Browsing by Subject "High dimensional"

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    Bayesian Modeling of Multi-Source Multi-Way Data
    (2023-11) Kim, Jonathan
    Biomedical research often involves data collected from multiple sources and these sources often have a multi-way (i.e.. multidimensional tensor) structure. Existing methods that can accommodate multi-source or multi-way data have various limitations on the exact structure of the data they are able to accommodate and in the type of predictions, if any, they are able to produce. Furthermore, few of these methods are able to handle data that are simultaneously multi-source and multi-way. We first introduce two such multi-source and multi-way datasets of molecular and hematological data from multiple sources, each measured over multiple developmental time points and in multiple tissues, as predictors of early-life iron deficiency (ID) in a rhesus monkey model. We describe preliminary analyses that were conducted on these datasets using existing methods. We then develop a Bayesian linear model that can perform prediction on a binary or continuous outcome and can accommodate data that are both multi-source and multi-way. We use a linear model with a low-rank structure on the coefficients to capture multi-way dependence and model the variance of the coefficients separately across each source to infer their relative contributions. Conjugate priors facilitate an efficient Gibbs sampling algorithm for posterior inference, assuming a continuous outcome with normal errors or a binary outcome with a probit link. Simulations demonstrate that our model performs as expected in terms of misclassification rates and correlation of estimated coefficients with true coefficients, with large gains in performance by incorporating multi-way structure and modest gains when accounting for differing signal sizes across the different sources. Moreover, it provides robust classification of ID monkeys for one of our motivating datasets. Finally, we propose a flexible method called Bayesian regression on numerous tensors (BRONTe) that can predict a continuous or binary outcome from data that are collected from an arbitrary number of sources with multi-way tensor structures of arbitrary, not necessarily equal, orders. Additionally, BRONTe is able to accommodate data where some sources partially share features within a dimension. Simulations show BRONTe to perform well at prediction when the data sources are of unequal dimensions. In an application to our other motivating dataset on multi-way measures of metabolomics and hematology parameters, BRONTe was capable of robust classification of early-life iron deficiency.
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    Likelihood ratio tests for high-dimensional normal distributions.
    (2011-12) Yang, Fan
    For a random sample of size n obtained from p-variate normal distributions, we consider the likelihood ratio tests (LRT) for their means and covariance matrices. Most of these test statistics have been extensively studied in the classical multivariate analysis and their limiting distributions under the null hypothesis were proved to be a Chi-Square distribution under the assumption that n goes to infinity while p remains fixed. In our research, we consider the high-dimensional case where both p and n go to infinity and their ratio p/n converges to a constant y in (0, 1]. We prove that the likelihood ratio test statistics under this assumption will converge in distribution to a normal random variable and we also give the explicit forms of its mean and variance. We run simulation study to show that the likelihood ratio test using this new central limit theorem outperforms the one using the traditional Chi-square approximation for analyzing high-dimensional data.