Scale-Invariant Sparse PCA on High-Dimensional Meta-Elliptical Data

REFERENCES

• Amini, A., and Wainwright, M. (2009), “High-Dimensional Analysis of Semidefinite Relaxations for Sparse Principal Components,” The Annals of Statistics, 37, 2877–2921.
   Web of Science ®Google Scholar
• Anderson, T.W. (1958), An Introduction to Multivariate Statistical Analysis (Vol. 2), New York: Wiley.
   Google Scholar
• Berthet, Q., and Rigollet, P. (2012), “Optimal Detection of Sparse Principal Components in High Dimension,” The Annals of Statistics, 41, 1780–1815.
   Google Scholar
• Bickel, P., and Levina, E. (2008), “Regularized Estimation of Large Covariance Matrices,” The Annals of Statistics, 36, 199–227.
   Web of Science ®Google Scholar
• Bühlmann, P., and van de Geer, S. (2011), Statistics for High-Dimensional Data: Methods, Theory and Applications, Berlin, Heidelberg: Springer.
   Google Scholar
• Chatfield, C., and Collins, A. (1980), Introduction to Multivariate Analysis, (Vol. 166), Boca Raton, FL: Chapman & Hall.
   Google Scholar
• Choi, K., and Marden, J. (1998), “A Multivariate Version of Kendall’s τ,” Journal of Nonparametric Statistics, 9, 261–293.
   Web of Science ®Google Scholar
• Croux, C., and Dehon, C. (2010), “Influence Functions of the Spearman and Kendall Correlation Measures,” Statistical Methods & Applications, 19, 497–515.
   Web of Science ®Google Scholar
• Croux, C., Filzmoser, P., and Fritz, H. (2013), “Robust Sparse Principal Component Analysis,” Technometrics, 55, 202–214.
   Web of Science ®Google Scholar
• Croux, C., and Haesbroeck, G. (2000), “Principal Component Analysis Based on Robust Estimators of the Covariance or Correlation Matrix: Influence Functions and Efficiencies,” Biometrika, 87, 603–618.
   Web of Science ®Google Scholar
• Croux, C., Ollila, E., and Oja, H. (2002), “Sign and Rank Covariance Matrices: Statistical Properties and Application to Principal Components Analysis,” in Statistical Data Analysis, Based on the L1 Norm and Related Methods (Y. Dodge, ed.), Birkhäuser, Basel, 257–269.
   Google Scholar
• Croux, C., and Ruiz-Gazen, A. (2005), “High Breakdown Estimators for Principal Components: The Projection-Pursuit Approach Revisited,” Journal of Multivariate Analysis, 95, 206–226.
   Web of Science ®Google Scholar
• d’Aspremont, A., Ghaoui, L., Jordan, M., and Lanckriet, G. (2004), “A Direct Formulation for Sparse pca Using Semidefinite Programming,” SIAM Review, 49, 434–448.
   Google Scholar
• Davies, P. (1987), “Asymptotic Behaviour of S-Estimates of Multivariate Location Parameters and Dispersion Matrices,” The Annals of Statistics, 15, 1269–1292.
   Web of Science ®Google Scholar
• Fang, H., Fang, K., and Kotz, S. (2002), “The Meta-Elliptical Distributions With Given Marginals,” Journal of Multivariate Analysis, 82, 1–16.
   Web of Science ®Google Scholar
• Fang, K., Kotz, S., and Ng, K.-W. (1990), Symmetric Multivariate and Related Distributions, London,UK: Chapman & Hall.
   Google Scholar
• Gibbons, J.D., and Chakraborti, S. (2003), Nonparametric Statistical Inference (Vol. 168), Boca Raton, FL: CRC Press.
   Google Scholar
• Gnanadesikan, R., and Kettenring, J.R. (1972), “Robust Estimates, Residuals, and Outlier Detection With Multiresponse Data,” Biometrics, 28, 81–124.
   Web of Science ®Google Scholar
• Hallin, M., Paindaveine, D., and Verdebout, T. (2010), “Optimal Rank-Based Testing for Principal Components,” The Annals of Statistics, 38, 3245–3299.
   Web of Science ®Google Scholar
• Hampel, F.R. (1974), “The Influence Curve and Its Role in Robust Estimation,” Journal of the American Statistical Association, 69, 383–393.
   Web of Science ®Google Scholar
• Han, F., and Liu, H. (2012), “Transelliptical Component Analysis,” in Proceedings of the Advances in Neural Information Processing Systems 25, pp. 368–376.
   Google Scholar
• Han, F., Zhao, T., and Liu, H. (2013), “CODA: High Dimensional Copula Discriminant Analysis,” Journal of Machine Learning Research, 14, 629–671.
   Web of Science ®Google Scholar
• Hoeffding, W. (1963), “Probability Inequalities for Sums of Bounded Random Variables,” Journal of the American Statistical Association, 58, 13–30.
   Web of Science ®Google Scholar
• Huber, P.J., and Ronchetti, E. (2009), Robust Statistics (2nd ed.), Hoboken,NJ: Wiley.
   Google Scholar
• Hubert, M., Rousseeuw, P.J., and Verboven, S. (2002), “A Fast Method for Robust Principal Components With Applications to Chemometrics,” Chemometrics and Intelligent Laboratory Systems, 60, 101–111.
   Web of Science ®Google Scholar
• Huffer, F.W., and Park, C. (2007), “A Test for Elliptical Symmetry,” Journal of Multivariate Analysis, 98, 256–281.
   Web of Science ®Google Scholar
• Jackson, D., and Chen, Y. (2004), “Robust Principal Component Analysis and Outlier Detection With Ecological Data,” Environmetrics, 15, 129–139.
   Web of Science ®Google Scholar
• Johnstone, I., and Lu, A. (2009), “On Consistency and Sparsity for Principal Components Analysis in High Dimensions,” Journal of the American Statistical Association, 104, 682–693.
   PubMed Web of Science ®Google Scholar
• Journée, M., Nesterov, Y., Richtárik, P., and Sepulchre, R. (2010), “Generalized Power Method for Sparse Principal Component Analysis,” Journal of Machine Learning Research, 11, 517–553.
   Web of Science ®Google Scholar
• Kendall, M.G. (1948), Rank Correlation Methods, London: Charles Griffin & Company Limited.
   Google Scholar
• Kruskal, W. (1958), “Ordinal Measures of Association,” Journal of the American Statistical Association, 53, 814–861.
   Web of Science ®Google Scholar
• Li, R., Fang, K., and Zhu, L. (1997), “Some Q-Q Probability Plots to Test Spherical and Elliptical Symmetry,” Journal of Computational and Graphical Statistics, 6, 435–450.
   Web of Science ®Google Scholar
• Lindskog, F., McNeil, A., and Schmock, U. (2003), Kendall’s Tau for Elliptical Distributions, Heidelberg: Springer.
   Google Scholar
• Liu, H., Han, F., Yuan, M., Lafferty, J., and Wasserman, L. (2012), “High Dimensional Semiparametric Gaussian Copula Graphical Models,” The Annals of Statistics, 40, 2293–2326.
   Web of Science ®Google Scholar
• Ma, Z. (2013), “Sparse Principal Component Analysis and Iterative Thresholding,” forthcoming in The Annals of Statistics, 41, 772–801.
   Web of Science ®Google Scholar
• Mackey, L. (2009), “Deflation Methods for Sparse PCA,” Advances in Neural Information Processing Systems, 21, 1017–1024.
   Google Scholar
• Marden, J. (1999), “Some Robust Estimates of Principal Components,” Statistics & Probability Letters, 43, 349–359.
   Web of Science ®Google Scholar
• Maronna, R.A. (1976), “Robust M-Estimators of Multivariate Location and Scatter,” The Annals of Statistics, 4, 51–67.
   Web of Science ®Google Scholar
• Maronna, R.A., and Zamar, R.H. (2002), “Robust Estimates of Location and Dispersion for High-Dimensional Datasets,” Technometrics, 44, 307–317.
   Web of Science ®Google Scholar
• Möttönen, J., and Oja, H. (1995), “Multivariate Spatial Sign and Rank Methods,” Journal of Nonparametric Statistics, 5, 201–213.
   Google Scholar
• Oja, H. (2010), Multivariate Nonparametric Methods With R: An Approach Based on Spatial Signs and Ranks (Vol. 199), Heidelberg: Springer.
   Google Scholar
• Paul, D., and Johnstone, I. (2012), “Augmented Sparse Principal Component Analysis for High Dimensional Data,” Arxiv preprint arXiv:1202.1242.
   Google Scholar
• Puri, M.L., and Sen, P.K. (1971), Nonparametric Methods in Multivariate Analysis, New York: Wiley.
   Google Scholar
• Rousseeuw, P., Croux, C., Todorov, V., Ruckstuhl, A., Salibian-Barrera, M., Verbeke, T., and Maechler, M. (2009), “Robustbase: Basic Robust Statistics,” . R package, available at http://CRAN. R-project. org/package= robustbase.
   Google Scholar
• Rousseeuw, P.J., and Croux, C. (1993), “Alternatives to the Median Absolute Deviation,” Journal of the American Statistical Association, 88, 1273–1283.
   Web of Science ®Google Scholar
• Sakhanenko, L. (2008), “Testing for Ellipsoidal Symmetry: A Comparison Study,” Computational Statistics & Data Analysis, 53, 565–581.
   Web of Science ®Google Scholar
• Shen, H., and Huang, J. (2008), “Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation,” Journal of Multivariate Analysis, 99, 1015–1034.
   Web of Science ®Google Scholar
• Visuri, S., Koivunen, V., and Oja, H. (2000), “Sign and Rank Covariance Matrices,” Journal of Statistical Planning and Inference, 91, 557–575.
   Web of Science ®Google Scholar
• Vu, V., and Lei, J. (2012), “Minimax Rates of Estimation for Sparse PCA in High Dimensions,” International Conference on Artificial Intelligence and Statistics (AISTATS), 15, 1278–1286.
   Google Scholar
• Witten, D., Tibshirani, R., and Hastie, T. (2009), “A Penalized Matrix Decomposition, With Applications to Sparse Principal Components and Canonical Correlation Analysis,” Biostatistics, 10, 515–534.
   PubMed Web of Science ®Google Scholar
• Xue, L., and Zou, H. (2012), “Regularized Rank-Based Estimation of High-Dimensional Nonparanormal Graphical Models,” The Annals of Statistics, 40, 2541–2571.
   Web of Science ®Google Scholar
• Yuan, X., and Zhang, T. (2013), “Truncated Power Method for Sparse Eigenvalue Problems,” Journal of Machine Learning Research, 14, 899–925.
   Web of Science ®Google Scholar
• Zhang, Y., and Ghaoui, L. (2011), “Large-Scale Sparse Principal Component Analysis With Application to Text Data,” in Proceedings of the Advances in Neural Information Processing Systems, 24, pp. 532–539.
   Google Scholar
• Zou, H., Hastie, T., and Tibshirani, R. (2006), “Sparse Principal Component Analysis,” Journal of Computational and Graphical Statistics, 15, 265–286.
   Web of Science ®Google Scholar