Deconvolution of transcript profiling data and asymptotic inference of crosscorrelation in L infinity
Creators: name: Sun, Die
Publisher: name: No Publisher Supplied
Year: 2017

description: In this dissertation, we consider two research projects:  deconvolution of transcript profiling data; and inferences for multivariate time series based on cross correlations, especially under high dimensionality.    The study of transcript profiling data such as macro-arrays or deep sequencing, has wide application in gene expression studies. A typical objective of gene expression study is to identify genes that are differentially expressed between groups of samples, such as normal vs. tumor tissue. However, most of the biological samples in scientific researches are heterogeneous: for the samples with identical cellular types, they may have very different proportions. Such variance in proportion will lead to confounding effects citep{Shen-Orr2013}. For example, the reflected gene expression variations are simply caused by the differences in proportions of cell subsets instead of the characteristic condition of a sample (e.g. disease). In order to eliminate the confounding effect, one solution might be to focus on the single cell subset. The isolation procedure, however, is limited by sample materials and financial budgets. Therefore, statistical deconvolution, which does not require any isolation, becomes necessary and practical. In the first project, we develop the Iterated Least Square (ILS) algorithms to estimate the cell specific signature and proportion matrix in complete blind case under homoscedasticity assumption, and theoretically justify the consistency of signature matrix estimate. We also find that the ILS estimate is equivalent to moment under homoscedasticity assumptions, and establish the central limit theorems for the moment estimates. In the heteroscedastic case, the ILS is no longer asymptotically unbiased. Thus, we propose to use the moment estimate, and develop the asymptotics of signature expression estimates. Both numerical examples and real data analysis are employed to illustrate the estimation methods and their asymptotic properties.    Cross correlations are of fundamental importance in multivariate time series analysis. We consider tests for independence of component series based on sample cross correlations. We start with a study of cross correlations between two time series. We derive the central limit theorems for sample cross correlations at large lags, establish convergence rates for maximum sample cross correlations, and demonstrate how they can be used to identify the lead lag relationship for a bivariate time series. We also propose a window sum approach to reduce the computational cost when the series is long. As a second problem, we consider tests for independence of components series under high dimensionality. We propose to use the maximum sample cross correlation over a large range of lags as the test statistic. We also consider an extension to Ljung-Box type statistics. We show that the limiting distributions of the test statistics are extreme value distribution of type I. Our results allow both the number of series, and the range of lags to grow as powers of the sample size, and reveal that how large they can be is determined by the dependence condition and moment condition. We also propose to use the moving blocks bootstrap to improve the finite sample performance of these test procedures.