Robust and nonparametric methods for complex data objects

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The research program proposed in this application is concerned with developing novel statistical and data science methods for the analysis of complex data arising in many areas of science, medicine, engineering, economics, and society. In most data analysis methods and all statistical modelling there are assumptions that are either implicitly or explicitly made. However, in practice and for real world datasets these assumptions are very unlikely to hold exactly. This is mainly because real world data are subject to error from many potentially unknown sources. So it is crucial to know how a method holds up if the underlying assumptions are violated, to what degree the method is resistant against these violations and how to develop methods that are resistant to these violations. This is the subject of robust and nonparametric statistics. When only one feature is measured from every subject under study (univariate data), the literature on robust and nonparametric methods is broad. On the other hand, when multiple features are measured (multivariate data), or one or more features are measured from every subject over time or space (functional data), the literature on robust and nonparametric methods is not well developed. This is even more crucial in the case of non-traditional datasets such as network and matrix valued data, where the literature on robust and nonparametric methods is almost of non-existant. The importance of robust and nonparametric methods is even more evident by noting that the quality verification of high-dimensional, functional, network, and matrix-valued datasets is an extremely difficult task and can not be done by manual or visual inspection. Therefore, there is a great need to develop statistical methodologies that are resistant to violations of assumptions, presence of outliers, and various model misspecification. In developing such robust and nonparametric methods, tools such as data depth, ranking or sorting of observations and various other procedures from robust statistics can play important roles. The objective of the proposed research program is to advance the current state-of-the-art in robust and nonparametric statistics by pioneering new tools or extending the existing methodology to more complex data structures. The emphasis is on performing theoretical studies and developing efficient computational algorithms. Some motivating application areas are neural spike trains, social networks, industrial process monitoring, additive manufacturing such as 3D printing, anomaly detection, environmental studies amongst other fields. In summary, the advancements achieved under the proposed research program is anticipated to have a significant impact on statistical data analysis and inference, and subsequently to science, technology and society through their application.