Nonparametric statistical methods based on graph theory

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High-dimensional data, or data spanning multiple dimensions such as time and space, pose unique challenges for data processing in an era of ever-increasing amounts of data availability. Graph theory can provide a structure of high-dimensional data. Currently, two kinds of graph structures have been studied in nonparametric statistics: minimum spanning tree and shortest Hamiltonian path, where the sum of edge distances attains the minimum among all of the trees or paths. My long-term objective is to explore a new class of nonparametric statistical methods based on graph theory. Over the next five years, I intend to bring graph theory into a fully developed and growing field of nonparametric statistics. With my team of HQP and collaborators, I will explore graph theory for change point detection (the change point refers to a location or time at which observations or data obey two different models: before and after), multiple-sample test, and clustering. My specific short-term objectives are focused on three threads. The first research thread will focus on three types of advances in change point detection: 1) proposing a weighted cusum statistic for univariate data, 2) using approximate shortest Hamiltonian path for high-dimensional data, and 3) developing resonance technology for multiple change points. These new methods will be applied to a library of images that can be considered as high-dimensional data. For example, medical images for functional detection of blood oxygen levels, webcams for motion detection, and Earthorbiting satellites for smoke/fire detection. Second, we will compare multiple independent samples using approximate shortest Hamiltonian path. Here, we will study how to efficiently adjust the experimental alignment or choose partial appropriate results for dependent-sample comparison. These techniques are well suited to address challenges in gene data, where the dimensionality of each observation is in the thousands, but there are only tens or hundreds of instances available for study. For the third area, we introduce a hierarchical clustering without predetermining the number of clusters, where the top cluster is the approximate shortest Hamiltonian path. We will propose an optimal clustering and show it is asymptotically identical to the correct clustering. These techniques will be applied for understanding how the COVID-19 outbreak spreads worldwide. A diverse group of HQP, including at least 1 PDF, 2 PhD, 1 MSc, and 4 UG will be trained as part of this proposal. Findings will have direct application for government agencies charged with disease control, wildfire detection, and medical abnormality detection. The team's research and training activities will thus help Canada make efficient and evidence-based decision in the face of unforeseen situations. For theoretical contributions, this proposal explains why and how graph theory works for high-dimensional data with applications to the three threads.