论文标题
双变量正常分布的径向和角度边缘化的工具箱
A Toolbox for the Radial and Angular Marginalization of Bivariate Normal Distributions
论文作者
论文摘要
双变量正常分布通常用于描述一对随机变量的关节概率密度。这些分布来自许多领域，从电信，气象，弹道和计算神经科学。在这些应用中，它通常可用于径向和角度边缘化（即在极性转化下）相对于坐标系的原点而言。这种边缘化对于零均值的各向同性分布来说是微不足道的，但对于具有非二进制的非双基因协方差矩阵的最普遍的情况，这是不平凡的。在整个领域，已经得出了一系列具有不同程度的通用性的解决方案。在这里，我们为双变量正常分布的极性边缘化提供了一个简明的分析解决方案摘要。该报告伴随MATLAB（Mathworks，Inc。）和R工具箱，该报告为本文所述的边缘化提供了封闭形式和数字实现。
Bivariate normal distributions are often used to describe the joint probability density of a pair of random variables. These distributions arise across many domains, from telecommunications, to meteorology, ballistics, and computational neuroscience. In these applications, it is often useful to radially and angularly marginalize (i.e.,~under a polar transformation) the joint probability distribution relative to the coordinate system's origin. This marginalization is trivial for a zero-mean, isotropic distribution, but is non-trivial for the most general case of a non-zero-mean, anisotropic distribution with a non-diagonal covariance matrix. Across domains, a range of solutions with varying degrees of generality have been derived. Here, we provide a concise summary of analytic solutions for the polar marginalization of bivariate normal distributions. This report accompanies a Matlab (Mathworks, Inc.) and R toolbox that provides closed-form and numeric implementations for the marginalizations described herein.
