Statistical Analysis for Models involving Riemannian Manifolds

这个应用程序的动机是实际生活中的问题，其建模和分析涉及非欧几里德黎曼流形。特别是，它是由分析扩散张量成像（DTI）数据和纵向数据分析所产生的问题所驱动的，两者在神经科学等各个科学领域都具有重要意义。尽管在涉及流形的统计模型方面已经有了相当多的理论发展，以及针对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，当数据在歧管上时，迫切需要开发新的和复杂的统计技术。还可以将可用的计算工具扩展到复杂的统计模型。在这个应用程序中，研究人员在涉及流形的模型中以统一的方式解决计算，估计以及统计推断的问题。特别是，研究者1)开发了一个框架，当数据位于一个特殊的流形上的非参数平滑；并研究了所得估计量的渐近性质；2)开发参数估计、非参数平滑和模型选择的计算算法和软件；3)建立了参数空间为特殊流形时统计推理的一般框架；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。总之，这项工作提高了对几何在统计建模和分析广泛问题中的作用的理解。它还有助于上述各个领域的计算工具的发展。在这一应用中，研究人员在认知神经科学等领域开发了定量工具，这些领域的问题具有特定的几何结构。他们的目标是通过显式地利用这些结构来提取数据的重要特征。这些研究者在统一的统计建模框架下解决这些问题的计算、估计和预测问题。这一应用的部分动机是通过扩散张量成像（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复杂的脑部疾病，如阿尔茨海默病和自闭症，意义重大。此外，这个应用程序导致了开源软件和定量技术的发展，这些技术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，对生物学、健康和医学领域许多问题的理解也可能受益于这一应用。通过统计学家和神经科学家之间的跨学科研究，该应用程序还具有更广泛的教育影响。

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.