Statistical Analysis for Models involving Riemannian Manifolds

这一应用是由现实生活中的问题推动的，这些问题的建模和分析涉及非欧几里德黎曼流形。特别是，它的动机是分析扩散张量成像(DTI)数据和纵向数据分析所产生的问题，这两个数据都在神经科学等各个科学领域具有重要意义。尽管涉及流形的统计模型已经有了相当多的理论发展，以及特殊流形的计算工具的发展，但在理论和计算方面仍有许多空白需要弥合。例如，当数据位于多种情况下时，迫切需要开发新的和复杂的统计技术。还可以将现有的计算工具扩展到复杂的统计模型。在这一应用中，研究人员以统一的方式解决了涉及流形的模型中的计算、估计以及统计推断问题。具体地说，研究人员1)建立了当数据位于特殊流形上时的非参数平滑框架；并研究了由此得到的估计量的渐近性质；2)开发了用于参数估计、非参数平滑以及模型选择的计算算法和软件；3)当参数空间是特殊流形时，开发了统计推断的一般框架；4)将分析和计算工具应用于包括(但不限于)认知神经科学、纵向研究和心理测量学等领域的问题。总而言之，这项工作加强了对几何学在统计建模和分析广泛问题中的作用的理解。在这一应用中，研究人员在认知神经科学等领域开发量化工具，这些领域的问题具有特定的几何结构。他们的目标是通过明确利用这些结构来提取数据的重要特征。这些研究人员在统一的统计建模框架下处理此类问题的计算、估计和预测问题。这一应用的部分动机是研究通过扩散张量成像(DTI)等技术生成的图像来发现脑组织的结构和功能。与大脑认知功能相关的基本问题成为一个丰富而又极其重要的研究领域。这一领域正在进行的研究对于理解和治疗阿尔茨海默病和自闭症等复杂的大脑疾病具有巨大的意义。此外，这一应用导致了开源软件和量化技术的发展，可以扩展到更广泛的复杂科学问题，包括纵向研究和心理测量学。因此，对生物、健康和医学领域的许多问题的理解也可能从这一应用中受益。这项应用还通过统计学家和神经学家之间的跨学科研究产生了更广泛的教育影响。

This application is motivated by real life problems whose modeling and analysis involve non-Euclidean Riemannian manifolds. In particular, it is motivated by problems arising from analyzing diffusion tensor imaging (DTI) data and longitudinal data analysis, with both having important implications in various scientific fields such as neuroscience. Although, there has been a considerable amount of theoretical developments for statistical models involving manifolds, as well as developments of computational tools for special manifolds, there are still many gaps that need to be bridged both in terms of theory and computation. For instance, there is a pressing need to develop new and sophisticated statistical techniques when data lie on a manifold. There is also a scope to extend available computational tools to complex statistical models. In this application, the investigators address the issues of computation, estimation as well statistical inference in a unified manner in the models involving manifolds. In particular, the investigators 1) develop a framework for non-parametric smoothing when data lie on a special manifold; and study the asymptotic properties of the resulting estimators; 2) develop computational algorithms and softwares for parameter estimation, non-parametric smoothing as well as model selection; 3) develop a general framework for statistical inference when the parameter space is a special manifold; 4) apply the analytical and computational tools to problems in areas including (but not limited to), cognitive neuroscience, longitudinal studies and psychometry. Together, this work enhances understanding of the role of geometry in statistical modeling and analysis for a broad range of problems. It also contributes to the development of computational tools for various fields as mentioned above.In this application, the investigators develop quantitative tools in areas such as cognitive neuroscience, where the problems have specific geometric structures. They aim to extract important features of the data by explicitly utilizing these structures. These investigators address the issues of computation, estimation and prediction for such problems under a unified statistical modeling framework. This application is partly motivated by studies in discovering the structure and functionality of brain tissues through images generated by technologies such as diffusion tensor imaging (DTI). The fundamental problems associated with the cognitive function of brains become a rich and extremely important field of study. The implication of ongoing research in this field is tremendous in terms of understanding and treating complex brain disorders such as Alzheimer's disease and autism. Moreover, this application results in developments of open source softwares and quantitative techniques that can be extended to a broader range of complex scientific problems, including longitudinal studies and psychometry. Hence, the understanding of many problems in the field of biology, health and medicine is also likely to be benefited from this application. This application also has a broader educational impact through interdisciplinary research between statisticians and neuroscientists.