Self-consistency and a generalized principal subspace theorem. (English) Zbl 1302.62145
Summary: Principal subspace theorems deal with the problem of finding subspaces supporting optimal approximations of multivariate distributions. The optimality criterion considered in this paper is the minimization of the mean squared distance between the given distribution and an approximating distribution, subject to some constraints. Statistical applications include, but are not limited to, cluster analysis, principal components analysis and projection pursuit. Most principal subspace theorems deal with elliptical distributions or with mixtures of spherical distributions. We generalize these results using the notion of self-consistency. We also show their connections with the skew-normal distribution and projection pursuit techniques. We also discuss their implications, with special focus on principal points and self-consistent points. Finally, we access the practical relevance of the theoretical results by means of several simulation studies.
MSC:
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
| 62H25 | Factor analysis and principal components; correspondence analysis |
Keywords:
location mixtures; principal points; projection pursuit; skewness; skew-normal distribution; spherical distributionReferences:
| [1] | Adcock, C.; Eling, M.; Loperfido, N., Skewed distributions in finance and actuarial science: a review, Eur. J. Finance (2012) |
| [2] | Arnold, B. C.; Beaver, R. J., Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion), Test, 11, 7-54 (2002) · Zbl 1033.62013 |
| [3] | Azzalini, A., The skew-normal distribution and related multivariate families (with discussion), Scand. J. Stat., 32, 159-188 (2005) · Zbl 1091.62046 |
| [4] | Azzalini, A.; Dalla Valle, A., The multivariate skew-normal distribution, Biometrika, 83, 715-726 (1996) · Zbl 0885.62062 |
| [5] | Barbosa-Cabral, C. R.; Lachos, V. H.; Prates, M. O., Multivariate mixture modeling using skew-normal independent distributions, Comput. Statist. Data Anal., 56, 126-142 (2012) · Zbl 1239.62058 |
| [6] | Bartoletti, S.; Loperfido, N., Modelling air pollution data by the skew-normal distribution, Stochastic Environ. Res. Risk Assessment, 24, 513-517 (2010) · Zbl 1462.62697 |
| [7] | Basso, R. M.; Lachos, V. H.; Barbosa-Cabral, C. R.; Ghosh, P., Robust mixture modeling based on scale mixtures of skew-normal distributions, Comput. Statist. Data Anal., 54, 2926-2941 (2010) · Zbl 1284.62193 |
| [8] | Bernardi, M.; Maruotti, A.; Petrella, L., Skew mixture models for loss distributions: a Bayesian approach, Insurance Math. Econom., 51, 617-623 (2012) · Zbl 1285.62027 |
| [9] | Chang, C. H.; Lin, J. J.; Pal, N.; Chiang, M. C., A note on improved approximation of the binomial distribution by the skew-normal distribution, Amer. Statist., 62, 167-170 (2008) |
| [10] | Christiansen, M.; Loperfido, N., Improved approximation of the sum of random vectors by the skew-normal distribution, J. Appl. Probab., 51, 466-482 (2014) · Zbl 1304.60030 |
| [11] | Daszykowski, M., From projection pursuit to other unsupervised chemometric techniques, J. Chemometrics, 21, 270-279 (2007) |
| [12] | Diaconis, P.; Freedman, D., Asymptotics of graphical projection pursuit, Ann. Statist., 12, 793-815 (1984) · Zbl 0559.62002 |
| [13] | Fang, K. T.; Kotz, S.; Ng, K. W., Symmetric Multivariate and Related Distributions (1990), Chapman and Hall: Chapman and Hall New York · Zbl 0699.62048 |
| [14] | Flury, B., Principal points, Biometrika, 77, 33-41 (1990) · Zbl 0691.62053 |
| [15] | Flury, B., Estimation of principal points, Appl. Stat., 42, 139-151 (1993) · Zbl 0825.62524 |
| [16] | Franceschini, C.; Loperfido, N., Testing for normality when the sampled distribution is extended skew-normal, (Corazza, M.; Pizzi, C., Mathematical and Statistical Methods for Actuarial Sciences and Finance (2013), Springer) |
| [17] | Friedman, J.; Tukey, J., A projection pursuit algorithm for exploratory data analysis, IEEE Trans. Comput., 881-889 (1974) · Zbl 0284.68079 |
| [18] | Frühwirth-Schnatter, S.; Pyne, S., Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-\(t\) distributions, Biostatistics, 11, 317-336 (2010) · Zbl 1437.62465 |
| [19] | Genton, M. G., Skew-symmetric and generalized skew-elliptical distributions, (Genton, M. G., Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality (2004), Chapman & Hall/CRC, Springer: Chapman & Hall/CRC, Springer Boca Raton, FL), 81-100 · Zbl 1069.62045 |
| [20] | Gonzalez-Farias, G.; Dominguez-Molina, J. A.; Gupta, A. K., Additive properties of skew-normal random vectors, J. Statist. Plann. Inference, 126, 521-534 (2003) · Zbl 1076.62052 |
| [21] | Gupta, A. K.; Kollo, T., Density expansions based on the multivariate skew normal distribution, Sankhya, 65, 821-835 (2003) · Zbl 1193.62091 |
| [22] | Hartigan, J. A., Asymptotic distributions for clustering criteria, Ann. Statist., 6, 117-131 (1978) · Zbl 0377.62033 |
| [23] | Hartigan, J. A.; Wong, M. A., A \(K\)-means clustering algorithm, Appl. Stat., 28, 100-108 (1979) · Zbl 0447.62062 |
| [24] | Hastie, T.; Stuetzle, W., Principal curves, J. Amer. Statist. Assoc., 84, 502-516 (1989) · Zbl 0679.62048 |
| [25] | Huber, P. J., Projection pursuit (with discussion), Ann. Statist., 13, 435-475 (1985) · Zbl 0595.62059 |
| [26] | Jones, M. C.; Sibson, R., What is projection pursuit? (with discussion), J. Roy. Statist. Soc. Ser. A, 150, 1-36 (1987) · Zbl 0632.62059 |
| [27] | Kotz, S.; Vicari, D., Survey of developments in the theory of continuous skewed distributions, Metron, 225-261 (2005) · Zbl 1416.62111 |
| [28] | Kurata, H., On principal points for location mixtures of multivariate spherically symmetric distributions, J. Statist. Plann. Inference, 138, 3405-3418 (2008) · Zbl 1145.62044 |
| [29] | Laha, R. G.; Rohatgi, V. K., Probability Theory (1979), Wiley: Wiley New York · Zbl 0409.60001 |
| [30] | Lin, T. I.; Lee, J. C.; Yen, S. Y., Finite mixture modelling using the skew-normal distribution, Statist. Sinica, 17, 909-927 (2007) · Zbl 1133.62012 |
| [31] | Loperfido, N., Canonical transformations of skew-normal variates, TEST, 19, 146-165 (2010) · Zbl 1203.62102 |
| [32] | Loperfido, N., Skewness and the linear discriminant function, Statist. Probab. Lett., 83, 93-99 (2013) · Zbl 1489.62183 |
| [34] | Malkovich, J.; Afifi, A., On tests for multivariate normality, J. Amer. Statist. Assoc., 68, 176-179 (1973) |
| [35] | Matsuuraa, S.; Kurata, H., A principal subspace theorem for 2-principal points of general location mixtures of spherically symmetric distributions, Statist. Probab. Lett., 80, 1863-1869 (2010) · Zbl 1202.62070 |
| [36] | Matsuuraa, S.; Kurata, H., Principal points of a multivariate mixture distribution, J. Multivariate Anal., 102, 213-224 (2011) · Zbl 1328.62077 |
| [37] | Matsuura, S.; Kurata, H., Definition and properties of \(m\)-dimensional \(n\)-principal points, Comm. Statist. Theory Methods, 42, 267-282 (2013) · Zbl 1298.62101 |
| [38] | McCabe, G., Principal variables, Technometrics, 826, 137-144 (1984) · Zbl 0548.62037 |
| [39] | R: A Language and Environment for Statistical Computing (2013), R Foundation for Statistical Computing: R Foundation for Statistical Computing Vienna, Austria, URL http://www.R-project.org |
| [40] | Sun, J., Projection pursuit, (Encyclopedia of Biostatistics (2005)) |
| [41] | Tarpey, T., Principal points (1992), Indiana University, (Ph.D. thesis) |
| [42] | Tarpey, T., Self-consistent patterns for symmetric multivariate distributions, J. Classification, 15, 57-79 (1998) · Zbl 0902.62057 |
| [43] | Tarpey, T., Self-consistency algorithms, J. Comput. Graph. Statist., 8, 889-905 (1999) |
| [44] | Tarpey, T., Self-consistency and principal component analysis, J. Amer. Statist. Assoc., 94, 456-467 (1999) · Zbl 0999.62042 |
| [45] | Tarpey, T.; Flury, B., Self-consistency: a fundamental concept in statistics, Statist. Sci., 11, 229-243 (1996) · Zbl 0955.62540 |
| [46] | Tarpey, T.; Kinateder, K. J., Clustering functional data, J. Classification, 20, 93-114 (2003) · Zbl 1112.62327 |
| [47] | Tarpey, T.; Li, L.; Flury, B., Principal points and self-consistent points of elliptical distributions, Ann. Statist., 23, 103-112 (1995) · Zbl 0822.62042 |
| [48] | Tarpey, T.; Petkova, E., Latent regression analysis, Stat. Model., 10, 138-158 (2010) · Zbl 07256819 |
| [49] | Yamamoto, W.; Shinozaki, N., Two principal points for multivariate location mixtures of spherically symmetric distributions, J. Japan Statist. Soc., 30, 53-63 (2000) · Zbl 0963.62046 |
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