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Multivariate Normal Distributions

Aspects of Multivariate Normal Distributions and their relation to ML

Cut-off BVN limited to an ellipsoidal core

Covariance matrix of a cut-off Multivariate Normal Distribution – I – intricate integrals with exponentials over the volumes and surfaces of n-dimensional ellipsoids?

In Machine Learning and statistics one sometimes has to work with a data sample whose underlying probability distribution approximates a Multivariate Normal Distribution [MVN] – but only within the ellipsoidal region of an innermost core with limited extensions. I.e., we have a MVN which either is abruptly cut off at some finite Mahalanobis distance D, or which quickly changes into… Read More »Covariance matrix of a cut-off Multivariate Normal Distribution – I – intricate integrals with exponentials over the volumes and surfaces of n-dimensional ellipsoids?

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