## Clustering and disjoint principal component analysis.(English)Zbl 1453.62230

Summary: A constrained principal component analysis, which aims at a simultaneous clustering of objects and a partitioning of variables, is proposed. The new methodology allows us to identify components with maximum variance, each one a linear combination of a subset of variables. All the subsets form a partition of variables. Simultaneously, a partition of objects is also computed maximizing the between cluster variance. The methodology is formulated in a semi-parametric least-squares framework as a quadratic mixed continuous and integer problem. An alternating least-squares algorithm is proposed to solve the clustering and disjoint PCA. Two applications are given to show the features of the methodology.

### MSC:

 62-08 Computational methods for problems pertaining to statistics 62H30 Classification and discrimination; cluster analysis (statistical aspects) 62H25 Factor analysis and principal components; correspondence analysis
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### References:

 [1] Cattell, R.B., The scree test for the number of factors, Multivariate behavioral research, 1, 245-276, (1966) [2] DeSarbo, W.S.; Jedidi, K.; Cool, K.; Schendel, D., Simultaneous multidimensional unfolding and cluster analysis: an investigation of strategic groups, Marketing letters, 2, 129-146, (1990) [3] De Soete, G.; Carroll, J.D., K-means clustering in a low-dimensional Euclidean space, (), 212-219 [4] De Soete, G.; Heiser, W.J., A latent class unfolding model for analyzing single stimulus preference ratings, Psychometrika, 58, 545-565, (1993) · Zbl 0826.62098 [5] Gabriel, K.R., The biplot graphic display of matrices with application to principal component analysis, Biometrika, 58, 453-467, (1971) · Zbl 0228.62034 [6] Heiser, W.J., Clustering in low-dimensional space, (), 162-173 [7] Heiser, W.J.; Groenen, P.J.F., Cluster differences scaling with a within-clusters loss component and a fuzzy successive approximation strategy to avoid local minima, Psychometrika, 62, 63-83, (1997) · Zbl 0889.92037 [8] Kaiser, H.F., The varimax criterion for analytic rotation in factor analysis, Psychometrika, 23, 187-200, (1958) · Zbl 0095.33603 [9] Milligan, G.W.; Cooper, M., An estimation of procedures for determining the number of clusters in a data set, Psychometrika, 50, 159-179, (1985) [10] Vichi, M.; Kiers, H.A.L., Factorial $$k$$-means analysis for two way data (2001), Computational statistics and data analysis, 37, 49-64, (2001) · Zbl 1051.62056 [11] Vichi, M., Double $$k$$-means clustering for simultaneous classification of objects and variables, (), 43-52 [12] Vichi, M., Discrete and continuous models for two way data (2002), (), 139-147 [13] Vichi, M.; Rocci, R; Kiers, H.A.L., Simultaneous component and clustering models for three-way data: within and between approaches, Journal of classification, 24, 1, 71-98, (2007) · Zbl 1144.62045 [14] Vigneau, E.; Qannari, E.M., Clustering of variables around latent component — application to sensory analysis, Communications in statistics, simulation and computation, 32, 4, 1131-1150, (2004) · Zbl 1100.62582 [15] Zou, H.; Hastie, T.; Tibshirani, R., Sparse principal component analysis, Journal of computational and graphical statistics, 15, 2, 262-286, (2006)
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