Dimension Reduction and Complex High-Dimensional Data

许多科学领域的最新技术进步导致了大量数据的常规收集。所收集的数据通常是高维且高度相关的。我的研究兴趣在于适应和扩展降维方法，以适应这些高通量生物数据的复杂性。作为我研究计划的一部分，我和我的学生将在未来五年内解决的挑战可以分为三个目标。首先，丢失数据在任何应用科学中都很常见，但对于高吞吐量数据来说，这尤其成问题。此外，多元数据的性质意味着完整的个案分析可能会丢弃大量信息。为了提高效率，我将研究如何将多个imputation过程与多变量方法一起使用。为此，我将描述多元分析中常见的检验统计总量的分布，并将重新制定这些方法的优化问题，以适应缺失的数据。其次，在我之前的一些工作的基础上，我将通过关联结构纳入先验知识。例如，在DNA甲基化数据中观察到的线性相关模式可以通过使用自回归协方差结构在降维方法中进行解释。然后，我将扩展这项工作，将空间相关性纳入其中，这在神经成像数据中很常见。我还将研究分层概率多元模型，以便模拟观测之间的相关性。第三，复相关性还表现为样本空间的复杂几何结构，如图像和文本数据。拓扑数据分析为研究数据的几何结构提供了工具。我将开发模拟框架来研究这种几何对多元方法的影响。我将利用这些知识来开发生成合成数据的方法来对不平衡数据进行过采样。我还将结合多元分析和拓扑数据分析的工具，将PCEV、CCA和PLS扩展到非线性降维方法。这里提出的研究计划将对多个学科产生影响。所有提出的方法都将在软件包中实现，以便应用研究人员可以更容易地在他们的工作中使用它们，并且方法研究人员可以更快地在它们的基础上进行构建。

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.