Shi Xiaoyan, Styner Martin, Lieberman Jeffrey, Ibrahim Joseph G, Lin Weili, Zhu Hongtu
Department of Biostatistics, Radiology, Psychiatry and Computer Science, University of North Carolina at Chapel Hill.
J Am Stat Assoc. 2009 Jan 1;5762:192-199. doi: 10.1007/978-3-642-04271-3_24.
In medical imaging analysis and computer vision, there is a growing interest in analyzing various manifold-valued data including 3D rotations, planar shapes, oriented or directed directions, the Grassmann manifold, deformation field, symmetric positive definite (SPD) matrices and medial shape representations (m-rep) of subcortical structures. Particularly, the scientific interests of most population studies focus on establishing the associations between a set of covariates (e.g., diagnostic status, age, and gender) and manifold-valued data for characterizing brain structure and shape differences, thus requiring a regression modeling framework for manifold-valued data. The aim of this paper is to develop an intrinsic regression model for the analysis of manifold-valued data as responses in a Riemannian manifold and their association with a set of covariates, such as age and gender, in Euclidean space. Because manifold-valued data do not form a vector space, directly applying classical multivariate regression may be inadequate in establishing the relationship between manifold-valued data and covariates of interest, such as age and gender, in real applications. Our intrinsic regression model, which is a semiparametric model, uses a link function to map from the Euclidean space of covariates to the Riemannian manifold of manifold data. We develop an estimation procedure to calculate an intrinsic least square estimator and establish its limiting distribution. We develop score statistics to test linear hypotheses on unknown parameters. We apply our methods to the detection of the difference in the morphological changes of the left and right hippocampi between schizophrenia patients and healthy controls using medial shape description.
在医学成像分析和计算机视觉中，分析各种流形值数据的兴趣日益浓厚，这些数据包括三维旋转、平面形状、有向方向、格拉斯曼流形、变形场、对称正定（SPD）矩阵以及皮质下结构的中间形状表示（m-rep）。特别地，大多数群体研究的科学兴趣集中在建立一组协变量（例如诊断状态、年龄和性别）与用于表征脑结构和形状差异的流形值数据之间的关联，因此需要一个用于流形值数据的回归建模框架。本文的目的是开发一种内在回归模型，用于分析作为黎曼流形中响应的流形值数据及其与欧几里得空间中一组协变量（如年龄和性别）的关联。由于流形值数据不构成向量空间，在实际应用中，直接应用经典多元回归可能不足以建立流形值数据与感兴趣的协变量（如年龄和性别）之间的关系。我们的内在回归模型是一种半参数模型，它使用一个链接函数从协变量的欧几里得空间映射到流形数据的黎曼流形。我们开发了一种估计程序来计算内在最小二乘估计量并建立其极限分布。我们开发了得分统计量来检验关于未知参数的线性假设。我们将我们的方法应用于使用中间形状描述来检测精神分裂症患者和健康对照之间左右海马形态变化的差异。
