /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 includes VAT (United Kingdom)

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

References

  • Agresti A (2010) Analysis of ordinal categorical data, 2nd edn. Wiley series in probability and statistics. Wiley, Hoboken

    Book  MATH  Google Scholar 

  • Agresti A (2013) Categorical data analysis, 3rd edn. Wiley series in probability and statistics. Wiley, Hoboken

    MATH  Google Scholar 

  • Agresti A, Lang JB (1993) Quasi-symmetric latent class models, with application to rater agreement. Biometrics 49(1):131–139

    Article  Google Scholar 

  • Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csaki F (eds) 2nd international symposium on information theory, pp 267–281

  • Anderson JA (1984) Regression and ordered categorical variables. J R Stat Soc Ser B 46(1):1–30

    MathSciNet  MATH  Google Scholar 

  • Arnold R, Hayakawa Y, Yip P (2010) Capture-recapture estimation using finite mixtures of arbitrary dimension. Biometrics 66(2):644–655

    Article  MathSciNet  MATH  Google Scholar 

  • Bartolucci F, Bacci S, Pennoni F (2014) Longitudinal analysis of self-reported health status by mixture latent auto-regressive models. J R Stat Soc Ser C (Appl Stat) 63(2):267–288

    Article  MathSciNet  Google Scholar 

  • Biernacki C, Celeux G, Govaert G (1998) Assessing a mixture model for clustering with the integrated completed likelihood. Technical Report 3521, INRIA, Rhne-Alpes

  • Böhning D, Seidel W, Alfò M, Garel B, Patilea V, Walther G (2007) Advances in mixture models. Comput Stat Data Anal 51(11):5205–5210

    Article  MathSciNet  MATH  Google Scholar 

  • Breen R, Luijkx R (2010) Assessing proportionality in the proportional odds model for ordinal logistic regression. Sociol Methods Res 39(1):3–24

    Article  MathSciNet  Google Scholar 

  • Browne RP, McNicholas PD (2012) Model-based clustering, classification, and discriminant analysis of data with mixed type. J Stat Plan Inference 142(11):2976–2984

    Article  MathSciNet  MATH  Google Scholar 

  • Burnham KP, Anderson DR (2002) Model selection and multi-model inference: a practical information-theoretic approach, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  • Cai JH, Song XY, Lam KH, Ip EHS (2011) A mixture of generalized latent variable models for mixed mode and heterogeneous data. Comput Stat Data Anal 55(11):2889–2907

    Article  MathSciNet  MATH  Google Scholar 

  • Cappé O, Robert C, Rydén T (2003) Reversible jump, birth-and-death, and more general continuous time MCMC samplers. J R Stat Soc Ser B 65(3):679–700

    Article  MathSciNet  MATH  Google Scholar 

  • Celeux G (1998) Bayesian inference for mixtures: the label switching problem. In: Proceedings in computational statistics 1998 (COMPSTAT98), Physica-Verlag HD, pp 227–232

  • Costilla R, Liu I, Arnold R (2015) A Bayesian model-based approach to estimate clusters in repeated ordinal data. In: JSM Proceedings, biometrics section, pp 545–556

  • Dellaportas P, Papageorgiou I (2006) Multivariate mixtures of normals with unknown number of components. Stat Comput 16(1):57–68

    Article  MathSciNet  Google Scholar 

  • Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B 39(1):1–38

    MathSciNet  MATH  Google Scholar 

  • DeSantis SM, Houseman EA, Coull BA, Stemmer-Rachamimov A, Betensky RA (2008) A penalized latent class model for ordinal data. Biostatistics 9(2):249–262

    Article  MATH  Google Scholar 

  • Diggle PJ, Heagerty PJ, Liang KY, Zeger SL (2002) Analysis of longitudinal data, 2nd edn. Oxford University Press, Oxford

    MATH  Google Scholar 

  • van Dijk B, van Rosmalen J, Paap R (2009) A Bayesian approach to two-mode clustering. Technical Report

  • Everitt BS, Landau S, Leese M, Stahl D (2011) Cluster analysis, 5th edn. Wiley, Chichester

    Book  MATH  Google Scholar 

  • Fernández D, Arnold R (2016) Model selection for mixture-based clustering for ordinal data. Aust NZ J Stat 58(4):437–472

    Article  MathSciNet  MATH  Google Scholar 

  • Fernández D, Liu I (2016) A goodness-of-fit test for the ordered stereotype model. Stat Med 35(25):4660–4696

    Article  MathSciNet  Google Scholar 

  • Fernández D, Pledger S (2016) Categorising count data into ordinal responses with application to ecological communities. J Agric Biol Environ Stat 21(2):348–362

    Article  MathSciNet  MATH  Google Scholar 

  • Fernández D, Pledger S, Arnold R (2014) Introducing spaced mosaic plots. Research Report Series. ISSN: 1174-2011. 14-3, School of Mathematics, Statistics and Operations Research, VUW. http://msor.victoria.ac.nz/foswiki/pub/Main/ResearchReportSeries/TechReport_Spaced_Mosaic_Plots.pdf

  • Fernández D, Arnold R, Pledger S (2016) Mixture-based clustering for the ordered stereotype model. Comput Stat Data Anal 93:46–75

    Article  MathSciNet  MATH  Google Scholar 

  • Fraley C, Raftery AE (1998) How many clusters? Which clustering method? Answers via model-based cluster analysis. Comput J 41(8):578–588

    Article  MATH  Google Scholar 

  • Fraley C, Raftery AE (2002) Model-based clustering, discriminant analysis, and density estimation. J Am Stat Assoc 97(458):611–631

    Article  MathSciNet  MATH  Google Scholar 

  • Fraley C, Raftery AE (2007) Bayesian regularization for normal mixture estimation and model-based clustering. J Classif 24(2):155–181

    Article  MathSciNet  MATH  Google Scholar 

  • Friedman HP, Rubin J (1967) On some invariant criteria for grouping data. J Amer Stat Assoc 62:1159–1178

    Article  MathSciNet  Google Scholar 

  • Friendly M (1991) Mosaic displays for multiway contingency tables. Technival Report 195, Department of Psychology Reports, New York University

  • Frühwirth-Schnatter S (2001) Markov chain Monte Carlo estimation of classical and dynamic switching and mixture models. J Am Stat Assoc 453(96):194–209

    Article  MathSciNet  MATH  Google Scholar 

  • Frühwirth-Schnatter S (2006) Finite mixture and Markov switching models. Wiley, New York

    MATH  Google Scholar 

  • Frühwirth-Schnatter S, Pamminger C, Weber A, Winter-Ebmer R (2012) Labor market entry and earnings dynamics: Bayesian inference using mixtures-of-experts markov chain clustering. J Appl Econom 27(7):1116–1137

    Article  MathSciNet  Google Scholar 

  • Frydman H (2005) Estimation in the mixture of markov chains moving with different speeds. J Am Stat Assoc 100(471):1046–1053

    Article  MathSciNet  MATH  Google Scholar 

  • Goodman LA (1974) Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61:215–231

    Article  MathSciNet  MATH  Google Scholar 

  • Gotelli NJ, Graves GR (1996) Null models in ecology. Smithsonian Institution Press, Washington

    Google Scholar 

  • Govaert G, Nadif M (2003) Clustering with block mixture models. Pattern Recognit 36(2):463–473

    Article  MATH  Google Scholar 

  • Govaert G, Nadif M (2005) An EM algorithm for the block mixture model. IEEE Trans Pattern Anal Mach Intell 27(4):643–647

    Article  MATH  Google Scholar 

  • Govaert G, Nadif M (2010) Latent block model for contingency table. Commun Stat Theory Methods 39(3):416–425

    Article  MathSciNet  MATH  Google Scholar 

  • Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4):711–732

    Article  MathSciNet  MATH  Google Scholar 

  • Haberman SJ (1979) Analysis of qualitative data, vol 2. Academic Press, New York

    Google Scholar 

  • Hartigan JA, Kleiner B (1981) Mosaics for contingency tables. In: Proceedings of the 13th symposium on the interface between computer sciencies and statistics, Springer, pp 268–273

  • Hartigan JA, Wong MA (1979) Algorithm as 136: a k-means clustering algorithm. J R Stat Soc Ser C (Appl Stat) 28(1):100–108

    MATH  Google Scholar 

  • Hasnat MA, Velcin J, Bonnevay S, Jacques J (2015) Simultaneous clustering and model selection for multinomial distribution: a comparative study. In: International symposium on intelligent data analysis, Springer, pp 120–131

  • Hui FK, Taskinen S, Pledger S, Foster SD, Warton DI (2015) Model-based approaches to unconstrained ordination. Methods Ecol Evol 6(4):399–411

    Article  Google Scholar 

  • Hurn M, Justel A, Robert CP (2003) Estimating mixture of regressions. J Comput Graph Stat 12(1):55–79

    Article  MathSciNet  Google Scholar 

  • Hurvich CM, Tsai CL (1989) Regression and time series model selection in small samples. Biometrika 76(2):297–307

    Article  MathSciNet  MATH  Google Scholar 

  • Jasra A, Holmes CC, Stephens DA (2005) MCMC and the label switching problem in Bayesian mixture models. Stat Sci 20(1):50–67

    Article  MATH  Google Scholar 

  • Jobson JD (1992) Applied multivariate data analysis: categorical and multivariate methods. Springer texts in statistics. Springer, Berlin

    Book  MATH  Google Scholar 

  • Johnson SC (1967) Hierarchical clustering schemes. Psychometrika 32(3):241–254

    Article  MATH  Google Scholar 

  • Lee K, Marin JM, Robert C, Mengersen K (2008) Bayesian inference on mixtures of distributions. In: Proceedings of the platinum jubilee of the Indian statistical institute, p 776

  • MacQueen J (1967) Some methods for classification and analysis of multivariate observations. In: Cam LML, Neyman J (eds) Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, University of California Press, pp 281–297

  • Manly BFJ (2005) Multivariate statistical methods: a primer. Chapman & Hall, London

    MATH  Google Scholar 

  • Manly BFJ (2007) Randomization, bootstrap and monte carlo methods in biology, 3rd edn. Chapman & Hall, London

    MATH  Google Scholar 

  • Marin JM, Robert C (2007) Bayesian core: a practical approach to computational Bayesian statistics. Springer texts in statistics. Springer, Berlin

    MATH  Google Scholar 

  • Marin JM, Mengersen K, Robert C (2005) Bayesian modelling and inferences on mixtures of distributions. In: Dey D, Rao CR (eds) Handbook of statistics, vol 25. Springer, New York

    Google Scholar 

  • Marrs AD (1998) An application of reversible-jump MCMC to multivariate spherical Gaussian mixtures. In: Jordan MI, Kearns MJ, Solla SA (eds) Advances in neural information processing systems, vol 10. MIT Press, Cambridge, pp 577–583

    Google Scholar 

  • Matechou E, Liu I, Pledger S, Arnold R (2011) Biclustering models for ordinal data, presentation at the NZ Statistical Assn. In: Annual conference, University of Auckland, 28–31 Aug 2011

  • Matechou E, Liu I, Fernández D, Farias M, Gjelsvik B (2016) Biclustering models for two-mode ordinal data. Psychometrika 81(3):611–624

    Article  MathSciNet  MATH  Google Scholar 

  • Maurizio V (2001) Double k-means clustering for simultaneous classification of objects and variables. Advances in classification and data analysis. Springer, Berlin, Heidelberg, pp 43–52

    Chapter  Google Scholar 

  • McCullagh P (1980) Regression models for ordinal data. J R Stat Soc 42(2):109–142

    MathSciNet  MATH  Google Scholar 

  • McCullagh P, Yang J (2008) How many clusters? Bayesian Anal 3(1):101–120

    Article  MathSciNet  MATH  Google Scholar 

  • McCune B, Grace JB (2002) Analysis of ecological communities. Struct Equ Model 28(2)

  • McCutcheon AL (1987) Latent class analysis. Sage Publications, Thousand Oaks

    Book  Google Scholar 

  • McLachlan G, Peel D (2004) Finite mixture models. Wiley series in probability and statistics. Wiley, New York

    MATH  Google Scholar 

  • McLachlan GJ (1982) The classification and mixture maximum likelihood approaches to cluster analysis. Handb Stat 2(299):199–208

    Article  MATH  Google Scholar 

  • McLachlan GJ (1987) On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Appl Stat 36(3):318–324

    Article  Google Scholar 

  • McLachlan GJ, Basford KE (1988) Mixture models: inference and applications to clustering. Statistics, textbooks and monographs. M. Dekker, New York

    MATH  Google Scholar 

  • McLachlan GJ, Krishnan T (1997) The EM algorithm and extensions. Wiley series in probability and statistics: applied probability and statistics. Wiley, Hoboken

    MATH  Google Scholar 

  • McParland D, Gormley IC (2013) Clustering ordinal data via latent variable models. In: Lausen B, Van den Poel D, Ultsch A (eds) Algorithms from and for nature and life, studies in classification, data analysis, and knowledge organization. Springer, Berlin, pp 127–135

    Google Scholar 

  • McParland D, Gormley IC (2016) Model based clustering for mixed data: clustMD. Adv Data Anal Classif 10(2):155–169

    Article  MathSciNet  MATH  Google Scholar 

  • Melnykov V (2013) Finite mixture modelling in mass spectrometry analysis. J R Stat Soc Ser C (Appl Stat) 62(4):573–592

    Article  MathSciNet  Google Scholar 

  • Melnykov V, Maitra R (2010) Finite mixture models and model-based clustering. Stat Surv 4(9):80–116

    Article  MathSciNet  MATH  Google Scholar 

  • Moustaki I (2000) A latent variable model for ordinal variables. Appl Psychol Meas 24(3):211–233

    Article  MathSciNet  Google Scholar 

  • Nadif M, Govaert G (2005) A comparison between block CEM and two-way CEM algorithms to cluster a contingency table. In: European conference on principles of data mining and knowledge discovery, Springer, pp 609–616

  • Pamminger C, Frühwirth-Schnatter S et al (2010) Model-based clustering of categorical time series. Bayesian Anal 5(2):345–368

    Article  MathSciNet  MATH  Google Scholar 

  • Pledger S (2000) Unified maximum likelihood estimates for closed capture-recapture models using mixtures. Biometrics 56(2):434–442

    Article  MATH  Google Scholar 

  • Pledger S, Arnold R (2014) Multivariate methods using mixtures: correspondence analysis, scaling and pattern-detection. Comput Stat Data Anal 71:241–261

    Article  MathSciNet  MATH  Google Scholar 

  • Quinn GP, Keough MJ (2002) Experimental design and data analysis for biologists. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Raftery AE, Dean N (2006) Variable selection for model-based clustering. J Am Stat Assoc 101(473):168–178

    Article  MathSciNet  MATH  Google Scholar 

  • Richardson S, Green PJ (1997) On Bayesian analysis of mixtures with an unknown number of components. J R Stat Soc Ser B 59(4):731–792

    Article  MathSciNet  MATH  Google Scholar 

  • Rocci R, Vichi M (2008) Two-mode multi-partitioning. Comput Stat Data Anal 52(4):1984–2003

    Article  MathSciNet  MATH  Google Scholar 

  • Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464

    Article  MathSciNet  MATH  Google Scholar 

  • Self SG, Liang KY (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J Am Stat Assoc 82(398):605–610

    Article  MathSciNet  MATH  Google Scholar 

  • Silvestre C, Cardoso MG, Figueiredo MA (2014) Identifying the number of clusters in discrete mixture models. arXiv:1409.7419

  • Skrondal A, Rabe-Hesketh S (2004) Generalized latent variable modeling: multilevel, longitudinal, and structural equation models. Monographs on statistics and applied probability. Chapman & Hall, London

    Book  MATH  Google Scholar 

  • Stahl D, Sallis H (2012) Model-based cluster analysis. Wiley Interdiscip Rev Comput Stat 4(4):341–358

    Article  Google Scholar 

  • Stephens M (2000a) Bayesian analysis of mixture models with an unknown number of components-an alternative to reversible jump methods. Ann Stat 28(1):40–74

    Article  MathSciNet  MATH  Google Scholar 

  • Stephens M (2000b) Dealing with label switching in mixture models. J R Stat Soc Ser B 62(4):795–809

    Article  MathSciNet  MATH  Google Scholar 

  • Sugar CA, James GM (2003) Finding the number of clusters in a dataset: an information-theoretic approach. J Am Stat Assoc 98(463):750–763

    Article  MathSciNet  MATH  Google Scholar 

  • Tibshirani R, Walther G (2005) Cluster validation by prediction strength. J Comput Graph Stat 14(3):511–528

    Article  MathSciNet  Google Scholar 

  • Vermunt JK (2001) The use of restricted latent class models for defining and testing nonparametric and parametric item response theory models. Appl Psychol Meas 25(3):283–294

    Article  MathSciNet  Google Scholar 

  • Vermunt JK, Hagenaars JA (2004) Ordinal longitudinal data analysis. In: Hauspie R, Cameron N, Molinari L (eds) Methods in human growth research. Cambridge University Press, Cambridge

    Google Scholar 

  • Vermunt JK, Van Dijk L (2001) A nonparametric random-coefficients approach: the latent class regression model. Multilevel Model Newsl 13(2):6–13

    Google Scholar 

  • Vichi M (2001) Double k-means clustering for simultaneous classification of objects and variables. In: Borra S, Rocci R, Vichi M, Schader M (eds) Studies in classification, data analysis, and knowledge organization. Springer, Berlin, pp 43–52

    Google Scholar 

  • Wagenmakers EJ, Lee M, Lodewyckx T, Iverson GJ (2008) Bayesian versus frequentist inference. Springer, Berlin

    Book  Google Scholar 

  • Wu X, Kumar V, Quinlan JR, Ghosh J, Yang Q, Motoda H, McLachlan GJ, Ng A, Liu B, Yu PS, Zhou ZH, Steinbach M, Hand DJ, Steinberg D (2008) Top 10 algorithms in data mining. Knowl Inf Syst 14(1):1–37

    Article  Google Scholar 

  • Wyse J, Friel N (2012) Block clustering with collapsed latent block models. Stat Comput 22(2):415–428

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Z, Chan KL, Wu Y, Chen C (2004) Learning a multivariate gaussian mixture model with the reversible jump MCMC algorithm. Stat Comput 14(4):343–355

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the Marsden Fund on “Dimension reduction for mixed type multivariate data” (Award Number E2987-3648) from New Zealand Government funding, administrated by the Royal Society of New Zealand.

Author information

Authors and Affiliations

  1. Institut de Recerca Sant Joan de Déu, Parc Sanitari Sant Joan de Déu, CIBERSAM, Dr. Antoni Pujades, 42, 08830, Sant Boi de Llobregat, Barcelona, Spain

    Daniel Fernández

  2. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

    Daniel Fernández, Richard Arnold, Shirley Pledger & Ivy Liu

  3. Institute for Molecular Bioscience, University of Queensland, Brisbane, Australia

    Roy Costilla

Authors
  1. Daniel Fernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard Arnold
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shirley Pledger
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ivy Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Roy Costilla
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Fernández.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 179 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fernández, D., Arnold, R., Pledger, S. et al. Finite mixture biclustering of discrete type multivariate data. Adv Data Anal Classif 13, 117–143 (2019). https://doi.org/10.1007/s11634-018-0324-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11634-018-0324-3

Keywords

Mathematics Subject Classification

Search

Navigation