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n-dimensional ellipsoid

Orthogonal projections of multidimensional ellipsoids – III – arbitrary vectors for the projection direction and cuts of the unit sphere

The first two posts of this series showed that the orthogonal projection of an ellipsoid in the ℝn onto a (n-1) dimensional sub-space has a (n-2)-dimensional ellipsoidal surface. The tangential points on the original ellipsoid which control the surface of the projection could be derived by linear transformation A of well defined, but special vectors of the (n-1)-dimensional unit sphere.… Read More »Orthogonal projections of multidimensional ellipsoids – III – arbitrary vectors for the projection direction and cuts of the unit sphere

Orthogonal projections of multidimensional ellipsoids – II – the surface of the projection image is a lower dimensional ellipsoid

In this post series we presently look at orthogonal projections of an ellipsoid in a n-dimensional (Euclidean) space onto a (n-1)-dimensional subspace. The ellipsoid is a closed surface in the ℝn and has a dimensionality of (n-1). Orthogonal projection means that the target subspace is orthogonal to a line of projection (defined by a projection vector) which is the same… Read More »Orthogonal projections of multidimensional ellipsoids – II – the surface of the projection image is a lower dimensional ellipsoid

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

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