Bayesian Fused Lasso regression for dynamic binary networks

We propose a multinomial logistic regression model for link prediction in a time series of directed binary networks. To account for the dynamic nature of the data we employ a dynamic model for the model parameters that is strongly connected with the fused lasso penalty. In addition to promoting sparseness, this prior allows us to explore the presence of change points in the structure of the network. We introduce fast computational algorithms for estimation and prediction using both optimization and Bayesian approaches. The performance of the model is illustrated using simulated data and data from a financial trading network in the NYMEX natural gas futures market. Supplementary material containing the trading network data set and code to implement the algorithms is available online

Nearly optimal Bayesian Shrinkage for High Dimensional Regression

During the past decade, shrinkage priors have received much attention in Bayesian analysis of high-dimensional data. In this paper, we study the problem for high-dimensional linear regression models. We show that if the shrinkage prior has a heavy and flat tail, and allocates a sufficiently large probability mass in a very small neighborhood of zero, then its posterior properties are as good as those of the spike-and-slab prior. While enjoying its efficiency in Bayesian computation, the shrinkage prior can lead to a nearly optimal contraction rate and selection consistency as the spike-and-slab prior. Our numerical results show that under posterior consistency, Bayesian methods can yield much better results in variable selection than the regularization methods, such as Lasso and SCAD. We also establish a Bernstein von-Mises type results comparable to Castillo et al (2015), this result leads to a convenient way to quantify uncertainties of the regression coefficient estimates, which has been beyond the ability of regularization methods

Covariance Estimation: The GLM and Regularization Perspectives

Finding an unconstrained and statistically interpretable reparameterization of a covariance matrix is still an open problem in statistics. Its solution is of central importance in covariance estimation, particularly in the recent high-dimensional data environment where enforcing the positive-definiteness constraint could be computationally expensive. We provide a survey of the progress made in modeling covariance matrices from two relatively complementary perspectives: (1) generalized linear models (GLM) or parsimony and use of covariates in low dimensions, and (2) regularization or sparsity for high-dimensional data. An emerging, unifying and powerful trend in both perspectives is that of reducing a covariance estimation problem to that of estimating a sequence of regression problems. We point out several instances of the regression-based formulation. A notable case is in sparse estimation of a precision matrix or a Gaussian graphical model leading to the fast graphical LASSO algorithm. Some advantages and limitations of the regression-based Cholesky decomposition relative to the classical spectral (eigenvalue) and variance-correlation decompositions are highlighted. The former provides an unconstrained and statistically interpretable reparameterization, and guarantees the positive-definiteness of the estimated covariance matrix. It reduces the unintuitive task of covariance estimation to that of modeling a sequence of regressions at the cost of imposing an a priori order among the variables. Elementwise regularization of the sample covariance matrix such as banding, tapering and thresholding has desirable asymptotic properties and the sparse estimated covariance matrix is positive definite with probability tending to one for large samples and dimensions.Comment: Published in at http://dx.doi.org/10.1214/11-STS358 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Network estimation in State Space Model with L1-regularization constraint

Biological networks have arisen as an attractive paradigm of genomic science ever since the introduction of large scale genomic technologies which carried the promise of elucidating the relationship in functional genomics. Microarray technologies coupled with appropriate mathematical or statistical models have made it possible to identify dynamic regulatory networks or to measure time course of the expression level of many genes simultaneously. However one of the few limitations fall on the high-dimensional nature of such data coupled with the fact that these gene expression data are known to include some hidden process. In that regards, we are concerned with deriving a method for inferring a sparse dynamic network in a high dimensional data setting. We assume that the observations are noisy measurements of gene expression in the form of mRNAs, whose dynamics can be described by some unknown or hidden process. We build an input-dependent linear state space model from these hidden states and demonstrate how an incorporated $L_{1}$ regularization constraint in an Expectation-Maximization (EM) algorithm can be used to reverse engineer transcriptional networks from gene expression profiling data. This corresponds to estimating the model interaction parameters. The proposed method is illustrated on time-course microarray data obtained from a well established T-cell data. At the optimum tuning parameters we found genes TRAF5, JUND, CDK4, CASP4, CD69, and C3X1 to have higher number of inwards directed connections and FYB, CCNA2, AKT1 and CASP8 to be genes with higher number of outwards directed connections. We recommend these genes to be object for further investigation. Caspase 4 is also found to activate the expression of JunD which in turn represses the cell cycle regulator CDC2.Comment: arXiv admin note: substantial text overlap with arXiv:1308.359

Functional Regression

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field