On the Euclidean Distance between Two Gaussian Points and the Normal Covariogram of $$\boldsymbol{\mathbb{R}}^{\boldsymbol{d}}$$
Pub Date : 2024-04-09
DOI:10.3103/s1068362324010059
引用次数: 0
Abstract
The concept of covariogram is extended from bounded convex bodies in \(\mathbb{R}^{d}\) to the entire space \(\mathbb{R}^{d}\) by obtaining integral representations for the distribution and probability density functions of the Euclidean distance between two \(d\)-dimensional Gaussian points that have correlated coordinates governed by a covariance matrix. When \(d=2\), a closed-form expression for the density function is obtained. Precise bounds for the moments of the considered distance are found in terms of the extreme eigenvalues of the covariance matrix.
论两个高斯点与 $$\boldsymbol\{mathbb{R}}^{boldsymbol{d}}$ 的正态协方差图之间的欧氏距离
摘要 协方差的概念从\(\mathbb{R}^{d}\)中的有界凸体扩展到了\(\mathbb{R}^{d}\)的整个空间，得到了两个\(d\)维高斯点之间欧氏距离的分布和概率密度函数的积分表示，这两个高斯点具有由协方差矩阵支配的相关坐标。当 \(d=2\) 时，可以得到密度函数的闭式表达式。根据协方差矩阵的极值特征值，可以找到所考虑的距离矩的精确边界。
本文章由计算机程序翻译，如有差异，请以英文原文为准。
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