Sufficient dimension reduction of high-dimensional data through regularized covariance estimation

许多用于降维、分类和预测的统计方法需要估计协方差或精度矩阵。在高维设置(其中变量的数量大于样本大小)中，众所周知，使用样本协方差的经典协方差估计的性能很差。这导致了大量可供选择的正则化高维协方差估计，其中许多是在过去十年中提出的。对这些估计量的分析主要是根据它们在直接估计总体协方差或精度矩阵时的表现，而不是它们如何影响需要正则化协方差估计的统计方法的性能。一类特别感兴趣的统计方法是那些执行充分降维(SDR)的方法，这是一种有效的方法来降低回归问题中预测器的维度。大多数SDR方法和理论要求变量的数量小于样本量，从而阻止其应用于高维数据。PI、Co-PI和他们的同事通过正则化协方差估计使足够的降维方法适应高维环境。具体地说，他们开发了另一种SDR方法、高维渐近分析(随着变量数量和样本量的增加)、高效的计算算法和数据应用程序。遗传学、光谱学、气候研究和遥感是产生高维数据的许多研究领域中的几个例子；这些数据具有比主题或案例更多的可测量特征。许多用于预测、分类和数据简化的标准统计方法在这种情况下要么不适用，要么执行得很差。作为响应，已经开发了用于提取用于预测模型的测量特征的子集的统计方法；然而，这些方法在相对较少数量的测量特征与预测相关的假设下运行。研究人员通过开发用于预测建模的高维数据缩减的新方法来解决这一不足，与许多现有方法不同，该方法能够从所有测量的特征中提取相关的预测信息。此外，研究人员还开发公开可用的计算机软件来实施这些新方法，使研究人员和从业者能够在许多领域应用这些方法。

Many statistical methods for dimensionality reduction, classification, and prediction, require an estimate of a covariance or precision matrix. In high-dimensional settings, (where the number of variables is larger than the sample size), it is known that classical covariance estimation with the sample covariance performs poorly. This has lead to a wealth of alternative regularized high-dimension covariance estimators, many of which have been proposed in the last decade. These estimators have been analyzed primarily in terms of how they perform when estimating the population covariance or precision matrix directly, rather than how they affect the performance of the statistical methods that require a regularized covariance estimate. A particular class of statistical methods of interest is those that perform sufficient dimension reduction (SDR), a powerful approach to reduce the dimensionality of the predictor in regression problems. Most of the SDR methodology and theory requires the number of variables to be less than the sample size, preventing its application to high-dimensional data. The PI, Co-PI, and their colleagues adapt sufficient dimension reduction methodology to high-dimensional settings via regularized covariance estimation. Specifically, they develop alternative SDR methodology, high-dimensional asymptotic analysis (as both the number of variables and the sample size grow), efficient computational algorithms, and applications to data.Genetics, spectroscopy, climate studies, and remote sensing are a few examples of the many research fields that produce high-dimensional data; these are data with many more measured characteristics than subjects or cases. Many standard statistical methods for prediction, classification, and data reduction are either inapplicable or perform poorly in this setting. In response, statistical methods to extract a subset of the measured characteristics for use in predictive models have been developed; however, these methods operate under the assumption that a relatively small number of measured characteristics are relevant for prediction. The investigators address this deficiency by developing new methods for the reduction of high-dimensional data for use in predictive modeling, which unlike many existing methods, are able to extract relevant predictive information from all of the measured characteristics. In addition, the investigators develop publicly available computer software to implement these new methods, enabling their application by researchers and practitioners in many fields.