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multidimensional ellipsoids

orthogonal projections of ellipsoid

Orthogonal projections of multidimensional ellipsoids – I – points on the ellipsoid that give us the surface points of the projection

A reader of a parallel post-series on n-ellipsoids has asked me how I could know what the matrix for the quadratic form of the orthogonal projection of a (n-1) dimensional-ellipsoid onto a (n-1)-dimensional sub-space of the ℝn looks like. I had stated in previous posts that we can directly derive the elements of the matrix describing projections of ellipsoids or… Read More »Orthogonal projections of multidimensional ellipsoids – I – points on the ellipsoid that give us the surface points of the projection

Concentric surfaces of ellipsoids

n-dimensional spheres and ellipsoids – II – volume of n-ellipsoids

This post series provides some insights into the nature of n-dimensional spheres and ellipsoids and the derivation of some special integrals over their volumes. In this 2nd post we look at the volume of a “(n-1)–ellipsoid“. This term refers to a closed ellipsoidal surface in a n-dimensional Euclidean space. Such a surface is a (n-1)-dimensional manifold. See the 1st post… Read More »n-dimensional spheres and ellipsoids – II – volume of n-ellipsoids

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

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