Lassoing Eigenvalues: A Classical and a Robust Approach

套索特征值：一种经典且稳健的方法

• 批准号: 1812198
• 负责人: David Tyler
• 金额: $ 15万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812198&HistoricalAwards=false
• 关键词: Lassoing; Eigenvalues; Classical; Robust; Approach

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.

分析多变量数据的需求出现在许多学科中，包括计算机科学、工程、气象学、化学计量学、心理学、社会学、生物学和遗传学等。多元统计分析的主要目标是建模和理解不同测量或变量之间的复杂相互关系。随着科学的当前趋势，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元的大量信息，即使样本点或实验单元本身的数量可能保持相对较小。这导致需要考虑大量的参数或变量之间的相互关系，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。该研究项目旨在研究基于相对较小的样本量对此类高维数据进行建模的新方法。当在每个采样点上记录许多测量值时出现的另一个问题是测量值中的大误差或异常值。这可能使基于经典统计方法的结论在未检测到异常值的情况下产生怀疑。 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鲁棒的统计方法，即即使数据包含错误的数据点也能产生有效结论的方法。本研究项目将对开发的统计方法的稳健性进行评估。本研究项目将使用在统计学中具有悠久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，随机矩阵的大样本特征值和小样本特征值分别严重向上和向下偏置，即使对于中等大的样本量。这个问题可以通过使用惩罚方法来解决，该方法将特征值收缩在一起。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。这个项目将采用测地凸处罚。此外，一些新的非光滑测地凸罚被引入，这不仅收缩特征值在一起，但也有一个套索型的效果，创建子集的相等的特征值。因此，这种非平滑惩罚方法产生了模型选择方法，或者更具体地说，产生了多尖峰协方差模型选择方法。测地线凸惩罚方法首先在经典的多元正态背景下得到发展。然而，众所周知，如果多变量正态模型不成立，在这种情况下开发的方法就会表现不佳。 一个简单而常用的方法，使经典的方法更强大的是插件的方法，也就是说，简单地取代的作用，样本协方差矩阵的方法与一个强大的替代品。对于相对于数据维度的适度样本大小，这种插件方法在性能上与利用样本协方差矩阵的方法没有太大差异。为了解决这一缺点，非光滑惩罚M-估计的协方差矩阵的开发和研究。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Lassoing eigenvalues

• DOI: 10.1093/biomet/asz076
• 发表时间: 2018-05
• 期刊: Biometrika
• 影响因子: 2.7
• 作者: David E. Tyler;Mengxi Yi

On the Variability of the Sample Covariance Matrix Under Complex Elliptical Distributions

复杂椭圆分布下样本协方差矩阵的变异性

• DOI: 10.1109/lsp.2021.3117443
• 发表时间: 2021
• 期刊: IEEE Signal Processing Letters
• 影响因子: 3.9
• 作者: Raninen, Elias;Ollila, Esa;Tyler, David
• 通讯作者: Tyler, David

Shrinking the Covariance Matrix Using Convex Penalties on the Matrix-Log Transformation

使用矩阵-对数变换上的凸惩罚来缩小协方差矩阵

• DOI: 10.1080/10618600.2020.1814788
• 发表时间: 2021
• 期刊: Journal of Computational and Graphical Statistics
• 影响因子: 2.4
• 作者: Yi, Mengxi;Tyler, David E.
• 通讯作者: Tyler, David E.

Asymptotic and bootstrap tests for subspace dimension

子空间维数的渐近和自举检验

• DOI: 10.1016/j.jmva.2021.104830
• 发表时间: 2022
• 期刊: Journal of Multivariate Analysis
• 影响因子: 1.6
• 作者: Nordhausen, Klaus;Oja, Hannu;Tyler, David E.
• 通讯作者: Tyler, David E.

Linear Pooling of Sample Covariance Matrices

• DOI: 10.1109/tsp.2021.3139207
• 发表时间: 2020-08
• 期刊: IEEE Transactions on Signal Processing
• 影响因子: 5.4
• 作者: Elias Raninen;David E. Tyler;E. Ollila