This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (SPD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. SPD matrices: the log-Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of SPD matrices. The limiting distributions of these averages are used to provide large-sample confidence regions and two-sample tests for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability.

中文翻译：

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对称正定矩阵的对数正态分布和几何平均值


本文给出了对称正定 (SPD) 矩阵集上的对数正态概率分布族的正式定义，将其视为单变量对数正态分布族的矩阵变量扩展。通过独立同分布 SPD 矩阵的两种几何平均数的中心极限定理，可以得到该分布的两种形式作为大样本极限分布：对数欧几里得平均数和规范几何平均数。这些平均值对应于 SPD 矩阵集上施加的两种不同几何形状。这些平均值的极限分布用于提供大样本置信区域和相应总体平均值的双样本检验。这些方法通过对扩散张量成像数据的体素分析进行​​说明，允许从采样变异性的角度对各种平均类型进行比较。