Skip to main content
SpringerLink
Log in

FINDCLUS: Fuzzy INdividual Differences CLUStering

  • Published:
Journal of Classification Aims and scope Submit manuscript

Abstract

ADditive CLUStering (ADCLUS) is a tool for overlapping clustering of two-way proximity matrices (objects × objects). In Simple Additive Fuzzy Clustering (SAFC), a variant of ADCLUS is introduced providing a fuzzy partition of the objects, that is the objects belong to the clusters with the so-called membership degrees ranging from zero (complete non-membership) to one (complete membership). INDCLUS (INdividual Differences CLUStering) is a generalization of ADCLUS for handling three-way proximity arrays (objects × objects × subjects). Here, we propose a fuzzified alternative to INDCLUS capable to offer a fuzzy partition of the objects by generalizing in a three-way context the idea behind SAFC. This new model is called Fuzzy INdividual Differences CLUStering (FINDCLUS). An algorithm is provided for fitting the FINDCLUS model to the data. Finally, the results of a simulation experiment and some applications to synthetic and real data are discussed.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • ARABIE, P., and CARROLL, J.D. (1980), “MAPCLUS: A Mathematical Programming Approach to Fitting the ADCLUS Model”, Psychometrika, 45, 211–235.

    Article  MATH  Google Scholar 

  • ARABIE, P., CARROLL, J.D., and DE SARBO, W.S. (1987), Three-way Scaling and Clustering, Newbury Park, California: Sage.

    Google Scholar 

  • BEZDEK, J.C. (1981), Pattern Recognition with Fuzzy Objective Function Algorithms, New York: Plenum Press.

    Book  MATH  Google Scholar 

  • CARROLL, J.D., and ARABIE, P., (1983), “INDCLUS: An Individual Differences Generalization of the ADCLUS Model and the MAPCLUS Algorithm”, Psychometrika, 48, 157–169.

    Article  Google Scholar 

  • CHATURVEDI, A.D., and CARROLL, J.D. (1994), “An Alternating Combinatorial Approach to Fitting the INDCLUS and Generalized INDCLUS Models”, Journal of Classification, 11, 155–170.

    Article  MATH  Google Scholar 

  • CHATURVEDI, A.D., and CARROLL, J.D. (2006), “CLUSCALE (“CLUstering and Multidimensional SCAL[E]ing”): A Three-Way Hybrid Model Incorporating Overlapping Clustering and Multidimensional Scaling Structure”, Journal of Classification, 23, 269–299.

    Article  MathSciNet  Google Scholar 

  • CICHOCKI, A., and ZDUNEK, R. (2007), “Regularized Alternating Least Squares Algorithms for Non-Negative Matrix/Tensor Factorization”, in Advances in Neural Networks (ISNN2007), Lecture Notes in Computer Science, 4493, eds. D. Liu, S. Fei, Z. Hou, H. Zhang, and C. Sun, pp. 793–802.

  • CICHOCKI, A., ZDUNEK, R., CHOI, S., PLEMMONS, R., and AMARI, S.I. (2007), “Non-Negative Tensor Factorization Using Alpha and Beta Divergences”, in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP2007), pp. III1393–III1396.

  • COPPI, R., D’URSO, P., and GIORDANI, P. (2010), “A Fuzzy Clustering Model for Multivariate Spatial Time Series”, Journal of Classification, 27, 54–88.

    Article  MathSciNet  Google Scholar 

  • CUTLER, A., and BREIMAN, L. (1994), “Archetypal Analysis”, Technometrics, 36, 338–347.

    MathSciNet  MATH  Google Scholar 

  • DEPRIL, D. (2009), Algorithms for Additive Overlapping Clustering, Ph.D. Thesis, Katholieke Universiteit Leuven.

  • GORDON, A.D., and VICHI, M. (2001), “Fuzzy Partition Models for Fitting a Set of Partitions”, Psychometrika, 66, 229–248.

    Article  MathSciNet  Google Scholar 

  • HEISER, W.J., and GROENEN, P.J.F. (1997), “Cluster Differences Scaling with a Within-Clusters Loss Component and a Fuzzy Successive Approximation Strategy to Avoid Local Minima”, Psychometrika, 62, 63–83.

    Article  MathSciNet  MATH  Google Scholar 

  • KIERS, H.A.L. (1997), “A Modification of the SINDCLUS Algorithm for Fitting the ADCLUS and INDCLUS models”, Journal of Classification, 14, 297–310.

    Article  MATH  Google Scholar 

  • LAWSON, C.L, and HANSON, R.J. (1995), Solving Least Squares Problems, (Classics in Applied Mathematics, Vol. 15), Philadelphia: SIAM.

    Book  Google Scholar 

  • MIRKIN, B.G. (1996), Mathematical Classification and Clustering, Dordrecht: Kluwer Academic Publishers.

    Book  MATH  Google Scholar 

  • MOOIJAART, A., VAN DER HEIJDEN, P.G.M., and VAN DER ARK, L.A. (1999), “A Least Squares Algorithm for a Mixture Model for Compositional Data”, Computational Statistics and Data Analysis, 30, 359–379.

    Article  MATH  Google Scholar 

  • RAO, C.R., and MITRA, S.K. (1971), Generalized Inverse of Matrices and its Applications, New York: Wiley.

    MATH  Google Scholar 

  • ROSENBORG, S., and KIM, M.P. (1975), “The Method of Sorting as a Data-Gathering Procedure in Multivariate Research”, Multivariate Behavioral Research, 10, 489–502.

    Article  Google Scholar 

  • SATO, M., and SATO, Y. (1994a), “An Additive Fuzzy Clustering Model”, Japanese Journal of Fuzzy Theory and Systems, 6, 185–204.

    Google Scholar 

  • SATO, M., and SATO, Y. (1994b), “On a Multicriteria Fuzzy Clustering Method for 3-Way Data”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2, 127–142.

    Article  Google Scholar 

  • SATO, M., and SATO, Y. (1995a), “A General Fuzzy Clustering Model Based on Aggregation Operators”, Behaviormetrika, 22, 115–128.

    Article  Google Scholar 

  • SATO, M., and SATO, Y. (1995b), “On A General Fuzzy Additive Clustering Model”, Intelligent Automation and Soft Computing, 1, 439-448.

    Google Scholar 

  • SATO, M., and SATO, Y. (1995c), “Extended Fuzzy Clustering Models for Asymmetric Similarity”, in Fuzzy Logic and Soft Computing, eds. B. Bouchon-Meunier, R.R. Yager, and L.A. Zadeh, World Scientific, pp. 228–237.

  • SATO, M., SATO, Y., and JAIN, L.C. (1997), Fuzzy Clustering Models and Applications, Berlin Heidelberg: Springer-Verlag.

    MATH  Google Scholar 

  • SHEPARD, R.N., and ARABIE, P. (1979), “Additive Clustering: Representation of Similarities as Combinations of Discrete Overlapping Properties”, Psychological Review, 86, 87–123.

    Article  Google Scholar 

  • TAKANE, Y., YOUNG, F.W., and DE LEEUW, J. (1977), “Nonmetric Individual Differences Multidimensional Scaling: An Alternating Least Squares Method with Optimal Scaling Features”, Psychometrika, 42, 7–67.

    Article  MATH  Google Scholar 

  • TEN BERGE, J., and KIERS, H.A.L. (2005), “A Comparison of Two Methods for Fitting the INDCLUS Model”, Journal of Classification, 22, 273–286.

    Article  MathSciNet  Google Scholar 

  • TER BRAAK, C.J.F., KOURMPETIS, Y., KIERS, H.A.L., and BINK, M.C.A.M. (2009), “Approximating a Similarity Matrix by a Latent Class Model: A Reappraisal of Additive Fuzzy Clustering”, Computational Statistics and Data Analysis, 53, 3183–3193.

    Article  MathSciNet  MATH  Google Scholar 

  • TER BRAAK, C.J.F., BOER, M.P., TOTIR, L.R., WINKLER, C.R., SMITH, O.S., and BINK, M.C.A.M. (2010), “Identity-By-Descent Matrix Decomposition Using Latent Ancestral Allele Models”, Genetics, 185, 1045–1057.

    Article  Google Scholar 

  • VAN ROSMALEN, J., GROENEN, P.J.F., TREJOS, J., and CASTILLO, W. (2009), “Optimization Strategies for Two-Mode Partitioning”, Journal of Classification, 26, 155–181.

    Article  MathSciNet  Google Scholar 

  • VICARI, D., and VICHI, M. (2009), “Structural Classification Analysis of Three-Way Dissimilarity Data”, Journal of Classification, 26, 121–154.

    Article  MathSciNet  Google Scholar 

  • VICHI, M., ROCCI, R., and KIERS, H.A.L. (2007), “Simultaneous Component and Clustering Models for Three-Way Data: Within and Between Approaches”, Journal of Classification, 24, 71–98.

    Article  MathSciNet  MATH  Google Scholar 

  • WANG W., CICHOCKI A., MØRUP M., SMARAGDIS P., and ZDUNEK R. (2008), “Advances in Nonnegative Matrix and Tensor Factorization”, Computational Intelligence and Neuroscience, 2008, 1–3.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistical Sciences, Sapienza University of Rome, P.le Aldo Moro, 5, 00185, Rome, Italy

    Paolo Giordani

  2. Heymans Institute, University of Groningen, Grote Kruisstraat 2/1, 9712, TS, Groningen, The Netherlands

    Henk A.L. Kiers

Authors
  1. Paolo Giordani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henk A.L. Kiers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paolo Giordani.

Additional information

We would like to thank the Editor and four anonymous reviewers whose comments and criticisms helped us improve the quality of the paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giordani, P., Kiers, H.A. FINDCLUS: Fuzzy INdividual Differences CLUStering. J Classif 29, 170–198 (2012). https://doi.org/10.1007/s00357-012-9109-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00357-012-9109-0

Keywords

Search

Navigation