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multivariate normal distribution

n-dimensional spheres and ellipsoids – I – Gamma-function, volume and surface of n-spheres

This post series summarizes some properties of spheres and ellipsoids in a n-dimensional Euclidean space. In addition we are going to study some special integrals over the volume of so called n-spheres and their surfaces. As a preparatory step we look at some useful properties of the so called Gamma-function, which almost automatically appears when one works with integrals in… Read More »n-dimensional spheres and ellipsoids – I – Gamma-function, volume and surface of n-spheres

Cut-off BVN limited to an ellipsoidal core

Covariance matrix of a cut-off Multivariate Normal Distribution – III – results for a 2-dimensional BVN-core and proper normalization of its cut-off distribution

In the math section of this blog, we try to cover interesting aspects of Multivariate Normal Distributions [MVNs]. The topic of this post series is the covariance matrix of a MVN-like distribution confined inside a hyper-surface of constant probability density. Outside of the surface we set the probability density to zero. This gives us a “cut-off” MVN- distribution. Contour surfaces… Read More »Covariance matrix of a cut-off Multivariate Normal Distribution – III – results for a 2-dimensional BVN-core and proper normalization of its cut-off distribution

3-dim projections of 4-dim MVN

Covariance matrix of a cut-off Multivariate Normal Distribution – II – integrals over volume and surface of an n-dimensional sphere

In the 1st post of this series, we have posed the following problem: Take the probability density of a Multivariate Normal Distribution [MVN], but set it to zero at Mahalanobis distances bigger than a finite distance D. Take a respective volume region enclosed by a contour surface of constant probability density (at the Mahalanobis distance dm=D). We have called such… Read More »Covariance matrix of a cut-off Multivariate Normal Distribution – II – integrals over volume and surface of an n-dimensional sphere

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 a central core. The core’s surface is assumed to reflect a contour surface of the MVN and would therefore be given by a finite Mahalanobis distance D. I.e.,… 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?