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颖的统计方法来分析许多科学和工程领域中出现的复杂数据。复杂性的性质因研究项目而异。复杂性可能指分布假设、数据集的高维和大小以及实验单元之间的依赖结构。所提出的方法是针对非传统形式的数据集定制的。非常需要适当的统计方法来分析这些类型的数据集。因此，预计计划的贡献将对统计（特别是统计）以及整个科学、工程和社会产生重大影响。 一些工具涉及使用称为数据深度的概念对多元数据进行排名和排序。当每个实验单元进行了多次不同的测量时，我将假设观察不完整，并使用基于深度的方法来比较两个或多个实验条件。我还介绍了变化点检测的方法。一个重要的方面是我对数据生成机制做出了最少的假设。 降维是处理高维数据时克服所谓的维度灾难的一种方法。尽管文献中已经介绍了许多降维方法，但缺乏评估这些方法性能的通用框架。在本提案中，我的目标是在存在异常值和模型错误指定的情况下开发降维算法的性能和鲁棒性度量。 网络是一类非传统数据集，近年来受到了多个科学界的广泛关注。这些大规模的复杂网络出现在许多科学技术领域，例如社交网络、社交媒体、万维网、疾病流行和生物网络。大多数研究都致力于静态网络的统计建模，它要么代表现象的单个时间快照，要么代表一段时间内的聚合。我打算开发研究动态网络的方法。 对于分析大脑中多个神经元同时记录的尖峰序列数据的统计方法有很大的需求。在这个提案中，我将开发新颖的技术来满足这一需求。 在许多实验中，实际上在大多数纵向研究中，函数回归中涉及的平滑随机过程的函数轨迹是无法直接观察到的。此外，观测到的数据是这些轨迹的噪声、稀疏且间隔不规则的测量结果。在这个提案中，我的重点是这个框架中的功能回归。 总之，拟议研究计划所取得的进展将对统计建模和推理产生重大影响，并通过其应用推动科学技术的发展。

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.