Nonparametric Methods for Analysis of Complex and High Dimensional Data

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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.