Geostatistical modeling for continuous point-referenced data has extensively been applied to neuroimaging because it produces efficient and valid statistical inference. However, diffusion tensor imaging (DTI), a neuroimaging technique characterizing the brain's anatomical structure, produces a positive-definite (p.d.) matrix for each voxel. Currently, only a few geostatistical models for p.d. matrices have been proposed because introducing spatial dependence among p.d. 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 random variable 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 way of modeling spatial dependence. Motivated by a DTI data set 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 an approximation method to obtain a feasible Cholesky decomposition model, which we show to be asymptotically equivalent to the spatial Wishart process model. A local likelihood approximation method is also applied to achieve fast computation. The simulation studies and real data application demonstrate that the Cholesky decomposition process model produces reliable inference and improved performance, compared to other methods.

中文翻译：

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正定矩阵的地统计建模：在扩散张量成像中的应用

连续点参考数据的地统计建模已广泛应用于神经影像学，因为它可以产生高效且有效的统计推断。然而，扩散张量成像 (DTI) 是一种表征大脑解剖结构的神经成像技术，可为每个体素生成正定 (pd) 矩阵。目前，仅提出了一些 pd 矩阵的地质统计模型，因为正确引入 pd 矩阵之间的空间依赖性具有挑战性。在本文中，我们使用空间 Wishart 过程，这是一种空间随机过程（随机场），其中每个 pd 矩阵变量随机变量都略微遵循 Wishart 分布，并且随机矩阵之间的空间依赖性是由潜在高斯过程引起的。这个过程对于不可数的空间位置集合是有效的，并且几乎肯定是连续的，从而产生了一种合理的空间依赖性建模方法。受可卡因使用者 DTI 数据集的启发，我们提出了一种基于空间 Wishart 过程的空间矩阵变量回归模型。一个有问题的问题是空间 Wishart 过程没有封闭形式的密度函数。因此，我们提出了一种近似方法来获得可行的 Cholesky 分解模型，我们证明该模型渐近等效于空间 Wishart 过程模型。还应用局部似然近似方法来实现快速计算。仿真研究和实际数据应用表明，与其他方法相比，Cholesky 分解过程模型可产生可靠的推理并提高性能。