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MVD

3-dim projections of 4-dim MVN

n-dimensional spheres and ellipsoids – IV – numerical check of Rivin’s formula for the surface areas of ellipsoids in 3 dimensions and the perimeters of ellipses

In the third post of this series we have discussed an idea of I. Rivin (see [1], [2]). Rivin has shown that the surface area of a general (n-1)-dimensional ellipsoid in a n-dimensional Euclidean space can be expressed in terms of an expectation value of a specific function weighted by a multivariate Gaussian probability density [pdf]. In contrast to (n-1)-spheres… Read More »n-dimensional spheres and ellipsoids – IV – numerical check of Rivin’s formula for the surface areas of ellipsoids in 3 dimensions and the perimeters of ellipses

Covariance matrix of a cut-off Multivariate Normal Distribution – IV – theoretical prediction for a 3-dimensional MVN-core

In this post series we study ellipsoidal cores of Multivariate Normal Distributions [MVNs]. We defined a “core” as the volume enclosed by a selected contour surface of constant probability density. We constructed cut-off distributions by setting the probability density to zero outside the core. In previous posts we have already discussed volume integrals which would give us a relation between… Read More »Covariance matrix of a cut-off Multivariate Normal Distribution – IV – theoretical prediction for a 3-dimensional MVN-core

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?