Multivariate Methods for High-Dimensional Transposable Data

高维数据在医学、金融、环境研究、成像、网络和互联网等领域很普遍。理解这些海量数据是关键科学问题的关键，例如发现与疾病相关的生物标志物和评估气候变化的影响。拟议的研究将使用统计学习技术来开发新的多变量方法，包括稀疏性（变量选择），平滑性和高维数据中已知结构的解释。扩展结构化数据的多变量方法，提高了信号恢复和特征选择。这些技术在时空和图像数据中将是有用的，例如，其中来自已知结构的强相关性使降维变得模糊。矩阵数据的多变量方法受到了广泛的关注。该研究将扩展许多现代多元技术，如稀疏主成分分析，多维或张量框架。最后，该建议旨在推广现有文献中的正则化多变量方法，如主成分分析，典型相关分析和线性判别分析，说明如何鼓励多种类型的正则化。该计划还寻求开发算法和计算框架，使研究人员能够将这些方法应用于现代海量数据集，随着多元分析技术的广泛应用，该计划中开发的理论和方法将在许多应用领域具有广泛的意义。开发的方法部分是由神经影像学研究，癌症基因组学和代谢组学研究的动机，并将产生直接的影响。 这种方法将证明有益的其他领域包括气候研究，遥感，网络，工程，金融和成像。这项研究的结果将通过发布公开源码的软件加以传播，并将纳入关于统计学习的高级研究生课程的教材。

High-dimensional data is prevalent in areas such as medicine, finance, environmental studies, imaging, networking, and the Internet. Making sense of this massive amount of data holds the key to critical scientific questions such as discovering biomarkers related to disease and evaluating the effects of climate change. The proposed research will use statistical learning techniques to develop novel multivariate approaches incorporating sparsity (variable selection), smoothness, and accounting for known structure in high-dimensional data. Extending multivariate methods for structured data improves signal recovery and feature selection. These techniques will be useful in spatio-temporal and image data, for example, where strong correlations from known structure obscure dimension reduction. Much attention has been given to multivariate methods for matrix data. The proposed research will extend many modern multivariate techniques such as sparse principal components analysis to the multi-dimensional or tensor framework. Finally, this proposal seeks to generalize much of the existing literature on regularized multivariate methods such as principal components analysis, canonical correlations analysis and linear discriminant analysis by illustrating how to encourage many types of regularization. This proposal also seeks to develop algorithmic and computational frameworks that will allow researchers to apply these methods to modern massive data sets.As multivariate analysis techniques enjoy nearly universal application, the theory and methods developed in this proposal will have wide ranging significance in many applied fields. The methods developed are in part motivated by and will have immediate impact in neuroimaging studies, cancer genomics, and metabolomics studies. Other areas where this methodology will prove beneficial include climate studies, remote sensing, networking, engineering, finance, and imaging. Results of this research will be disseminated through the release of open-source, publicly available software and will be incorporated in course material of an advanced graduate course on statistical learning.