Statistical Analysis of Manifold-Valued Data

摘要广义上解释的回归方法使用户能够测量感兴趣的响应变量对一组协变量的依赖性，即预期会影响响应变量的可测量变量。这种方法的功效在于，给定协变量值，回归模型可用于预测响应变量值的可能范围，并评估哪些协变量是响应行为的主要驱动因素。该项目关注具有复杂非线性结构的响应变量类型（在数学术语中，响应是流形值）。对于这类数据，不存在回归建模的一般框架。我们希望考虑的响应变量类型的一个例子是物体的形状;形状是一个高度非线性的实体。我们将开发的回归方法有许多潜在的应用，其中许多（但不是全部）是在生物学和医学中。例如，在可预见的未来，我们希望我们的项目的输出，以帮助外科医生在以下情况下作出决定。假设一个病人有一个肿瘤，外科医生希望决定哪种类型的手术（如果有的话）是最好的。合适的回归模型将使得能够在每种类型的手术下预测手术后肿瘤的生长轨迹。相关协变量信息将包括诸如手术前肿瘤的大小和形状、肿瘤的位置、患者的年龄和性别等变量。然后外科医生将能够评估哪种轨迹，从而评估哪种类型的手术对患者最有利。第二个应用，这次是神经科学，与扩散张量成像有关。该项目的产出之一将是在空间环境中对多值数据进行内插的方法。在脑扩散张量成像的背景下，扩散张量数据的空间插值将提供更准确的脑图，这将给出对白色物质纤维结构的改进的和更合理的解释，以帮助理解脑功能。第三个应用是在法医学中。我们开发的模型将允许预测面部形状的发展，这取决于协变量信息，例如父母的面部形状，以及性别和年龄等其他信息。例如，这种方法在儿童绑架案件中将是有用的。虽然目前确实存在推断脸部形状的方法，但它们没有将协变量信息纳入模型中。还有许多其他研究领域自然产生流形值响应数据，并且我们预计项目输出将产生重大影响，包括植物生物学（最终，相关性，的实际问题，突出的一般问题，在回归建模的多方面-有价值的数据都来自我们与其他领域合作者的合作。因此，成功实施本提案中的新颖和令人兴奋的想法将提供一个框架，不仅可以解决激发本提案的问题，而且还对许多科学学科的研究产生重大影响，此外还对统计学，计算机科学，数学和相关领域的研究人员具有方法和理论兴趣。拟议的研究还将大大增加英国现有的专业知识库，并保持其在形状统计分析方面的国际领先地位，更普遍的是，对象数据。

SummaryRegression methods, interpreted broadly, enable the user to measure dependence of a response variable of interest on a set of covariates, i.e. measurable variables that are expected to affect the response variable. The power of this approach is due to the fact that, given the covariate values, the regression model can be used to predict a likely range of values of the response variable, and to assess which covariates are the main drivers in the behaviour of the response. This project is concerned with types of response variable which have complicated nonlinear structure (in mathematical terminology, the response is manifold-valued). For such data, no general framework for regression modelling exists. An example of the type of response variable that we wish to consider is the shape of an object; shape is a highly nonlinear entity. There are numerous potential applications of the regression methodology that we will develop, many (but not all) of which are in biology and medicine. For example, within the forseeable future we expect the outputs of our project to assist surgeons in making decisions in the following situation. Suppose a patient has a tumour and the surgeon wishes to decide which type of operation (if any) would be best. A suitable regression model would enable prediction, under each type of operation, of the growth trajectory of the tumour after the operation. Relevant covariate information would include variables such as size-and-shape of the tumour before the operation, location of the tumour, age and gender of the patient. The surgeon would then be able to assess which trajectory, and therefore which type of operation, would be most favourable for the patient.A second application, this time for neuroscience, relates to diffusion tensor imaging. One output of the project will be methodology for interpolating manifold-valued data in a spatial setting. In the context of diffusion tensor imaging of the brain, spatial interpolation of the diffusion tensor data will provide more accurate maps of the brain which will give improved and more soundly-based interpretations of the white matter fibre structure to help understand brain function.A third application is in forensic science. The models we develop will allow prediction of the development of the shape of a face, depending on covariate information, such as the shapes of the parents' faces, and other information such as gender and age. This methodology will be useful in child abduction cases for example. While it is certainly the case that methods for extrapolating face shape currently exist, they do not incorporate covariate information in the model.There are many other research areas in which manifold-valued response data arise naturally and where we expect the project outputs to have a major impact, including plant biology (of relevance, ultimately, to food security) and protein modelling.The practical problems which highlight generic issues in regression modelling for manifold-valued data have all arisen from our work with collaborators in other fields. Therefore the successful implementation of the novel and exciting ideas in this proposal will provide a framework for addressing not only the problems that motivated this proposal, but also have a major impact on research in many scientific disciplines, in addition to being of methodological and theoretical interest to researchers in statistics, computer science, mathematics and related fields. The proposed research will also add in a substantial way to the available pool of UK expertise and to maintain its position as internationally-leading in the statistical analysis of shape and, more generally, object data.