Inference for High-dimensional Differential Correlation Matrices

Motivated by differential co-expression analysis in genomics, we consider in this paper estimation and testing of high-dimensional differential correlation matrices. An adaptive thresholding procedure is introduced and theoretical guarantees are given. Minimax rate of convergence is established and the proposed estimator is shown to be adaptively rate-optimal over collections of paired correlation matrices with approximately sparse differences. Simulation results show that the procedure significantly outperforms two other natural methods that are based on separate estimation of the individual correlation matrices. The procedure is also illustrated through an analysis of a breast cancer dataset, which provides evidence at the gene co-expression level that several genes, of which a subset has been previously verified, are associated with the breast cancer. Hypothesis testing on the differential correlation matrices is also considered. A test, which is particularly well suited for testing against sparse alternatives, is introduced. In addition, other related problems, including estimation of a single sparse correlation matrix, estimation of the differential covariance matrices, and estimation of the differential cross-correlation matrices, are also discussed.Comment: Accepted for publication in Journal of Multivariate Analysi

Modelling structural coverage and the number of failure occurrences with non-homogeneous Markov chains

Most software reliability growth models specify the expected number of failures experienced as a function of testing effort or calendar time. However, there are approaches to model the development of intermediate factors driving failure occurrences. This paper starts out with presenting a model framework consisting of four consecutive relationships. It is shown that a differential equation representing this framework is a generalization of several finite failures category models. The relationships between the number of test cases executed and expected structural coverage, and between expected structural coverage and the expected number of failure occurrences are then explored further. A non-homogeneous Markov model allowing for partial redundancy in sampling code constructs is developed. The model bridges the gap between setups related to operational testing and systematic testing, respectively. Two extensions of the model considering the development of the number of failure occurrences are discussed. The paper concludes with showing that the extended models fit into the structure of the differential equation presented at the beginning, which permits further interpretation. --