Reduced rank stochastic regression with a sparse singular value decomposition

Journal of the Royal Statistical Society. Series B: Statistical Methodology

Volumn 74, Issue 2, 2012, Pages 203-221

  Chen, Kun ^a   Chan, Kung Sik ^b   Stenseth, Nils Chr ^c  

^a KANSAS STATE UNIVERSITY   (United States)

^b UNIVERSITY OF IOWA   (United States)

^c UNIVERSITY OF OSLO   (Norway)

Author keywords

Biclustering; Microarrary gene expression data; Multivariate regression; Oracle property; Regularization

Indexed keywords

EID: 84858281659     PISSN: 13697412     EISSN: 14679868     Source Type: Journal    
DOI: 10.1111/j.1467-9868.2011.01002.x     Document Type: Article

References (37)

1. Anderson, T. W. (1951) Estimating linear restrictions on regression coefficients for multivariate normal distributions. Ann. Math. Statist., 22, 327-351.
2. Anderson, T. W. (2002) Specification and misspecification in reduced rank regression. Sankhya A, 64, 193-205.
3. Anderson, T. W. (2003) An Introduction to Multivariate Statistical Analysis, 3nd edn. New York: Wiley.
4. Bhattacharjee, A., Richards, W. G., Staunton, J., Li, C., Monti, S., Vasa, P., Ladd, C., Beheshti, J., Bueno, R., Gillette, M., Loda, M., Weber, G., Mark, E. J., Lander, E. S., Wong, W., Johnson, B. E., Golub, T. R., Sugarbaker, D. J. and Meyerson, M. (2001) Classification of human lung carcinomas by mrna expression profiling reveals distinct adenocarcinoma subclasses. Proc. Natn. Acad. Sci. USA, 98, 13790-13795.
5. Breheny, P. and Huang, J. (2009) Penalized methods for bi-level variable selection. Statist. Interface, 2, 369-380.
6. Bunea, F., She, Y. and Wegkamp, M. (2011) Optimal selection of reduced rank estimators of high-dimensional matrices. Ann. Statist., 39, 1282-1309.
7. Busygin, S., Prokopyev, O. and Pardalos, P. M. (2008) Biclustering in data mining. Comput. Oper. Res., 35, 2964-2987.
8. Camba-Mendez, G., Kapetanios, G., Smith, R. J. and Weale, M. R. (2003) Tests of rank in reduced rank regression models. J. Bus. Econ. Statist., 21, 145-155.
9. Donoho, D. L. and Johnstone, I. M. (1995) Adapting to unknown smoothness via wavelet shrinkage. J. Am. Statist. Ass., 90, 1200-1224.
10. Efron, B., Hastie, T., Johnstone, I. and Tibshirani, R. (2004) Least angle regression. Ann. Statist., 32, 407-499.
11. Fan, J. and Li, R. (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Statist. Ass., 96, 1348-1360.
12. Friedman, J., Hastie, T., Höfling, H. and Tibshirani, R. (2007) Pathwise coordinate optimization. Ann. Appl. Statist., 2, 302-332.
13. Friedman, J. H., Hastie, T. and Tibshirani, R. (2010) Regularization paths for generalized linear models via coordinate descent. J. Statist. Softwr., 33, 1-22.
14. Gorski, J., Pfeuffer, F. and Klamroth, K. (2007) Biconvex sets and optimization with biconvex functions: a survey and extensions. Math. Meth. Oper. Res., 66, 373-407.
15. Izenman, A. J. (1975) Reduced-rank regression for the multivariate linear model. J. Multiv. Anal., 5, 248-264.
16. Lee, M., Shen, H., Huang, J. Z. and Marron, J. S. (2010) Biclustering via sparse singular value decomposition. Biometrics, 66, 1087-1095.
17. Leek, J. T. and Storey, J. D. (2007) Capturing heterogeneity in gene expression studies by surrogate variable analysis. PLOS Genet., 3, article e161.
18. Liu, Y., Hayes, D. N. N., Nobel, A. and Marron, J. S. (2008) Statistical significance of clustering for high-dimension, low sample size data. J. Am. Statist. Ass., 103, 1281-1293.
19. Obozinski, G., Wainwright, M. J. and Jordan, M. I. (2008) Union support recovery in high-dimensional multivariate regression. Technical Report. Department of Statistics, University of California, Berkeley. (Available from).
20. Peng, J., Zhu, J., Bergamaschi, A., Han, W., Noh, D.-Y., Pollack, J. R. and Wang, P. (2010) Regularized multivariate regression for identifying master predictors with application to integrative genomics study of breast cancer. Ann. Appl. Statist., 4, 53-77.
21. R Development Core Team (2008) R: a Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing.
22. Reinsel, G. C. and Velu, P. (1998) Multivariate Reduced-rank Regression: Theory and Applications. New York: Springer.
23. Schott, J. R. (2005) Matrix Analysis for Statistics, 2nd edn. Hoboken: Wiley.
24. Stenseth, N. C., Jordel, P. E., Chan, K.-S., Hansen, E., Knutsen, H., Andre, C., Skogen, M. D. and Lekve, K. (2006) Ecological and genetic impact of Atlantic cod larval drift in the Skagerrak. Proc. R. Soc. Lond. B, 273, 1085-1092.
25. Stone, M. (1974) Cross-validation and multinomial prediction. Biometrika, 61, 509-515.
26. Tibshirani, R. (1996) Regression shrinkage and selection via the lasso. J. R. Statist. Soc. B, 58, 267-288.
27. Toh, K.-C. and Yun, S. (2009) An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. Pacif. J. Optim., 6, 615-640.
28. Turlach, B. A., Venables, W. N. and Wright, S. J. (2005) Simultaneous variable selection. Technometrics, 47, 349-363.
29. van der Vaart, A. W. (2000) Asymptotic Statistics. Cambridge: Cambridge University Press.
30. Vounou, M., Nichols, T. E. and Montana, G. (2010) Discovering genetic associations with high-dimensional neuroimaging phenotypes: a sparse reduced-rank regression approach. NeuroImage, 53, 1147-1159.
31. Witten, D. M., Tibshirani, R. and Hastie, T. (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics, 10, 515-534.
32. Yuan, M., Ekici, A., Lu, Z. and Monteiro, R. (2007) Dimension reduction and coefficient estimation in multivariate linear regression. J. R. Statist. Soc. B, 69, 329-346.
33. Zhang, C.-H. and Huang, J. (2008) The sparsity and bias of the lasso selection in high-dimensional linear regression. Ann. Statist., 36, 1567-1594.
34. Zhao, P. and Yu, B. (2006) On model selection consistency of lasso. J. Mach. Learn. Res., 7, 2541-2563.
35. Zou, H. (2006) The adaptive lasso and its oracle properties. J. Am. Statist. Ass., 101, 1418-1429.
36. Zou, H. and Hastie, T. (2005) Regularization and variable selection via the elastic net. J. R. Statist. Soc. B, 67, 301-320.
37. Zou, H., Hastie, T. and Tibshirani, R. (2007) On the degree of freedom of the lasso. Ann. Statist., 35, 2173-2192.