Nonparametric statistical methods based on graph theory

在数据可用性不断增加的时代，高维数据或跨越时间和空间等多个维度的数据对数据处理构成了独特的挑战。图论可以提供高维数据的结构。目前，在非参数统计中已经研究了两种图结构：最小生成树和最短哈密尔顿路，其中边距离之和在所有树或路中达到最小。我的长期目标是探索一类基于图论的新的非参数统计方法。在接下来的五年里，我打算将图论带入一个充分发展和不断增长的非参数统计领域。与我的HQP团队和合作者一起，我将探索用于变化点检测(变化点是指观察或数据遵循两种不同模型的位置或时间：之前和之后)、多样本测试和集群的图论。我的具体短期目标集中在三条主线上。第一个研究线索将集中在三种类型的变点检测方面的进展：1)提出一种针对单变量数据的加权和统计量，2)针对高维数据使用近似最短哈密顿路径，以及3)发展针对多个变点的共振技术。这些新方法将被应用于可被视为高维数据的图像库。例如，用于功能检测血氧水平的医学图像，用于运动检测的网络摄像头，以及用于烟雾/火灾检测的地球探测卫星。其次，我们将使用近似最短哈密顿路径来比较多个独立样本。在这里，我们将研究如何有效地调整实验比对或选择部分合适的结果进行相依样本比较。这些技术非常适合于解决基因数据中的挑战，在基因数据中，每一次观测的维度都在数千维，但可供研究的实例只有数十或数百个。对于第三个区域，我们引入了一种不预先确定聚类数目的层次聚类，其中顶层聚类是近似的最短哈密顿路径。我们将提出一个最优聚类，并证明它与正确的聚类渐近相同。这些技术将被用于了解新冠肺炎疫情是如何在全球传播的。作为该计划的一部分，包括至少1名PDF、2名博士、1名硕士和4名UG在内的不同HQP群体将接受培训。研究结果将直接应用于负责疾病控制、野火检测和医疗异常检测的政府机构。因此，该小组的研究和培训活动将帮助加拿大在面对不可预见的情况时作出有效和有证据的决定。对于理论贡献，该建议解释了图论为什么以及如何通过应用到三个线程来处理高维数据。

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.