A Constrained L1 Minimization Approach to Sparse Precision Matrix Estimation

A constrained L1 minimization method is proposed for estimating a sparse inverse covariance matrix based on a sample of $n$ iid $p$-variate random variables. The resulting estimator is shown to enjoy a number of desirable properties. In particular, it is shown that the rate of convergence between the estimator and the true $s$-sparse precision matrix under the spectral norm is $s\sqrt{\log p/n}$ when the population distribution has either exponential-type tails or polynomial-type tails. Convergence rates under the elementwise $L_{\infty}$ norm and Frobenius norm are also presented. In addition, graphical model selection is considered. The procedure is easily implementable by linear programming. Numerical performance of the estimator is investigated using both simulated and real data. In particular, the procedure is applied to analyze a breast cancer dataset. The procedure performs favorably in comparison to existing methods.Comment: To appear in Journal of the American Statistical Associatio

Profiling time course expression of virus genes---an illustration of Bayesian inference under shape restrictions

There have been several studies of the genome-wide temporal transcriptional program of viruses, based on microarray experiments, which are generally useful in the construction of gene regulation network. It seems that biological interpretations in these studies are directly based on the normalized data and some crude statistics, which provide rough estimates of limited features of the profile and may incur biases. This paper introduces a hierarchical Bayesian shape restricted regression method for making inference on the time course expression of virus genes. Estimates of many salient features of the expression profile like onset time, inflection point, maximum value, time to maximum value, area under curve, etc. can be obtained immediately by this method. Applying this method to a baculovirus microarray time course expression data set, we indicate that many biological questions can be formulated quantitatively and we are able to offer insights into the baculovirus biology.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS258 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org