Geostatistical modeling for continuous point-referenced data has been extensively applied to neuroimaging because it produces efficient and valid statistical inference. However, diffusion tensor imaging (DTI), a neuroimaging characterizing the brain structure produces a positive definite (p.d.) matrix for each voxel. Current geostatistical modeling has not been extended to p.d. matrices because introducing spatial dependence among positive definite matrices properly is challenging. In this paper, we use the spatial Wishart process, a spatial stochastic process (random field) where each p.d. matrix-variate marginally follows a Wishart distribution, and spatial dependence between random matrices is induced by latent Gaussian processes. This process is valid on an uncountable collection of spatial locations and is almost surely continuous, leading to a reasonable means of modeling spatial dependence. Motivated by a DTI dataset of cocaine users, we propose a spatial matrix-variate regression model based on the spatial Wishart process. A problematic issue is that the spatial Wishart process has no closed-form density function. Hence, we propose approximation methods to obtain a feasible working model. A local likelihood approximation method is also applied to achieve fast computation. The simulation studies and real data analysis demonstrate that the working model produces reliable inference and improved performance compared to other methods.
Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices
LAMSADE, CNRS, PSL Univ. Paris-Dauphine, France（巴黎-萨克雷大学巴黎-达菲学院、国家科学研究中心、巴黎-达菲大学）；Inria center at the University of Bordeaux / LaBRI, France（波尔多大学/LaBRI与Inria中心）；TAU, LISN, University Paris-Saclay, France（巴黎-萨克雷大学、LISN、TAU）
Random matrix theory improved Fréchet mean of symmetric positive definite matrices
Université Paris Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；CentraleSupélec（中央理工-高等电力学院）；Université Savoie Mont Blanc（萨瓦勃朗峰大学）；Huawei Paris Research Center（华为巴黎研究中心）
Division of Biostatistics Department of Population Health Grossman School of Medicine New York University（生物统计学系 人口健康部 Grossman 医学院 新 York 大学）；Department of Biostatistics Columbia University（生物统计学系 Columbia 大学）；Division of Biostatistics Department of Population Health Weill Cornell Medicine Cornell University（生物统计学系 人口健康部 Weill Cornell 医学院 Cornell 大学）
