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Bivariate Normal Distributions

Contour ellipses from Cholesky decomp ot various covariance matrices

Bivariate Normal Distributions – parameterization of contour ellipses in terms of the Mahalanobis distance and an angle

In my last post about Bivariate Normal Distributions [BVD] I have discussed why contour lines of a BVD’s probability density function [pdf] are concentric ellipses. These contour ellipses are defined by constant values of the so called Mahalanobis distance. In addition, I have discussed a method to create these ellipses from values of the coefficients of the BVD’s (variance-) covariance… Read More »Bivariate Normal Distributions – parameterization of contour ellipses in terms of the Mahalanobis distance and an angle

BVD contour ellipses

Bivariate Normal Distribution – Mahalanobis distance and contour ellipses

I continue with my posts on Bivariate Normal Distributions [BVDs]. In this post we consider the exponent of a BVD’s probability density function [pdf]. This function is governed by a central matrix Σ-1, the inverse of the variance-covariance matrix of the BVD’s random vector. We define the so called Mahalanobis distance dm for BVD vectors. A constant value of the… Read More »Bivariate Normal Distribution – Mahalanobis distance and contour ellipses

confidence ellipses of bivariate normal distribution

Eigenvalues and eigenvector of a positive-definite, real valued and symmetric matrix

A bivariate normal distributions [BVD] is governed by a central positive symmetric matrix. This matrix is a covariance matrix which describes the variances and correlation of the BVD’s marginal distributions. The contour lines of the probabilty density function of a BVD are ellipses. The half axes and the orientation of these ellipses are controlled by the eigenvalues and eigenvectors of… Read More »Eigenvalues and eigenvector of a positive-definite, real valued and symmetric matrix

Contours of a bivariate normal distribution

Bivariate normal distribution – explicit reconstruction of a BVD random vector via Cholesky decomposition of the covariance matrix

In other posts of this blog I have discussed the general form of a Bivariate Normal Distribution [BVD] . For a centered Cartesian coordinate system [CCS] (see below), we have already seen the following: In this post I will give you a recipe to explicitly construct two random variables X, Y of a BVD from 1-dimensional Gaussians Z1, Z2 with… Read More »Bivariate normal distribution – explicit reconstruction of a BVD random vector via Cholesky decomposition of the covariance matrix