×

Estimating common principal components in high dimensions. (English) Zbl 1474.62183

Summary: We consider the problem of minimizing an objective function that depends on an orthonormal matrix. This situation is encountered, for example, when looking for common principal components. The Flury method is a popular approach but is not effective for higher dimensional problems. We obtain several simple majorization-minimization (MM) algorithms that provide solutions to this problem and are effective in higher dimensions. We use mixture model-based clustering applications to illustrate our MM algorithms. We then use simulated data to compare them with other approaches, with comparisons drawn with respect to convergence and computational time.

MSC:

62H12 Estimation in multivariate analysis
62H30 Classification and discrimination; cluster analysis (statistical aspects)

Software:

Mixmod; Rmixmod; mixture; R
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Absil P-A, Mahony R, Sepulchre R (2008) Optimization algorithms on matrix manifolds. Princeton University Press, Princeton · Zbl 1147.65043
[2] Andrews JL, McNicholas PD (2012) Model-based clustering, classification, and discriminant analysis via mixtures of multivariate t-distributions. Stat Comput 22(5):1021-1029 · Zbl 1252.62062
[3] Arnold S, Phillips P (1999) Hierarchical comparison of genetic variance-covariance matrices. II. Coastal-inland divergence in the garter snake, Thamnophis elegans. Evolution 53:1516-1527
[4] Banfield JD, Raftery AE (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics 49(3): 803-821 · Zbl 0794.62034
[5] Biernacki C, Celeux G, Govaert G, Langrognet F (2006) Model-based cluster analysis and discriminant analysis with the MIXMOD software. Comput Stat Data Anal 51:587-600 · Zbl 1157.62431
[6] Boik RJ (2003) Principal component models for correlation matrices. Biometrika 90:679-701 · Zbl 1436.62221
[7] Boik RJ (2007) Spectral models for covariance matrices. Biometrika 89:159-182 · Zbl 0997.62046
[8] Bouveyron C, Girard S, Schmid C (2007) High-dimensional data clustering. Comput Stat and Data Anal 52:502-519 · Zbl 1452.62433
[9] Browne RP, McNicholas PD (2012) Orthogonal Stiefel manifold optimization for eigen-decomposed covariance parameter estimation in mixture models. Statistics and Computing. To appear. doi:10.1007/s11222-012-9364-2
[10] Browne RP, McNicholas PD (2013) mixture: Mixture models for clustering and classification. R package version 1.0
[11] Celeux G, Govaert G (1995) Gaussian parsimonious clustering models. Pattern Recogn 28(5):781-793
[12] Dasgupta A, Raftery AE (1998) Detecting features in spatial point processes with clutter via model-based clustering. J Am Stat Assoc 93:294-302 · Zbl 0906.62105
[13] Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J Royal Stat Soc Series B 39(1):1-38 · Zbl 0364.62022
[14] Flury BW, Gautschi W (1984) Common principal components in k groups. J Am Stat Assoc 79(388): 892-898
[15] Flury BW, Gautschi W (1986) An algorithm for simultaneous orthogonal transformation of several positive definite symmetric matrices to nearly diagonal form. J Sci Stat Comput 7(1):169-184 · Zbl 0614.65043
[16] Hunter D (2004) MM algorithms for generalized Bradley-Terry models. Ann Stat 32:386-408 · Zbl 1105.62359
[17] Hunter D, Lange K (2000) Quantile regression via an MM algorithm. J Comput Graph Stat 9:60-77
[18] Kiers H (2002) Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems. Comput Stat Data Anal 41:157-170 · Zbl 1018.65074
[19] Klingenberg C, Neuenschwander B, Flury B (1996) Ontogeny and individual variation: Analysis of patterned covariance matrices with common principal components. Syste Biol 45:135-150
[20] Krzanowski WJ (1990) Between-group analysis with heterogeneous covariance. matrices: The common principal component model. J Classif 7:81-98 · Zbl 0714.62051
[21] Kulkarni B, Rao G (2000) The common principal components approach for clustering under multiple sampling. J Indian Soc Agric Stat 53:1-11 · Zbl 1188.62183
[22] Lebret R, Iovleff S, Langrognet F (2012) Rmixmod: MIXture MODelling Package. R package version 1.1.1
[23] Lefkomtch LP (2004) Consensus principal components. Biometrical J 35:567-580
[24] Merbouha A, Mkhadri A (2004) Regularization of the location model in discrimination with mixed discrete and continuous variables. Comput Stat Data Anal 45:463-576 · Zbl 1429.62254
[25] Oksanen J, Huttunen P (1989) Finding a common ordination for several data sets by individual differences scaling. Plant Ecol 83:137-145
[26] R Development Core Team (2012) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna
[27] Schott J (1998) Estimating correlation matrices that have common eigenvectors. Comput Stat Data Anal 27:445-459 · Zbl 1042.62561
[28] Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461-464 · Zbl 0379.62005
[29] Sengupta S, Boyle J (1998) Using common principal components for comparing GCM simulations. J Climate 11:816-830
[30] von Mises R, Pollaczek-Geiringer H (1929) Praktische verfahren der gleichungsauflösung. Zeitschrift für Angewandte Mathematik und Mechanik 9(1):58-77 · JFM 55.0305.01
[31] Yang K, Shahabi C (2006) An efficient k nearest neighbor search for multivariate time series. Info Comput 205:65-98 · Zbl 1109.68040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.