Dimension Reduction and Complex High-Dimensional Data

点击翻译按钮获取中文摘要

Recent technological advances in many fields of science have led to the routine collection of vast amounts of data. The collected data is typically high--dimensional and highly correlated. My research interests lie in adapting and extending dimension reduction methods to accommodate the complexities of these high--throughput biological data. As part of my research program, the challenges that my students and I will tackle over the next five years can be grouped into three objectives. First, missing data is common in any applied science, but it is particularly problematic with high--throughput data. Moreover, the nature of multivariate data implies that complete--case analysis can discard a significant amount of information. To increase efficiency, I will look at how multiple imputation procedures can be used with multivariate methods. To this end, I will describe the distribution of aggregates of test statistics common in multivariate analysis, and I will reformulate the optimisation problem of these methods to accommodate missing data. Second, building on some of my previous work, I will incorporate a priori knowledge through correlation structures. For example, the linear correlation patterns observed in DNA methylation data can be accounted for in dimension reduction methods by using autoregressive covariance structures. I will then look at extending this work to incorporate spatial correlation, which is common in neuroimaging data. I will also study hierarchical probabilistic multivariate models in order to model correlation between observations. Third, complex correlation also manifests through the complex geometry of the sample space, such as with image and text data. Topological data analysis provides tools to study the geometric structure of our data. I will develop simulation frameworks to study the impact of this geometry on multivariate methods. I will use this knowledge to develop methods to generate synthetic data to oversample imbalanced data. I will also combine tools from multivariate analysis and topological data analysis to extend PCEV, CCA and PLS to nonlinear dimension reduction methods. The research program proposed here will have an impact across multiple disciplines. All proposed methodologies will be implemented in software packages, so that applied researchers can more easily use them in their work and methodological researchers can more rapidly build upon them.