High-dimensional Data Analysis: Modeling Unobserved Heterogeneity in Data, and Studying Imbalanced Classification Problems

由于当今世界不断扩大的数据收集手段，数据科学已成为众多科学学科关注的中心。在许多应用中，当前数据的规模和结构复杂性前所未有，因此需要从这些数据中提取有用信息的计算效率高和统计可靠的方法。为了实现这一目标，我的研究项目的总体主题是分析高维数据。更具体地说，在本提案的五年中，我的短期目标是：1)异质高维数据的统计建模：在健康科学、工程与环境、社会科学和金融计量经济学等应用中，高维数据通常来自由多个隐藏的同质亚种群组成的异质种群。有限混合回归（FMR）和马尔可夫状态切换自回归（MSAR）模型为捕获数据中未观察到的异质性提供了灵活的工具。后一种模型用于时间序列数据的建模。在实践中，当将这样的模型拟合到数据集时，人们面临三个推理问题：顺序选择或隐藏子种群或制度数量的估计，变量选择，以及所谓的选择后统计推断，如数据驱动的选择模型的参数的假设检验或置信区间。尽管它们的广泛应用，在高维统计的不断增长的文献中，解决上述问题的严格的方法发展非常有限。在我的短期目标中，我将研究新的基于似然的正则化技术，用于：FMR和MSAR中的顺序选择，以及具有固定顺序和高维设置的稀疏动态FMR和矢量MSAR中的变量选择。建立这样的结果将为选择后推理问题铺平道路，这是我长期目标的主题。II)高维不平衡分类问题：在欺诈检测、医疗诊断、设备故障检测等应用中，分类任务在训练数据中某些类的观测频率往往存在高维性和不平衡性。后者要么是由于数据收集过程，要么是因为某些类别在种群中确实很罕见。由于少数类别的数据稀缺性，传统的判别方法往往偏向于多数类别，导致少数类别的错误分类率更高。不平衡分类问题通常很难，所以我从研究不平衡的线性二元情况开始。我将研究分而治之技术与硬阈值变量选择方法在高维少数类的标准线性判别分析中用于偏差校正的效用。我也会研究多类问题。

Data science has become the center of attention in a wide range of scientific disciplines, thanks to ever-expanding means of data collection in today's world. Unprecedented size and structural complexity of current data in many applications call for computationally efficient and statistically sound methodologies for extracting useful information from such data. Toward this goal, the general theme of my research program focuses on analyzing high-dimensional data. More specifically, over the five years of this proposal, my short-term objectives are: I) Statistical modeling of heterogeneous high-dimensional data: In applications such as health sciences, engineering and environment, social sciences, and financial econometrics, high-dimensional data often arise from heterogeneous populations consisting of multiple hidden homogeneous sub-populations. Finite mixture of regressions (FMR) and Markov regime-switching autoregressive (MSAR) models provide flexible tools for capturing unobserved heterogeneity in data. The later models are used for modeling time series data. In practice, when fitting such models to a dataset, one faces three inferential problems: order selection or estimation of the number of hidden sub-populations or regimes, variable selection, and so-called post-selection statistical inference such as hypothesis testing or confidence intervals for parameters of a data-driven selected model. Despite their wide applications, rigorous methodological developments addressing the aforementioned problems in the growing literature on high-dimensional statistics have been very limited. In my short-term objectives, I will investigate new likelihood-based regularization techniques for: order selection in FMR and MSAR, and variable selection in sparse dynamic FMR and vector MSAR with fixed order and in high-dimensional settings. Establishment of such results will pave the way toward post-selection inference problems which are the subjects of my long-term objectives. II) High-dimensional imbalanced classification problems: In applications such as fraud detection, medical diagnosis, or equipment malfunction detection, classification tasks often suffer from both high-dimensionality and imbalance in the observed frequency of some classes in the training data. The latter is due to either data collection process or because some classes are indeed rare in the population. Due to data scarcity in minority class(es), conventional discriminative methods are often biased toward the majority class(es) resulting in much higher misclassification rates for the minority class(es). Imbalanced classification problems are generally hard, so I begin by studying imbalanced linear binary cases. I will investigate the utility of divide-and-conquer techniques coupled with hard-thresholding variable selection methods for bias correction in the standard linear discriminant analysis toward the minority class in high-dimensions. I will also study multi-class problems.