Clustering of functional data in a low-dimensional subspace. (English) Zbl 1254.62077

Yamamoto, Michio

Adv. Data Anal. Classif., ADAC 6, No. 3, 219-247 (2012).

Summary: To find optimal clusters of functional objects in a lower-dimensional subspace of data, a sequential method, called tandem analysis, is often used, though such a method is problematic. A new procedure is developed to find optimal clusters of functional objects and also to find an optimal subspace for clustering simultaneously. The method is based on the \(k\)-means criterion for functional data and seeks the subspace that is maximally informative of the clustering structure in the data. An efficient alternating least-squares algorithm is described, and the proposed method is extended to a regularized method. Analyses of artificial and real data examples demonstrate that the proposed method gives correct and interpretable results.

MSC:

62H30	Classification and discrimination; cluster analysis (statistical aspects)
65C60	Computational problems in statistics (MSC2010)

Keywords:

dimension reduction; smoothing

Software:

funHDDC; R; fda (R)

References:

[1]	Abraham C, Cornillon PA, Matzner-Lober E, Molinari N (2003) Unsupervised curve clustering using B-splines. Scand J Statist 30: 581–595 · Zbl 1039.91067
[2]	Arabie P, Hubert L (1994) Cluster analysis in marketing research. In: Bagozzi RP (eds) Advanced methods of marketing research. Blackwell Business, Cambridge, pp 160–189
[3]	Besse PC, Cardot H, Ferraty F (1997) Simultaneous non-parametric regressions of unbalanced longitudinal data. Comput Stat Data Anal 24: 255–270 · Zbl 0900.62199
[4]	Besse PC, Ramsay JO (1986) Principal components analysis of sampled functions. Psychometorika 51: 285–311 · Zbl 0623.62048
[5]	Boente G, Fraiman R (2000) Kernel-based functional principal components. Stat Probab Lett 48: 335–345 · Zbl 0997.62024
[6]	Bouveyron C, Jacques J (2011) Model-based clustering of time series in group-specific functional subspaces. Adv Data Anal Classif 5: 281–300 · Zbl 1274.62416
[7]	De Boor C (2001) A practical guide to splines, revised edition. Springer, New York · Zbl 0987.65015
[8]	de Leeuw J, Young FW, Takane Y (1976) Additive structure in qualitative data: An alternating least squares method with optimal scaling features. Psychometorika 41: 471–503 · Zbl 0351.92031
[9]	DeSarbo WS, Jedidi K, Cool K, Schendel D (1990) Simultaneous multidimensional unfolding and cluster analysis: an investigation of strategic groups. Mark Lett 2: 129–146
[10]	De Soete G, Carroll JD (1994) K-means clustering in a low-dimensional Euclidean space. In: Diday E, Lechevallier Y, Schader M, Bertrand P, Burtschy B (eds) New approaches in classification and data analysis. Springer, Heidelberg, pp 212–219
[11]	Dunford N, Schwartz JT (1988) Linear operators, spectral theory, self adjoint operators in Hilbert space, part 2. Interscience, NewYork · Zbl 0635.47002
[12]	Green PJ, Silverman BW (1994) Nonparametric regression and generalized linear models: a roughness penalty approach. Chapman and Hall, London · Zbl 0832.62032
[13]	Hardy A (1996) On the number of clusters. Comput Stat Data Anal 23: 83–96 · Zbl 0900.92186
[14]	Hartigan J (1975) Clustering algorithms. Wiley, New York · Zbl 0372.62040
[15]	Hubert L, Arabie P (1985) Comparing partitions. J Classif 2: 193–218 · Zbl 0587.62128
[16]	Kneip A (1994) Nonparametric estimation of common regressors for similar curve data. Ann Stat 22: 1386–1427 · Zbl 0817.62029
[17]	Illian JB, Prosser JI, Baker KL, Rangel-Castro JI (2009) Functional principal component data analysis: A new method for analysing microbial community fingerprints. J Microbiol Methods 79: 89–95
[18]	Lloyd SP (1982) Least squares quantization in PCM. IEEE Trans Inf Theory 28: 128–137 · Zbl 0504.94015
[19]	Milligan GW, Cooper MC (1985) An examination of procedures for determining the number of clusters in a data set. Psychometrika 50: 159–179
[20]	Ocaña FA, Aguilera AM, Valderrama MJ (1982) Functional principal components analysis by choice of norm. J Multivariate Anal 71: 262–276 · Zbl 0944.62059
[21]	Pezzulli SD, Silverman BW (1993) Some properties of smoothed principal components analysis for functional data. Comput Stat 8: 1–16 · Zbl 0775.62146
[22]	R Development Core Team (2005) R: A language and environment for statistical computing. R Foundation for Statistical Computing. Austria. ISBN 3-900051-07-0, URL http://www.R-project.org
[23]	Ramsay JO, Wang X, Flanagan R (1995) A functional data analysis of the pinch force of human fingers. J Roy Stat Soc Ser C 44: 17–30 · Zbl 0821.62085
[24]	Ramsay JO, Silverman BW (2005) Functional data analysis, 2nd Edn. Springer, New York
[25]	Rice JA, Silverman BW (1991) Estimating the mean and covariance structure nonparametrically when the data are curves. J Roy Stat Soc Ser B 53: 233–243 · Zbl 0800.62214
[26]	Rossi F, Conan-Guez B, Golli AE (2004) Clustering functional data with the SOM algorithm. ESANN’2004 proceedings, pp 305–312
[27]	Silverman BW (1996) Smoothed functional principal components analysis by choice of norm. Ann Stat 24: 1–24 · Zbl 0853.62044
[28]	Steinley D (2003) K-means clustering: What you don’t know may hurt you. Psychol Methods 8: 294–304
[29]	Steinley D, Henson R (2005) OCLUS: an analytic method for generating clusters with known overlap. J Classif 22: 221–250 · Zbl 1336.62191
[30]	Suyundykov R, Puechmorel S, Ferre L (2010) Multivariate functional data clusterization by PCA in Sobolev space using wavelets. Hyper Articles en Ligne:inria-00494702
[31]	Tarpey T (2007) Linear transformations and the k-means clustering algorithm: Applications to clustering curves. Am Stat 61: 34–40 · Zbl 05680714
[32]	Timmerman ME, Ceulemans E, Kiers HAL, Vichi M (2010) Factorial and reduced K-means reconsidered. Comput Stat Data Anal 54: 1858–1871 · Zbl 1284.62396
[33]	Vichi M, Kiers HAL (2001) Factorial k-means analysis for two-way data. Comput Stat Data Anal 37: 49–64 · Zbl 1051.62056
[34]	Wahba G (1990) Spline models for observational data. Society for Industrial and Applied Mathematics, Philadelphia · Zbl 0813.62001