Regularized multivariate regression for identifying master predictors with application to integrative genomics study of breast cancer

Abstract

In this paper, we propose a new method remMap - REgularized Multivariate regression for identifying MAster Predictors - for fitting multivariate response regression models under the high-dimension-low-sample-size setting. remMap is motivated by investigating the regulatory relationships among different biological molecules based on multiple types of high dimensional genomic data. Particularly, we are interested in studying the influence of DNA copy number alterations on RNA transcript levels. For this purpose, we model the dependence of the RNA expression levels on DNA copy numbers through multivariate linear regressions and utilize proper regularization to deal with the high dimensionality as well as to incorporate desired network structures. Criteria for selecting the tuning parameters are also discussed. The performance of the proposed method is illustrated through extensive simulation studies. Finally, remMap is applied to a breast cancer study, in which genome wide RNA transcript levels and DNA copy numbers were measured for 172 tumor samples. We identify a trans-hub region in cytoband 17q12-q21, whose amplification influences the RNA expression levels of more than 30 unlinked genes. These findings may lead to a better understanding of breast cancer pathology.

Multi-view multi-sparsity kernel reconstruction for multi-class image classification

“…For example, the 1 -norm regularizer (e.g., [9]) leads to the element sparsity; the 2,1 -norm regularizer (e.g., [36]) for the group sparsity and the mixed-norm regularizer (concatenating a 1 -norm regularizer with a 2,1 -norm regularizer, e.g., [22]) for the mixed sparsity. To generate the sparsity, the 1 -norm regularizer makes each code as a singleton, then generates four codes in the first column of Fig.2.…”

“…However, it still generates sparse codes one sample once. The mixed sparsity has been explained (e.g., [22]) as first generating the group sparsity for each sample, e.g., sparsity in the second red box (i.e., group) in the first column of Fig.2. (c), and then generating the element sparsity in the dense (i.e., non-sparse) groups, e.g., the second element in the first column of Fig.2.(c).…”

Regularized multivariate regression models with skew-t error distributions

“…Generating B in this way, we expect (1−s 2 )p predictors to be irrelevant for all q responses, and each predictor to be relevant for s 1 q of all the response variables. An n × p predictor matrix X is also generated with rows drawn independently from N (0, Σ X ), where (Σ X ) ij = 0.7 |i−j| , as in Yuan and Lin (2007) and Peng et al (2010). We consider the AR(1) covariance model as the scale matrix of the errors with (Σ) ij = ρ |i−j| .…”

“…In these situations, the traditional estimators for B and Σ with pq and q(q + 1)/2 parameters, respectively, have rather poor performances and are not suitable for prediction and other applications, so that one must seek workable alternatives by regularizing these parameters. Traditionally, this has been done individually by focusing on regularizing B or Σ alone under the headings of regularized multivariate regression (Peng et al, 2010) or regularized covariance estimation , respectively.…”

Segmented regression, a versatile tool to analyze mRNA levels in relation to DNA copy number aberrations

“…Correlation gives insight into the strength of the relationship between CNA and relative mRNA levels. Similarly, regression analysis finds its place in integrative genomic analysis (Pollack et al, 2002;Stranger et al, 2007;Menezes et al, 2009;Peng et al, 2010;Asimit et al, 2011). Some authors choose to combine the two-step analysis with an assessment of relationship strength (Pollack et al, 2002;Heidenblad et al, 2005;van Wieringen et al, 2006;Tsukamoto et al, 2008;Bicciato et al, 2009).…”