Robust Estimation for Structured Covariance Models

分析多变量数据的需求出现在许多不同的学科中，例如计算机科学与工程、信号处理、心理学、气象学、化学计量学、社会学和生物学。由于收集数据的方法不断变化，在一次观察中或为一个单一的主题测量的变量或属性的数量变得非常大，它们可以大大超过观察或主题本身的数量。这样的数据集通常被称为大型稀疏数据集。对于这类数据集，记录不良数据点或异常值的可能性越来越大。异常值往往对数据的解释产生不成比例的影响，除非使用可靠的方法，即可以容纳坏数据的方法。开发分析大型稀疏数据集的方法已成为统计学领域的一个主要研究课题。然而，对于大型稀疏数据集的鲁棒方法的发展，这是本研究项目的主要目标，相对较少关注。本研究项目的目的是为其他研究人员提供适用于大型稀疏数据集稳健分析的基本结果、理论方法和统计方法。大多数稳健的多变量统计方法主要适用于样本量远大于变量数的情况，而不是特别适用于大型稀疏数据集。特别是，对于相对于变量数量适中的样本大小，多变量位置和散布的稳健仿射等变估计在性能上与经典样本均值向量和样本协方差矩阵相似，因此不会产生此类数据集的稳健结果。分析相对稀疏的多变量数据往往需要假设某些协方差结构，例如图形模型，因子分析或其他降秩模型中出现的协方差结构，或者开发通过正则化方法优先考虑某些协方差结构的方法。这些特殊的协方差结构通常不被考虑在大多数强大的多元方法。为了解决这一缺点，该研究项目旨在开发考虑到假设协方差结构的稳健方法，特别是开发和研究结构协方差模型的直接M估计方法和S估计方法，以及开发和研究协方差矩阵的惩罚M估计。解决结构化协方差模型和惩罚方法的鲁棒性问题是一个基本问题，在数学和计算上比在经典设置或在不受限制的鲁棒估计设置更具挑战性。在这里，最近的一些工作测地线凸性信号处理社区内预计将发挥重要作用，在解决这些问题。

The need to analyze multivariate data arises in many diverse disciplines, such as computer science and engineering, signal processing, psychology, meteorology, chemometrics, sociology, and biology. Due to the changing methods for collecting data, the number of variables or attributes measured in a single observation or for a single subject are becoming exceptionally large, and they can be considerably larger than the number of observations or subjects themselves. Such data sets are commonly referred to as large sparse data sets. For such data sets, the possibility of recording bad data points or outliers is increasingly likely. Outliers tend to have a disproportionate impact on the interpretation of the data unless one uses robust methods, that is, methods that can accommodate bad data. Developing methods to analyze large sparse data sets has become a major research topic within the field of statistics. There has been, however, relatively little attention given to the development of robust methods for large sparse data sets, which is the primary goal of this research project. The research project aims to produce fundamental results, theoretical approaches and statistical methods applicable to the robust analysis of large sparse data sets, upon which other researchers can build.Most robust multivariate statistical methods are mainly applicable whenever the sample size is considerably larger than the number of variables, and are not particularly applicable to large sparse data sets. In particular, for sample sizes that are modest relative to the number of variables, robust affine equivariant estimates of multivariate location and scatter are similar in performance to the classical sample mean vector and sample covariance matrix, and consequently do not yield robust results for such data sets. Analyzing relatively sparse multivariate data tends to require either presuming certain covariance structures, such as those arising in graphical models, factor analysis or other reduced rank models, or developing methods which give preference to certain covariance structures via regularization methods. These special covariance structures are usually not considered in most robust multivariate methods. To address this shortcoming, the research project aims to develop robust methods which take into account a presumed covariance structure, and in particular to develop and study direct M-estimation methods and S-estimation methods for structured covariance models, as well as to develop and study penalized M-estimates of the covariance matrix. Addressing robustness issues for structured covariance models and for penalization methods are fundamental problems which is more mathematically and computationally challenging than in the classical setting or in the unrestricted robust estimation setting. Here, some recent work on geodesic convexity within the signal processing community is expected to play an important role in addressing these problems.