Challenges and Models for the Analysis of Functional Data in Sports

随着体育技术的新发展，数据被连续地记录在越来越精细的网格上。例如，详细的球员跟踪数据以高达30 Hz的速率捕获每次比赛中所有球员和球的二维坐标。功能数据分析（FDA）是分析此类数据的一个越来越有用的工具。更一般地说，FDA处理功能、表面或任何多维功能的分析和理论。我的研究项目主题是开发新的统计模型和方法来分析运动中的时空数据和大规模复杂的功能数据。我的研究计划的第一个长期目标是开发新颖实用的FDA方法来分析运动中的时空数据。我们将研究时空跟踪数据和事件数据。事件数据是定义良好的比赛事件（如传球、射门和犯规）的时间顺序记录，并使用时间戳进行记录。这项研究计划将提供一系列新的发展。例如，分析体育运动中时空数据的传统方法是将比赛区域离散为细分，通过计算每个区域的事件数量来构建强度矩阵，并对强度矩阵进行矩阵分解。在我们的研究中，我们假设一个连续和光滑的强度函数，我们的目标是开发新的FDA方法来估计强度函数，同时实现低秩结构。我们希望从低阶结构中发现空间表征，并提供诸如球员进攻和防守角色等体育见解。我们还将提出涉及空间功能对象的回归模型，并开发用于降维的正则化方法。现代体育中大规模和复杂的功能数据引入了计算和统计建模的挑战。我研究计划的第二个长期目标是提出一系列可解释的FDA模型，以适应体育运动中大规模和复杂的功能数据。例如，我们将开发基于子抽样的函数回归模型。另一个例子是，我们将提出一种多分辨率方法来聚类大规模空间功能数据。该研究主题有望建立分析运动中各种大规模和复杂功能数据的框架，同时开发更灵活且可解释的FDA模型，以深入了解不同类型的运动。我们与加拿大国家和省级体育组织有联系，我们希望拟议的研究项目有助于为加拿大运动队和个人运动员提供竞争优势。提出的目标将引起体育分析领域的研究人员的兴趣。同时，提出的FDA模型在自然科学和其他科学领域也有许多应用。

With new developments in sports technology, data are recorded continuously on a large-scale over finer and finer grids. For example, detailed player tracking data captures the two-dimensional coordinates of all players and the ball during each play at a rate up to 30 Hz. Functional data analysis (FDA) is an increasingly useful tool for analyzing such data. More generally, FDA deals with the analysis and theory of functions, surfaces, or any multidimensional functions. The theme of my research program is the development of new statistical models and methods to analyze spatio-temporal data and large-scale and complex functional data in sports. The first long-term objective of my research program is to develop novel and practical FDA methods to analyze spatio-temporal data in sports. We will study spatio-temporal tracking data and event data. Event data are chronological records of well-defined match events, such as passes, shots, and fouls, and are recorded with timestamps. The research program will provide a series of novel developments. For example, a conventional method for analyzing spatio-temporal data in sports is to discretize the playing area into subdivisions, construct an intensity matrix by counting the number of events in each region and apply matrix factorization on the intensity matrix. In our research, we assume a continuous and smooth intensity function and we aim to develop novel FDA methods for estimating the intensity function and simultaneously achieve a low-rank structure. We expect to discover a spatial representation from the low-rank structure and provide sports insights such as the offensive and defensive roles of players. We will also propose regression models that involve spatial functional objects and develop regularization methods for dimension reduction. The large-scale and complex functional data in modern sports introduce both computational and statistical modelling challenges. The second long-term objective of my research program is to propose a collection of interpretable FDA models to accommodate the large-scale and complex functional data in sports. For instance, we will develop subsampling-based functional regression models. Another example is that we will propose a multiresolution approach to cluster large-scale spatial functional data. The research theme is expected to establish frameworks for analyzing a variety of large-scale and complex functional data in sports and simultaneously develop more flexible yet interpretable FDA models to gain insights into different types of sports. We have connections to national and provincial Canadian sports organizations and we expect the proposed research program is useful in providing a competitive advantage to Canadian sports teams and individual players. The proposed objectives will be of interest to researchers in the field of sports analytics. At the same time, the proposed FDA models also have many applications in natural sciences and other scientific areas.