Reduced rank stochastic regression with a sparse singular value decomposition

Spectra in low-rank localized layers (SpeLLL) for interpretable time-frequency analysis

The time-varying frequency characteristics of many biomedical time series contain important scientific information. However, the high-dimensional nature of the time-varying power spectrum as a surface in time and frequency limits its direct use by applied researchers and clinicians for elucidating complex mechanisms. In this article, we introduce a new approach to time-frequency analysis that decomposes the time-varying power spectrum in to orthogonal rank-one layers in time and frequency to provide a parsimonious representation that illustrates relationships between power at different times and frequencies. The approach can be used in fully nonparametric analyses or in semiparametric analyses that account for exogenous information and time-varying covariates. An estimation procedure is formulated within a penalized reduced-rank regression framework that provides estimates of layers that are interpretable as power localized within time blocks and frequency bands. Empirical properties of the procedure are illustrated in simulation studies and its practical use is demonstrated through an analysis of heart rate variability during sleep.

Stability Approach to Regularization Selection for Reduced-Rank Regression

The reduced-rank regression model is a popular model to deal with multivariate response and multiple predictors, and is widely used in biology, chemometrics, econometrics, engineering, and other fields. In the reduced-rank regression modelling, a central objective is to estimate the rank of the coefficient matrix that represents the number of effective latent factors in predicting the multivariate response. Although theoretical results such as rank estimation consistency have been established for various methods, in practice rank determination still relies on information criterion based methods such as AIC and BIC or subsampling based methods such as cross validation. Unfortunately, the theoretical properties of these practical methods are largely unknown. In this paper, we present a novel method called StARS-RRR that selects the tuning parameter and then estimates the rank of the coefficient matrix for reduced-rank regression based on the stability approach. We prove that StARS-RRR achieves rank estimation consistency, i.e., the rank estimated with the tuning parameter selected by StARS-RRR is consistent to the true rank. Through a simulation study, we show that StARS-RRR outperforms other tuning parameter selection methods including AIC, BIC, and cross validation as it provides the most accurate estimated rank. In addition, when applied to a breast cancer dataset, StARS-RRR discovers a reasonable number of genetic pathways that affect the DNA copy number variations and results in a smaller prediction error than the other methods with a random-splitting process.

Robust Sparse Reduced-Rank Regression with Response Dependency

In multiple response regression, the reduced rank regression model is an effective method to reduce the number of model parameters and it takes advantage of interrelation among the response variables. To improve the prediction performance of the multiple response regression, a method for the sparse robust reduced rank regression with covariance estimation(Cov-SR4) is proposed, which can carry out variable selection, outlier detection, and covariance estimation simultaneously. The random error term of this model follows a multivariate normal distribution which is a symmetric distribution and the covariance matrix or precision matrix must be a symmetric matrix that reduces the number of parameters. Both the element-wise penalty function and row-wise penalty function can be used to handle different types of outliers. A numerical algorithm with a covariance estimation method is proposed to solve the robust sparse reduced rank regression. We compare our method with three recent reduced rank regression methods in a simulation study and real data analysis. Our method exhibits competitive performance both in prediction error and variable selection accuracy.

Response Best-subset Selector for Multivariate Regression with High-dimensional Response Variables

This article investigates the statistical problem of response-variable selection with high-dimensional response variables and a diverging number of predictor variables with respect to the sample size in the framework of multivariate linear regression. A response best-subset selection model is proposed by introducing a 0–1 selection indictor for each response variable, then a response best-subset selector is developed by introducing a separation parameter and a novel penalized least-squares function. The developed procedure can perform response-variable selection and regression-coefficient estimation simultaneously, and the proposed response best-subset selector has model consistency under mild conditions for both fixed and diverging numbers of predictor variables. Also, consistency and asymptotic normality of regression-coefficient estimators are presented for cases with a fixed dimension, and it is discovered that the Bonferroni test is a special response best-subset selector. Finite-sample simulations show that the response best-subset selector has strong advantages over existing competitors in terms of the Matthews correlation coefficient, a criterion aimed at balancing accuracies for both true and false response variables. An analysis of actual data demonstrates the effectiveness of the response best-subset selector in an application involving the identification of dosage-sensitive genes.

Negative binomial factor regression with application to microbiome data analysis

The human microbiome provides essential physiological functions and helps maintain host homeostasis via the formation of intricate ecological host‐microbiome relationships. While it is well established that the lifestyle of the host, dietary preferences, demographic background, and health status can influence microbial community composition and dynamics, robust generalizable associations between specific host‐associated factors and specific microbial taxa have remained largely elusive. Here, we propose factor regression models that allow the estimation of structured parsimonious associations between host‐related features and amplicon‐derived microbial taxa. To account for the overdispersed nature of the amplicon sequencing count data, we propose negative binomial reduced rank regression (NB‐RRR) and negative binomial co‐sparse factor regression (NB‐FAR). While NB‐RRR encodes the underlying dependency among the microbial abundances as outcomes and the host‐associated features as predictors through a rank‐constrained coefficient matrix, NB‐FAR uses a sparse singular value decomposition of the coefficient matrix. The latter approach avoids the notoriously difficult joint parameter estimation by extracting sparse unit‐rank components of the coefficient matrix sequentially, effectively delivering interpretable bi‐clusters of taxa and host‐associated factors. To solve the nonconvex optimization problems associated with these factor regression models, we present a novel iterative block‐wise majorization procedure. Extensive simulation studies and an application to the microbial abundance data from the American Gut Project (AGP) demonstrate the efficacy of the proposed procedure. In the AGP data, we identify several factors that strongly link dietary habits and host life style to specific microbial families.

Sparse Reduced Rank Huber Regression in High Dimensions

We propose a sparse reduced rank Huber regression for analyzing large and complex high-dimensional data with heavy-tailed random noise. The proposed method is based on a convex relaxation of a rank- and sparsity-constrained nonconvex optimization problem, which is then solved using a block coordinate descent and an alternating direction method of multipliers algorithm. We establish nonasymptotic estimation error bounds under both Frobenius and nuclear norms in the high-dimensional setting. This is a major contribution over existing results in reduced rank regression, which mainly focus on rank selection and prediction consistency. Our theoretical results quantify the tradeoff between heavy-tailedness of the random noise and statistical bias. For random noise with bounded ( 1 + δ ) th moment with δ ∈ ( 0 , 1 ) , the rate of convergence is a function of δ , and is slower than the sub-Gaussian-type deviation bounds; for random noise with bounded second moment, we obtain a rate of convergence as if sub-Gaussian noise were assumed. We illustrate the performance of the proposed method via extensive numerical studies and a data application. Supplementary materials for this article are available online.

Sparse reduced-rank regression for simultaneous rank and variable selection via manifold optimization

We consider the problem of constructing a reduced-rank regression model whose coefficient parameter is represented as a singular value decomposition with sparse singular vectors. The traditional estimation procedure for the coefficient parameter often fails when the true rank of the parameter is high. To overcome this issue, we develop an estimation algorithm with rank and variable selection via sparse regularization and manifold optimization, which enables us to obtain an accurate estimation of the coefficient parameter even if the true rank of the coefficient parameter is high. Using sparse regularization, we can also select an optimal value of the rank. We conduct Monte Carlo experiments and a real data analysis to illustrate the effectiveness of our proposed method.

An efficient framework to identify key miRNA-mRNA regulatory modules in cancer

MOTIVATION

Micro-RNAs (miRNAs) are known as the important components of RNA silencing and post-transcriptional gene regulation, and they interact with messenger RNAs (mRNAs) either by degradation or by translational repression. miRNA alterations have a significant impact on the formation and progression of human cancers. Accordingly, it is important to establish computational methods with high predictive performance to identify cancer-specific miRNA-mRNA regulatory modules.

RESULTS

We presented a two-step framework to model miRNA-mRNA relationships and identify cancer-specific modules between miRNAs and mRNAs from their matched expression profiles of more than 9000 primary tumors. We first estimated the regulatory matrix between miRNA and mRNA expression profiles by solving multiple linear programming problems. We then formulated a unified regularized factor regression (RFR) model that simultaneously estimates the effective number of modules (i.e. latent factors) and extracts modules by decomposing regulatory matrix into two low-rank matrices. Our RFR model groups correlated miRNAs together and correlated mRNAs together, and also controls sparsity levels of both matrices. These attributes lead to interpretable results with high predictive performance. We applied our method on a very comprehensive data collection by including 32 TCGA cancer types. To find the biological relevance of our approach, we performed functional gene set enrichment and survival analyses. A large portion of the identified modules are significantly enriched in Hallmark, PID and KEGG pathways/gene sets. To validate the identified modules, we also performed literature validation as well as validation using experimentally supported miRTarBase database.

AVAILABILITY AND IMPLEMENTATION

Our implementation of proposed two-step RFR algorithm in R is available at https://github.com/MiladMokhtaridoost/2sRFR together with the scripts that replicate the reported experiments.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

A fully Bayesian approach to sparse reduced-rank multivariate regression

In the context of high-dimensional multivariate linear regression, sparse reduced-rank regression (SRRR) provides a way to handle both variable selection and low-rank estimation problems. Although there has been extensive research on SRRR, statistical inference procedures that deal with the uncertainty due to variable selection and rank reduction are still limited. To fill this research gap, we develop a fully Bayesian approach to SRRR. A major difficulty that occurs in a fully Bayesian framework is that the dimension of parameter space varies with the selected variables and the reduced-rank. Due to the varying-dimensional problems, traditional Markov chain Monte Carlo (MCMC) methods such as Gibbs sampler and Metropolis-Hastings algorithm are inapplicable in our Bayesian framework. To address this issue, we propose a new posterior computation procedure based on the Laplace approximation within the collapsed Gibbs sampler. A key feature of our fully Bayesian method is that the model uncertainty is automatically integrated out by the proposed MCMC computation. The proposed method is examined via simulation study and real data analysis.

Sparse reduced-rank regression for integrating omics data

BACKGROUND

The problem of assessing associations between multiple omics data including genomics and metabolomics data to identify biomarkers potentially predictive of complex diseases has garnered considerable research interest nowadays. A popular epidemiology approach is to consider an association of each of the predictors with each of the response using a univariate linear regression model, and to select predictors that meet a priori specified significance level. Although this approach is simple and intuitive, it tends to require larger sample size which is costly. It also assumes variables for each data type are independent, and thus ignores correlations that exist between variables both within each data type and across the data types.

RESULTS

We consider a multivariate linear regression model that relates multiple predictors with multiple responses, and to identify multiple relevant predictors that are simultaneously associated with the responses. We assume the coefficient matrix of the responses on the predictors is both row-sparse and of low-rank, and propose a group Dantzig type formulation to estimate the coefficient matrix.

CONCLUSION

Extensive simulations demonstrate the competitive performance of our proposed method when compared to existing methods in terms of estimation, prediction, and variable selection. We use the proposed method to integrate genomics and metabolomics data to identify genetic variants that are potentially predictive of atherosclerosis cardiovascular disease (ASCVD) beyond well-established risk factors. Our analysis shows some genetic variants that increase prediction of ASCVD beyond some well-established factors of ASCVD, and also suggest a potential utility of the identified genetic variants in explaining possible association between certain metabolites and ASCVD.

A regression framework to uncover pleiotropy in large-scale electronic health record data

OBJECTIVE

Pleiotropy, where 1 genetic locus affects multiple phenotypes, can offer significant insights in understanding the complex genotype-phenotype relationship. Although individual genotype-phenotype associations have been thoroughly explored, seemingly unrelated phenotypes can be connected genetically through common pleiotropic loci or genes. However, current analyses of pleiotropy have been challenged by both methodologic limitations and a lack of available suitable data sources.

MATERIALS AND METHODS

In this study, we propose to utilize a new regression framework, reduced rank regression, to simultaneously analyze multiple phenotypes and genotypes to detect pleiotropic effects. We used a large-scale biobank linked electronic health record data from the Penn Medicine BioBank to select 5 cardiovascular diseases (hypertension, cardiac dysrhythmias, ischemic heart disease, congestive heart failure, and heart valve disorders) and 5 mental disorders (mood disorders; anxiety, phobic and dissociative disorders; alcohol-related disorders; neurological disorders; and delirium dementia) to validate our framework.

RESULTS

Compared with existing methods, reduced rank regression showed a higher power to distinguish known associated single-nucleotide polymorphisms from random single-nucleotide polymorphisms. In addition, genome-wide gene-based investigation of pleiotropy showed that reduced rank regression was able to identify candidate genetic variants with novel pleiotropic effects compared to existing methods.

CONCLUSION

The proposed regression framework offers a new approach to account for the phenotype and genotype correlations when identifying pleiotropic effects. By jointly modeling multiple phenotypes and genotypes together, the method has the potential to distinguish confounding from causal genotype and phenotype associations.

SOFAR: Large-Scale Association Network Learning

Many modern big data applications feature large scale in both numbers of responses and predictors. Better statistical efficiency and scientific insights can be enabled by understanding the large-scale response-predictor association network structures via layers of sparse latent factors ranked by importance. Yet sparsity and orthogonality have been two largely incompatible goals. To accommodate both features, in this paper we suggest the method of sparse orthogonal factor regression (SOFAR) via the sparse singular value decomposition with orthogonality constrained optimization to learn the underlying association networks, with broad applications to both unsupervised and supervised learning tasks such as biclustering with sparse singular value decomposition, sparse principal component analysis, sparse factor analysis, and spare vector autoregression analysis. Exploiting the framework of convexity-assisted nonconvex optimization, we derive nonasymptotic error bounds for the suggested procedure characterizing the theoretical advantages. The statistical guarantees are powered by an efficient SOFAR algorithm with convergence property. Both computational and theoretical advantages of our procedure are demonstrated with several simulations and real data examples.

Integrative multi-view regression: Bridging group-sparse and low-rank models

Multi-view data have been routinely collected in various fields of science and engineering. A general problem is to study the predictive association between multivariate responses and multi-view predictor sets, all of which can be of high dimensionality. It is likely that only a few views are relevant to prediction, and the predictors within each relevant view contribute to the prediction collectively rather than sparsely. We cast this new problem under the familiar multivariate regression framework and propose an integrative reduced-rank regression (iRRR), where each view has its own low-rank coefficient matrix. As such, latent features are extracted from each view in a supervised fashion. For model estimation, we develop a convex composite nuclear norm penalization approach, which admits an efficient algorithm via alternating direction method of multipliers. Extensions to non-Gaussian and incomplete data are discussed. Theoretically, we derive non-asymptotic oracle bounds of iRRR under a restricted eigenvalue condition. Our results recover oracle bounds of several special cases of iRRR including Lasso, group Lasso, and nuclear norm penalized regression. Therefore, iRRR seamlessly bridges group-sparse and low-rank methods and can achieve substantially faster convergence rate under realistic settings of multi-view learning. Simulation studies and an application in the Longitudinal Studies of Aging further showcase the efficacy of the proposed methods.

Bayesian sparse reduced rank multivariate regression

Many modern statistical problems can be cast in the framework of multivariate regression, where the main task is to make statistical inference for a possibly sparse and low-rank coefficient matrix. The low-rank structure in the coefficient matrix is of intrinsic multivariate nature, which, when combined with sparsity, can further lift dimension reduction, conduct variable selection, and facilitate model interpretation. Using a Bayesian approach, we develop a unified sparse and low-rank multivariate regression method to both estimate the coefficient matrix and obtain its credible region for making inference. The newly developed sparse and low-rank prior for the coefficient matrix enables rank reduction, predictor selection and response selection simultaneously. We utilize the marginal likelihood to determine the regularization hyperparameter, so our method maximizes its posterior probability given the data. For theoretical aspect, the posterior consistency is established to discuss an asymptotic behavior of the proposed method. The efficacy of the proposed approach is demonstrated via simulation studies and a real application on yeast cell cycle data.

Sequential Co-Sparse Factor Regression

In multivariate regression models, a sparse singular value decomposition of the regression component matrix is appealing for reducing dimensionality and facilitating interpretation. However, the recovery of such a decomposition remains very challenging, largely due to the simultaneous presence of orthogonality constraints and co-sparsity regularization. By delving into the underlying statistical data generation mechanism, we reformulate the problem as a supervised co-sparse factor analysis, and develop an efficient computational procedure, named sequential factor extraction via co-sparse unit-rank estimation (SeCURE), that completely bypasses the orthogonality requirements. At each step, the problem reduces to a sparse multivariate regression with a unit-rank constraint. Nicely, each sequentially extracted sparse and unit-rank coefficient matrix automatically leads to co-sparsity in its pair of singular vectors. Each latent factor is thus a sparse linear combination of the predictors and may influence only a subset of responses. The proposed algorithm is guaranteed to converge, and it ensures efficient computation even with incomplete data and/or when enforcing exact orthogonality is desired. Our estimators enjoy the oracle properties asymptotically; a non-asymptotic error bound further reveals some interesting finite-sample behaviors of the estimators. The efficacy of SeCURE is demonstrated by simulation studies and two applications in genetics.

LINKING LUNG AIRWAY STRUCTURE TO PULMONARY FUNCTION VIA COMPOSITE BRIDGE REGRESSION

The human lung airway is a complex inverted tree-like structure. Detailed airway measurements can be extracted from MDCT-scanned lung images, such as segmental wall thickness, airway diameter, parent-child branch angles, etc. The wealth of lung airway data provides a unique opportunity for advancing our understanding of the fundamental structure-function relationships within the lung. An important problem is to construct and identify important lung airway features in normal subjects and connect these to standardized pulmonary function test results such as FEV1%. Among other things, the problem is complicated by the fact that a particular airway feature may be an important (relevant) predictor only when it pertains to segments of certain generations. Thus, the key is an efficient, consistent method for simultaneously conducting group selection (lung airway feature types) and within-group variable selection (airway generations), i.e., bi-level selection. Here we streamline a comprehensive procedure to process the lung airway data via imputation, normalization, transformation and groupwise principal component analysis, and then adopt a new composite penalized regression approach for conducting bi-level feature selection. As a prototype of composite penalization, the proposed composite bridge regression method is shown to admit an efficient algorithm, enjoy bi-level oracle properties, and outperform several existing methods. We analyze the MDCT lung image data from a cohort of 132 subjects with normal lung function. Our results show that, lung function in terms of FEV1% is promoted by having a less dense and more homogeneous lung comprising an airway whose segments enjoy more heterogeneity in wall thicknesses, larger mean diameters, lumen areas and branch angles. These data hold the potential of defining more accurately the "normal" subject population with borderline atypical lung functions that are clearly influenced by many genetic and environmental factors.

Analysis of Double Single Index Models

Motivated from problems in canonical correlation analysis, reduced rank regression and sufficient dimension reduction, we introduce a double dimension reduction model where a single index of the multivariate response is linked to the multivariate covariate through a single index of these covariates, hence the name double single index model. Since nonlinear association between two sets of multivariate variables can be arbitrarily complex and even intractable in general, we aim at seeking a principal one-dimensional association structure where a response index is fully characterized by a single predictor index. The functional relation between the two single-indices is left unspecified, allowing flexible exploration of any potential nonlinear association. We argue that such double single index association is meaningful and easy to interpret, and the rest of the multi-dimensional dependence structure can be treated as nuisance in model estimation. We investigate the estimation and inference of both indices and the regression function, and derive the asymptotic properties of our procedure. We illustrate the numerical performance in finite samples and demonstrate the usefulness of the modeling and estimation procedure in a multi-covariate multi-response problem concerning concrete.

A note on rank reduction in sparse multivariate regression

A reduced-rank regression with sparse singular value decomposition (RSSVD) approach was proposed by Chen et al. for conducting variable selection in a reduced-rank model. To jointly model the multivariate response, the method efficiently constructs a prespecified number of latent variables as some sparse linear combinations of the predictors. Here, we generalize the method to also perform rank reduction, and enable its usage in reduced-rank vector autoregressive (VAR) modeling to perform automatic rank determination and order selection. We show that in the context of stationary time-series data, the generalized approach correctly identifies both the model rank and the sparse dependence structure between the multivariate response and the predictors, with probability one asymptotically. We demonstrate the efficacy of the proposed method by simulations and analyzing a macro-economical multivariate time series using a reduced-rank VAR model.

Regularization Methods for High-Dimensional Instrumental Variables Regression With an Application to Genetical Genomics

In genetical genomics studies, it is important to jointly analyze gene expression data and genetic variants in exploring their associations with complex traits, where the dimensionality of gene expressions and genetic variants can both be much larger than the sample size. Motivated by such modern applications, we consider the problem of variable selection and estimation in high-dimensional sparse instrumental variables models. To overcome the difficulty of high dimensionality and unknown optimal instruments, we propose a two-stage regularization framework for identifying and estimating important covariate effects while selecting and estimating optimal instruments. The methodology extends the classical two-stage least squares estimator to high dimensions by exploiting sparsity using sparsity-inducing penalty functions in both stages. The resulting procedure is efficiently implemented by coordinate descent optimization. For the representative L₁ regularization and a class of concave regularization methods, we establish estimation, prediction, and model selection properties of the two-stage regularized estimators in the high-dimensional setting where the dimensionality of co-variates and instruments are both allowed to grow exponentially with the sample size. The practical performance of the proposed method is evaluated by simulation studies and its usefulness is illustrated by an analysis of mouse obesity data. Supplementary materials for this article are available online.

On the degrees of freedom of reduced-rank estimators in multivariate regression

We study the effective degrees of freedom of a general class of reduced-rank estimators for multivariate regression in the framework of Stein's unbiased risk estimation. A finite-sample exact unbiased estimator is derived that admits a closed-form expression in terms of the thresholded singular values of the least-squares solution and hence is readily computable. The results continue to hold in the high-dimensional setting where both the predictor and the response dimensions may be larger than the sample size. The derived analytical form facilitates the investigation of theoretical properties and provides new insights into the empirical behaviour of the degrees of freedom. In particular, we examine the differences and connections between the proposed estimator and a commonly-used naive estimator. The use of the proposed estimator leads to efficient and accurate prediction risk estimation and model selection, as demonstrated by simulation studies and a data example.

Learning regulatory programs by threshold SVD regression

We formulate a statistical model for the regulation of global gene expression by multiple regulatory programs and propose a thresholding singular value decomposition (T-SVD) regression method for learning such a model from data. Extensive simulations demonstrate that this method offers improved computational speed and higher sensitivity and specificity over competing approaches. The method is used to analyze microRNA (miRNA) and long noncoding RNA (lncRNA) data from The Cancer Genome Atlas (TCGA) consortium. The analysis yields previously unidentified insights into the combinatorial regulation of gene expression by noncoding RNAs, as well as findings that are supported by evidence from the literature.

Network-guided regression for detecting associations between DNA methylation and gene expression

MOTIVATION

High-throughput profiling in biological research has resulted in the availability of a wealth of data cataloguing the genetic, epigenetic and transcriptional states of cells. These data could yield discoveries that may lead to breakthroughs in the diagnosis and treatment of human disease, but require statistical methods designed to find the most relevant patterns from millions of potential interactions. Aberrant DNA methylation is often a feature of cancer, and has been proposed as a therapeutic target. However, the relationship between DNA methylation and gene expression remains poorly understood.

RESULTS

We propose Network-sparse Reduced-Rank Regression (NsRRR), a multivariate regression framework capable of using prior biological knowledge expressed as gene interaction networks to guide the search for associations between gene expression and DNA methylation signatures. We use simulations to show the advantage of our proposed model in terms of variable selection accuracy over alternative models that do not use prior network information. We discuss an application of NsRRR to The Cancer Genome Atlas datasets on primary ovarian tumours.

AVAILABILITY AND IMPLEMENTATION

R code implementing the NsRRR model is available at http://www2.imperial.ac.uk/∼gmontana

CONTACT

giovanni.montana@kcl.ac.uk

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Reconstructing source-sink dynamics in a population with a pelagic dispersal phase

For many organisms, the reconstruction of source-sink dynamics is hampered by limited knowledge of the spatial assemblage of either the source or sink components or lack of information on the strength of the linkage for any source-sink pair. In the case of marine species with a pelagic dispersal phase, these problems may be mitigated through the use of particle drift simulations based on an ocean circulation model. However, when simulated particle trajectories do not intersect sampling sites, the corroboration of model drift simulations with field data is hampered. Here, we apply a new statistical approach for reconstructing source-sink dynamics that overcomes the aforementioned problems. Our research is motivated by the need for understanding observed changes in jellyfish distributions in the eastern Bering Sea since 1990. By contrasting the source-sink dynamics reconstructed with data from the pre-1990 period with that from the post-1990 period, it appears that changes in jellyfish distribution resulted from the combined effects of higher jellyfish productivity and longer dispersal of jellyfish resulting from a shift in the ocean circulation starting in 1991. A sensitivity analysis suggests that the source-sink reconstruction is robust to typical systematic and random errors in the ocean circulation model driving the particle drift simulations. The jellyfish analysis illustrates that new insights can be gained by studying structural changes in source-sink dynamics. The proposed approach is applicable for the spatial source-sink reconstruction of other species and even abiotic processes, such as sediment transport.

Reduced rank regression via adaptive nuclear norm penalization

We propose an adaptive nuclear norm penalization approach for low-rank matrix approximation, and use it to develop a new reduced rank estimation method for high-dimensional multivariate regression. The adaptive nuclear norm is defined as the weighted sum of the singular values of the matrix, and it is generally non-convex under the natural restriction that the weight decreases with the singular value. However, we show that the proposed non-convex penalized regression method has a global optimal solution obtained from an adaptively soft-thresholded singular value decomposition. The method is computationally efficient, and the resulting solution path is continuous. The rank consistency of and prediction/estimation performance bounds for the estimator are established for a high-dimensional asymptotic regime. Simulation studies and an application in genetics demonstrate its efficacy.

Analysis of genome-wide association studies with multiple outcomes using penalization

Genome-wide association studies have been extensively conducted, searching for markers for biologically meaningful outcomes and phenotypes. Penalization methods have been adopted in the analysis of the joint effects of a large number of SNPs (single nucleotide polymorphisms) and marker identification. This study is partly motivated by the analysis of heterogeneous stock mice dataset, in which multiple correlated phenotypes and a large number of SNPs are available. Existing penalization methods designed to analyze a single response variable cannot accommodate the correlation among multiple response variables. With multiple response variables sharing the same set of markers, joint modeling is first employed to accommodate the correlation. The group Lasso approach is adopted to select markers associated with all the outcome variables. An efficient computational algorithm is developed. Simulation study and analysis of the heterogeneous stock mice dataset show that the proposed method can outperform existing penalization methods.