Lassoing Eigenvalues: A Classical and a Robust Approach

分析多变量数据的需求出现在许多学科，包括计算机科学、工程学、气象学、化学计量学、心理学、社会学、生物学和遗传学等。多变量统计分析的一个主要目标是对不同测量或变量之间的复杂相互关系进行建模和理解。在目前的科学趋势下，一个越来越常见的情况是收集关于每个单独的样本点或实验单位的大量信息，即使样本点或实验单位本身的数量可能仍然相对较少。这导致要考虑的参数或变量之间的相互关系非常多，但没有足够的数据来使用经典统计方法对这些关系进行充分建模。该研究项目旨在探索基于相对较小样本量的高维数据建模的新方法。当在每个采样点上记录许多测量时产生的另一个问题是测量中的大误差或离群值。这可能使基于经典统计方法的结论变得可疑，如果没有检测到离群值的话。然而，对于高维数据，检测离群值是有问题的，因此另一种选择是使用稳健的统计方法，即即使数据包含错误的数据点也能产生有效结论的方法。将评估研究项目内开发的统计方法的稳健性。该项目将使用在统计学中有很长历史的惩罚方法，来开发高维协方差矩阵的模型和估计程序。长期以来，人们已经认识到，随机矩阵的较大和较小的样本特征值分别严重地向上和向下倾斜，即使对于中等大的样本大小也是如此。这个问题可以通过使用惩罚方法来解决，这些方法将特征值一起缩小。然而，这种收缩不能使用通常的惩罚来完成，这些惩罚是精度矩阵的凸函数。该项目将采用测地线凸罚。此外，还引入了一些新的非光滑测地凸罚，它们不仅使特征值一起收缩，而且具有产生相等特征值子集的套索式效应。因此，该非平滑惩罚方法产生模型选择方法，或者更具体地，产生多尖峰协方差模型选择方法。在经典的多元正态分布下，首先发展了测地凸罚函数法。然而，众所周知，在这种情况下开发的方法在多变量正态模型不成立的情况下表现不佳。使经典方法更稳健的一种简单且经常使用的方法是插入式方法，即用稳健的替代方法简单地取代样本协方差矩阵在方法中的作用。对于相对于数据维度的适度样本大小，这种插件方法往往与利用样本协方差矩阵的方法在性能上没有太大差别。为了克服这一缺点，需要发展和研究协方差矩阵的非光滑惩罚M-估计。在这里，测地线凸性的概念起着至关重要的作用。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The need to analyze multivariate data arises in many disciplines, including computer science, engineering, meteorology, chemometrics, psychology, sociology, biology, and genetics, among others. A primary goal of multivariate statistical analysis is to model and understand the complex interrelationships between different measurements or variables. With current trends in the sciences, an increasingly common occurrence is the collection of large amounts of information on each individual sample point or experimental unit, even though the number of sample points or experimental units themselves may remain relatively small. This results in an extremely large number of parameters or interrelationships between variables to consider, but with insufficient data to adequately model these relationships using classical statistical methods. This research project aims to investigate novel ways to model such high-dimensional data based on relatively small sample sizes. Another issue that arises when many measurements are recorded on each sample point is that of large errors or outliers in the measurements. This may make the conclusion based on classical statistical methods suspect if the outliers are not detected. For high-dimensional data, though, detecting outliers is known to be problematic, and so an alternative is to use robust statistical methods, that is, methods producing valid conclusions even if the data contains bad data points. The robustness of the statistical methods developed within the research project will be evaluated.This project will use penalization methods, which have a long history within statistics, for developing models and estimation procedures for high-dimensional covariance matrices. It has long been recognized that the larger and smaller sample eigenvalues of random matrices are heavily biased upwards and downwards respectively, even for moderately large sample sizes. This problem can be addressed by using penalization methods, which shrink eigenvalues together. Such shrinkage, though, cannot be accomplished using the usual penalties which are convex functions of the precision matrix. This project will employ geodesic convex penalties. Furthermore, some novel non-smooth geodesic convex penalties are to be introduced, which not only shrink eigenvalues together but also have a lasso-type effect of creating subsets of equal eigenvalues. This non-smooth penalization approach thus yields a model selection method, or more specifically a multi-spiked covariance model selection method. The geodesic convex penalization approach is to be first developed under the classical multivariate normal setting. Methods developed under this setting, though, are well known to perform poorly if the multivariate normal model does not hold. A simple and often used approach for making classical methods more robust is the plug-in method, that is, to simply replace the role of the sample covariance matrix in a method with a robust alternative. For modest sample sizes relative to the dimension of the data, such plug-in methods tend not to differ greatly in performance from those utilizing the sample covariance matrix. To address this shortcoming, non-smooth penalized M-estimators of the covariance matrix are to be developed and studied. Here, the concept of geodesic convexity plays a crucial role.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.