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Multivariate Normal Distributions

Aspects of Multivariate Normal Distributions and their relation to ML

Iterative method to compute the covariance-matrix of normal MVN-like inner cores of multivariate distributions with strongly asymmetric outer layers – I

In other posts in this blog (see [1] to [3]), I have discussed multiple methods to calculate and construct confidence ellipses of “Bivariate Normal Distributions” [BVNs]. BVNs are the marginal distributions of “Multivariate Normal Distributions” [MVNs] in e.g. n dimensions ( n > 2). Therefore, two-dimensional confidence ellipses appear as projections of n-dimensional concentric confidence ellipsoids of MVNs onto (2-dim) coordinate planes. The properties of the confidence ellipsoids, which also give us contours of the probability density, are defined by the variance-covariance matrix Σ of the MVN. This post discusses a method to compute the confidence ellipsoids and ellipses for an inner MVN-like core of an otherwise largely asymmetric distribution, which in its overall shape and structure deviates strongly from a MVN.

Read More »Iterative method to compute the covariance-matrix of normal MVN-like inner cores of multivariate distributions with strongly asymmetric outer layers – I

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

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