Nonparametric Methods for Analysis of Complex and High Dimensional Data

本申请中提出的研究计划涉及开发新的统计方法，用于分析科学和工程的许多领域中出现的复杂数据。复杂性的性质因研究项目的不同而不同。复杂性可能涉及分布假设、数据集的高维和大小以及实验单元之间的依赖结构。建议的方法是为非传统形式的数据集量身定做的。需要适当的统计方法来分析这些类型的数据集。因此，预计计划中的捐款将对统计、特别是对科学、工程和整个社会产生重大影响。 其中一些工具涉及使用称为数据深度的概念对多变量数据进行排名和排序。我将假设观察是不完整的，并使用基于深度的方法来比较两个或更多的实验条件，当每个实验单元进行了几次不同的测量时。本文还介绍了变点检测的方法。一个重要的方面是，我对数据生成机制做了最小的假设。 降维是在处理高维数据时克服所谓的维灾的一种方法。虽然文献中已经介绍了许多降维方法，但缺乏一个评价这些方法性能的通用框架。在这个方案中，我的目标是在存在离群点和模型错误指定的情况下，为降维算法开发性能和稳健性的度量。 网络是近年来受到多个科学界关注的一类非传统数据集。这些大规模的复杂网络出现在许多科学和技术领域，如社交网络、社交媒体、万维网、疾病流行病和生物网络。大多数研究都致力于静态网络的统计建模，静态网络要么代表现象的单一时间快照，要么代表一段时间的聚合。我打算开发研究动态网络的方法。 对于从大脑中的多个神经元同时记录的棘波序列数据进行分析的统计方法有很大的需求。在这项提议中，我将开发新的技术来回应这一需求。 在许多实验中，实际上大多数纵向研究中，函数回归中涉及的光滑随机过程的函数轨迹是不能直接观察到的。此外，观测到的数据是这些轨迹的噪声、稀疏和不规则间隔的测量。在这个提案中，我的重点是这个框架中的函数回归。 总而言之，在拟议研究方案下取得的进展将对统计建模和推断产生重大影响，并通过其应用促进科学技术的发展。

The research programs proposed in this application are concerned with developing novel statistical methods for the analysis of complex data arising in many areas of science and engineering. The nature of the complexity varies from one research program to another. The complexity may refer to distributional assumptions, high dimensionality and size of the dataset, and the dependence structure among experimental units. The proposed methodologies are tailored to datasets of non-traditional form. There is great demand for appropriate statistical methodology to analyze these types of datasets. It is therefore expected that the planned contributions will have a substantial impact to statistics, in particular, and science, engineering, and society, in general. Some of the tools involve ranking and sorting of multivariate data by using a concept called data depth. I will assume that observations are incomplete and use depth-based methodology for comparing two or more experimental conditions, when several different measurements have been made from each experimental unit. I also introduce methods for change point detection. An important aspect is that I make minimal assumptions regarding the data generating mechanism. Dimensionality reduction is a way of overcoming the so-called curse of dimensionally when dealing with high dimensional data. Although many dimensionality reduction methods have been introduced in the literature, a general framework for evaluating the performance of these methods is lacking. In this proposal, I aim to develop measures of performance and robustness for dimensionality reduction algorithms in the presence of outliers and model misspecification. Networks are a class of non-traditional datasets that have received a lot of attention from several scientific communities in recent years. These large-scale complex networks arise in many areas of science and technology, such as social networks, social media, the world wide web, disease epidemics, and biological networks. Most research has been devoted to statistical modelling of static networks, which either represent a single time snapshot of the phenomena or an aggregate over time. I intend to develop methods for studying dynamic networks. There is a great demand for statistical methods for analysis of spike train data recorded simultaneously from multiple neurons in the brain. In this proposal I will develop novel techniques in response to this demand. In many experiments, and in fact most longitudinal studies, the functional trajectories of the involved smooth random processes in functional regression are not directly observable. In addition, the observed data are noisy, sparse and irregularly spaced measurements of these trajectories. In this proposal my focus is on functional regression in this framework. In summary, the advancements achieved under the proposed research program will have significant impact in statistical modelling and inference, and advance science and technology through their application.