Covariance-regularized regression and classification for high dimensional problems
We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse covariance matrix of the features to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing the log-...
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| Published in | Journal of the Royal Statistical Society. Series B, Statistical methodology Vol. 71; no. 3; pp. 615 - 636 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Oxford, UK
Oxford, UK : Blackwell Publishing Ltd
01.06.2009
Blackwell Publishing Ltd Blackwell Publishing Blackwell Royal Statistical Society Oxford University Press |
| Series | Journal of the Royal Statistical Society Series B |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1369-7412 1467-9868 |
| DOI | 10.1111/j.1467-9868.2009.00699.x |
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| Abstract | We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse covariance matrix of the features to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing the log-likelihood of the data, under a multivariate normal model, subject to a penalty; it is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyse gene expression data sets with multiple class and survival outcomes. |
|---|---|
| AbstractList | We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse covariance matrix of the features to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing the log-likelihood of the data, under a multivariate normal model, subject to a penalty; it is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyse gene expression data sets with multiple class and survival outcomes. Copyright (c) 2009 Royal Statistical Society. We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse covariance matrix of the features to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing the log-likelihood of the data, under a multivariate normal model, subject to a penalty; it is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyse gene expression data sets with multiple class and survival outcomes. Reprinted by permission of Blackwell Publishers We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse covariance matrix of the features to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing the log-likelihood of the data, under a multivariate normal model, subject to a penalty; it is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyse gene expression data sets with multiple class and survival outcomes. We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse covariance matrix of the features to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing the log-likelihood of the data, under a multivariate normal model, subject to a penalty; it is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyse gene expression data sets with multiple class and survival outcomes. [PUBLICATION ABSTRACT] In recent years, many methods have been developed for regression in high-dimensional settings. We propose covariance-regularized regression, a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing its log likelihood, under a multivariate normal model, subject to a constraint on its elements; this estimate is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso, and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyze gene expression data sets with multiple class and survival outcomes.In recent years, many methods have been developed for regression in high-dimensional settings. We propose covariance-regularized regression, a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing its log likelihood, under a multivariate normal model, subject to a constraint on its elements; this estimate is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso, and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyze gene expression data sets with multiple class and survival outcomes. In recent years, many methods have been developed for regression in high-dimensional settings. We propose covariance-regularized regression, a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing its log likelihood, under a multivariate normal model, subject to a constraint on its elements; this estimate is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso, and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyze gene expression data sets with multiple class and survival outcomes. |
| Author | Tibshirani, Robert Witten, Daniela M. |
| Author_xml | – sequence: 1 givenname: Daniela M. surname: Witten fullname: Witten, Daniela M. – sequence: 2 givenname: Robert surname: Tibshirani fullname: Tibshirani, Robert |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=21516506$$DView record in Pascal Francis https://www.ncbi.nlm.nih.gov/pubmed/20084176$$D View this record in MEDLINE/PubMed http://econpapers.repec.org/article/blajorssb/v_3a71_3ay_3a2009_3ai_3a3_3ap_3a615-636.htm$$DView record in RePEc |
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| Keywords | Ridge regression Rank statistic Generalized linear model Covariance regularization Prediction theory Multivariate analysis Stochastic process Linear model Penalty method Parametric method Covariance Survival function Inverse matrix n « p Log likelihood Classification Ill posed problem Variable selection Censored data Discriminant analysis Numerical linear algebra Prediction Regression Statistical estimation Covariance matrix Survival Regularization method Statistical method Statistical regression Regression coefficient Selection problem Numerical analysis Filtering theory Regularization |
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| Publisher | Oxford, UK : Blackwell Publishing Ltd Blackwell Publishing Ltd Blackwell Publishing Blackwell Royal Statistical Society Oxford University Press |
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| References | Frank, I. and Friedman, J. (1993) A statistical view of some chemometrics regression tools (with discussion). Technometrics, 35, 109-148. Rothman, A., Levina, E. and Zhu, J. (2008) Sparse permutation invariant covariance estimation. Electr. J. Statist., 2, 494-515. Zhu, J. and Hastie, T. (2004) Classification of gene microarrays by penalized logistic regression. Biostatistics, 5, 427-443. Kalbfleisch, J. and Prentice, R. (1980) The Statistical Analysis of Failure Time Data. New York: Wiley. Liang, F., Mukherjee, S. and West, M. (2007) The use of unlabeled data in predictive modeling. Statist. Sci., 22, 189-205. Zhao, P. and Yu, B. (2006) On model selection consistency of lasso. J. Mach. Learn. Res., 7, 2541-2563. Shipp, M. A., Ross, K. N., Tamayo, P., Weng, A. P., Kutok, J. L., Aguiar, R. C., Gaasenbeek, M., Angelo, M., Reich, M., Pinkus, G. S., Ray, T. S., Koval, M. A., Last, K. W., Norton, A., Lister, T. A., Mesirov, J., Neuberg, D. S., Lander, E. S., Aster, J. 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L., Haralambieva, E., Harder, L., Hasenclever, D., Kuhn, M., Lenze, D., Lichter, P., Martin-Subero, J. I., Moller, P., Muller-Hermelink, H.-K., Ott, G., Parwaresch, R. M., Pott, C., Rosenwald, A., Rosolowski, M., Schwaenen, C., Sturzenhofecker, B., Szczepanowski, M., Trautmann, H., Wacker, H.-H., Spang, R., Loefler, M., Trumper, L., Stein, H. and Siebert, R. (2006) A biological definition of Burkitt's lymphoma from transcriptional and genomic profiling. New Engl. J. Med., 354, 2419-2430. Mardia, K., Kent, J. and Bibby, J. (1979) Multivariate Analysis. London: Academic Press. Banerjee, O., El Ghaoui, L. E. and D'Aspremont, A. (2008) Model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data. J. Mach. Learn. Res., 9, 485-516. Bair, E. and Tibshirani, R. (2004) Semi-supervised methods to predict patient survival from gene expression data. PLOS Biol., 2, 511-522. Tibshirani, R., Hastie, T., Narasimhan, B. and Chu, G. (2002) Diagnosis of multiple cancer types by shrunken centroids of gene expression. Proc. Natn. Acad. Sci. USA, 99, 6567-6572. Hoerl, A. E. and Kennard, R. (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics, 12, 55-67. O'Neill, T. (1978) Normal discrimination with unclassified observations. J. Am. Statist. Ass., 73, 821-826. Hinton, G., Osindero, S. and Teh, Y. (2006) A fast learning algorithm for deep belief nets. Neur. Computn, 18, 1527-1553. Bickel, P. and Levina, E. (2008) Covariance regularization by thresholding. Ann. Statist., to be published. Zou, H. and Hastie, T. (2005) Regularization and variable selection via the elastic net. J. R. Statist. Soc. B, 67, 301-320. Tibshirani, R., Hastie, T., Narasimhan, B. and Chu, G. (2003) Class prediction by nearest shrunken centroids, with applications to DNA microarrays. Statist. Sci., 18, 104-117. Dey, D. and Srinivasan, C. (1985) Estimation of a covariance matrix under Stein's loss. Ann. Statist., 13, 1581-1591. Monti, S., Savage, K. J., Kutok, J. L., Feuerhake, F., Kurtin, P., Mihm, M., Wu, B., Pasqualucci, L., Neuberg, D., Aguiar, R. C. T., Dal Cin, P., Ladd, C., Pinkus, G. S., Salles, G., Harris, N. L., Dalla-Favera, R., Habermann, T. M., Aster, J. C., Golub, T. R. and Shipp, M. A. (2005) Molecular profiling of diffuse large B-cell lymphoma identifies robust subtypes including one characterized by host inflammatory response. Blood, 105, 1851-1861. Rosenwald, A., Wright, G., Chan, W. C., Connors, J. M., Campo, E., Fisher, R. I., Gascoyne, R. D., Muller-Hermelink, H. K., Smeland, E. B. and Staudt, L. M. (2002) The use of molecular profiling to predict survival after chemotherapy for diffuse large B-cell lymphoma. New Engl. J. Med., 346, 1937-1947. Guo, Y., Hastie, T. and Tibshirani, R. (2007) Regularized linear discriminant analysis and its application in microarrays. Biostatistics, 8, 86-100. Haff, L. (1979) Estimation of the inverse covariance matrix: random mixtures of the inverse Wishart matrix and the identity. Ann. Statist., 7, 1264-1276. Park, M. Y. and Hastie, T. (2007) L1-regularization path algorithm for generalized linear models. J. R. Statist. Soc. B, 69, 659-677. Ramaswamy, S., Tamayo, P., Rifkin, R., Mukherjee, S., Yeang, C., Angelo, M., Ladd, C., Reich, M., Latulippe, E., Mesirov, J., Poggio, T., Gerald, W., Loda, M., Lander, E. and Golub, T. (2001) Multiclass cancer diagnosis using tumor gene expression signature. Proc. Natn. Acad. Sci. USA, 98, 15149-15154. Breiman, L. (2001) Random forests. Mach. Learn., 45, 5-32. McLachlan, G. J. (1992) Discriminant Analysis and Statistical Pattern Recognition. New York: Wiley. Tusher, V. G., Tibshirani, R. and Chu, G. (2001) Significance analysis of microarrays applied to the ionizing radiation response. Proc. Natn. Acad. Sci. USA, 98, 5116-5121. 1989; 84 2006; 34 1978; 73 1984; 46 2002; 8 2002; 99 2008; 9 1970; 12 2006; 7 2008 2006; 18 2004; 5 2003; 18 2004; 2 1992 1996; 58 2001; 45 2008; 2 2006; 354 1979 2005; 67 1993; 35 2005; 105 2007; 8 2007; 9 2002; 346 1961 1980 2007; 22 2007; 69 1985; 13 1979; 7 2001; 98 James (2023033000105762400_) 1961 Meinshausen (2023033000105762400_) 2006; 34 Hoerl (2023033000105762400_) 1970; 12 Tusher (2023033000105762400_) 2001; 98 Bair (2023033000105762400_) 2004; 2 Friedman (2023033000105762400_) 1989; 84 Breiman (2023033000105762400_) 2001; 45 Guo (2023033000105762400_) 2007; 8 Haff (2023033000105762400_) 1979; 7 Zhao (2023033000105762400_) 2006; 7 Dey (2023033000105762400_) 1985; 13 Banerjee (2023033000105762400_) 2008; 9 Mardia (2023033000105762400_) 1979 Tibshirani (2023033000105762400_) 1996; 58 Shipp (2023033000105762400_) 2002; 8 Rosenwald (2023033000105762400_) 2002; 346 Ramaswamy (2023033000105762400_) 2001; 98 Tibshirani (2023033000105762400_) 2003; 18 Hummel (2023033000105762400_) 2006; 354 McLachlan (2023033000105762400_) 1992 Hinton (2023033000105762400_) 2006; 18 Monti (2023033000105762400_) 2005; 105 Park (2023033000105762400_) 2007; 69 Tibshirani (2023033000105762400_) 2002; 99 Zhu (2023033000105762400_) 2004; 5 O’Neill (2023033000105762400_) 1978; 73 Frank (2023033000105762400_) 1993; 35 Green (2023033000105762400_) 1984; 46 Friedman (2023033000105762400_) 2007; 9 Rothman (2023033000105762400_) 2008; 2 Bickel (2023033000105762400_) 2008 Liang (2023033000105762400_) 2007; 22 Zou (2023033000105762400_) 2005; 67 Friedman (2023033000105762400_) 2008 Kalbfleisch (2023033000105762400_) 1980 |
| References_xml | – reference: Hinton, G., Osindero, S. and Teh, Y. (2006) A fast learning algorithm for deep belief nets. Neur. Computn, 18, 1527-1553. – reference: Green, P. J. (1984) Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives. J. R. Statist. Soc. B, 46, 149-192. – reference: Monti, S., Savage, K. J., Kutok, J. L., Feuerhake, F., Kurtin, P., Mihm, M., Wu, B., Pasqualucci, L., Neuberg, D., Aguiar, R. C. T., Dal Cin, P., Ladd, C., Pinkus, G. S., Salles, G., Harris, N. L., Dalla-Favera, R., Habermann, T. M., Aster, J. C., Golub, T. R. and Shipp, M. A. (2005) Molecular profiling of diffuse large B-cell lymphoma identifies robust subtypes including one characterized by host inflammatory response. Blood, 105, 1851-1861. – reference: Zou, H. and Hastie, T. (2005) Regularization and variable selection via the elastic net. J. R. Statist. Soc. B, 67, 301-320. – reference: Dey, D. and Srinivasan, C. (1985) Estimation of a covariance matrix under Stein's loss. Ann. Statist., 13, 1581-1591. – reference: Meinshausen, N. and Bühlmann, P. (2006) High dimensional graphs and variable selection with the lasso. Ann. Statist., 34, 1436-1462. – reference: Park, M. Y. and Hastie, T. (2007) L1-regularization path algorithm for generalized linear models. J. R. Statist. Soc. B, 69, 659-677. – reference: Friedman, J., Hastie, T. and Tibshirani, R. (2007) Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9, 432-441. – reference: Zhao, P. and Yu, B. (2006) On model selection consistency of lasso. J. Mach. Learn. Res., 7, 2541-2563. – reference: Tibshirani, R., Hastie, T., Narasimhan, B. and Chu, G. (2002) Diagnosis of multiple cancer types by shrunken centroids of gene expression. Proc. Natn. Acad. Sci. USA, 99, 6567-6572. – reference: Tibshirani, R., Hastie, T., Narasimhan, B. and Chu, G. (2003) Class prediction by nearest shrunken centroids, with applications to DNA microarrays. Statist. Sci., 18, 104-117. – reference: Shipp, M. A., Ross, K. N., Tamayo, P., Weng, A. P., Kutok, J. L., Aguiar, R. C., Gaasenbeek, M., Angelo, M., Reich, M., Pinkus, G. S., Ray, T. S., Koval, M. A., Last, K. W., Norton, A., Lister, T. A., Mesirov, J., Neuberg, D. S., Lander, E. S., Aster, J. C. and Golub, T. R. (2002) Diffuse large B-cell lymphoma outcome prediction by gene-expression profiling and supervised machine learning. Nat. Med., 8, 68-74. – reference: Zhu, J. and Hastie, T. (2004) Classification of gene microarrays by penalized logistic regression. Biostatistics, 5, 427-443. – reference: Friedman, J. (1989) Regularized discriminant analysis. J. Am. Statist. Ass., 84, 165-175. – reference: Hummel, M., Bentink, S., Berger, H., Klappwe, W., Wessendorf, S., Barth, F. T. E., Bernd, H.-W., Cogliatti, S. B., Dierlamm, J., Feller, A. C., Hansmann, M. L., Haralambieva, E., Harder, L., Hasenclever, D., Kuhn, M., Lenze, D., Lichter, P., Martin-Subero, J. I., Moller, P., Muller-Hermelink, H.-K., Ott, G., Parwaresch, R. M., Pott, C., Rosenwald, A., Rosolowski, M., Schwaenen, C., Sturzenhofecker, B., Szczepanowski, M., Trautmann, H., Wacker, H.-H., Spang, R., Loefler, M., Trumper, L., Stein, H. and Siebert, R. (2006) A biological definition of Burkitt's lymphoma from transcriptional and genomic profiling. New Engl. J. Med., 354, 2419-2430. – reference: Haff, L. (1979) Estimation of the inverse covariance matrix: random mixtures of the inverse Wishart matrix and the identity. Ann. Statist., 7, 1264-1276. – reference: Ramaswamy, S., Tamayo, P., Rifkin, R., Mukherjee, S., Yeang, C., Angelo, M., Ladd, C., Reich, M., Latulippe, E., Mesirov, J., Poggio, T., Gerald, W., Loda, M., Lander, E. and Golub, T. (2001) Multiclass cancer diagnosis using tumor gene expression signature. Proc. Natn. Acad. Sci. USA, 98, 15149-15154. – reference: Rothman, A., Levina, E. and Zhu, J. (2008) Sparse permutation invariant covariance estimation. Electr. J. Statist., 2, 494-515. – reference: Kalbfleisch, J. and Prentice, R. (1980) The Statistical Analysis of Failure Time Data. New York: Wiley. – reference: Bair, E. and Tibshirani, R. (2004) Semi-supervised methods to predict patient survival from gene expression data. PLOS Biol., 2, 511-522. – reference: Mardia, K., Kent, J. and Bibby, J. (1979) Multivariate Analysis. London: Academic Press. – reference: Hoerl, A. E. and Kennard, R. (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics, 12, 55-67. – reference: McLachlan, G. J. (1992) Discriminant Analysis and Statistical Pattern Recognition. New York: Wiley. – reference: Breiman, L. (2001) Random forests. Mach. Learn., 45, 5-32. – reference: Tibshirani, R. (1996) Regression shrinkage and selection via the lasso. J. R. Statist. Soc. 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| Snippet | We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse... We propose covariance‐regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse... In recent years, many methods have been developed for regression in high-dimensional settings. We propose covariance-regularized regression, a family of... |
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| SubjectTerms | Classification Coefficients Covariance Covariance matrices Covariance regularization data collection Datasets Discriminant analysis Estimating techniques Estimation Estimation methods Estimators Exact sciences and technology Gene expression General topics Generalized linear models Least squares Linear inference, regression Linear models Linear regression Mathematical procedures Mathematics Modeling Multivariate analysis n[double less-than sign]p n≪p prediction Probability and statistics Probability theory and stochastic processes Regression Regression analysis Regression coefficients Regulation Sciences and techniques of general use Statistical methods Statistics Stochastic processes Studies Variable selection |
| Title | Covariance-regularized regression and classification for high dimensional problems |
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