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Mathematics

Some math topics with relation to ML

Concentric surfaces of ellipsoids

Multivariate Normal Distributions – III – Variance-Covariance Matrix and a distance measure for vectors of non-degenerate distributions

In previous posts of this series I have motivated the functional form of the probability density of a so called “non-degenerate Multivariate Normal Distribution“. In this post we will have a closer look at the matrix Σ that controls the probability density function [pdf] of such a distribution. We will show that it actually is the covariance matrix of the… Read More »Multivariate Normal Distributions – III – Variance-Covariance Matrix and a distance measure for vectors of non-degenerate distributions

Linear transformed 3-dim Z-distribution

Multivariate Normal Distributions – II – Linear transformation of a random vector with independent standardized normal components

In Machine Learning we typically deal with huge, but finite vector distributions defined in the ℝn. At least in certain regions of the ℝn these distributions may approximate an underlying continuous distribution. In the first post of this series we worked with a special type of continuous vector distribution based on independent 1-dimensional standardized normal distributions for the vector components.… Read More »Multivariate Normal Distributions – II – Linear transformation of a random vector with independent standardized normal components

contour ellipsoids of a projected MND

Multivariate Normal Distributions – I – Basics and a random vector of independent Gaussians

This post series is about mathematical aspects of so called “Multivariate Normal Distributions“. In the literature two abbreviations are common: MNDs or MVNs. I will use both synonymously. To get an easy access I want to introduce MNDs as the result of a linear transformations applied to random vectors whose components can be described by independent 1-dimensional normal distributions. Afterward… Read More »Multivariate Normal Distributions – I – Basics and a random vector of independent Gaussians

Bivariate Normal Distribution from face data encoded by a CAE

Bivariate Normal Distribution – derivation of the covariance and correlation by integration of the probability density

In a previous post of this blog we have derived the functional form of a bivariate normal distribution [BND] of a two 1-dimensional random variables X and Y). By rewriting the probability density function [pdf] in terms of vectors (x, y)T and a matrix Σ-1 we recognized that a coefficient appearing in a central exponential of the pdf could be… Read More »Bivariate Normal Distribution – derivation of the covariance and correlation by integration of the probability density