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Mathematics

Some math topics with relation to ML

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

Bivariate Normal Distribution

Bivariate normal distribution – derivation by linear transformation of a random vector of two independent Gaussians

In an another post on properties of a Bivariate Normal Distribution [BVD] I have motivated the form of its probability density function [pdf] by symmetry arguments and the underlying probability density functions of its marginals, namely 1-dimensional Gaussians. In this post we will derive the probability density function by following the line of argumentation for a general Multivariate Normal Distribution… Read More »Bivariate normal distribution – derivation by linear transformation of a random vector of two independent Gaussians

Multivariate Normal Distributions – IV – Spectral decomposition of the covariance matrix and rotation of the coordinate system

In the preceding posts of this series we have considered a comprehensible definition and basic properties of a non-degenerate “Multivariate Normal Distribution” of vectors in the ℝn [N-MND]. In this post we will make a step in the direction of a numerical analysis of some given finite vector distribution with properties that indicate an underlying N-MND. We want to find… Read More »Multivariate Normal Distributions – IV – Spectral decomposition of the covariance matrix and rotation of the coordinate system