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ellipse

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

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

BVD contour ellipses with varying negative Pearson coefficient

Properties of BVD confidence ellipses – II – dependency of the half-axes on the correlation coefficient

If you have read my last post on confidence ellipses, you may have tried to derive the result on the longer half-axis for maximum correlation by following an eigenvalue analysis of the (inverse) covariance matrix of a Bivariate Normal Distribution [BVD]. If you have succeeded, jump over this post. If not, the contents my be interesting for you. Its is… Read More »Properties of BVD confidence ellipses – II – dependency of the half-axes on the correlation coefficient

BVD confidence ellipses for varying correlation

Properties of BVD confidence ellipses – I – constant limits and tangents in x- and y-direction during variation of the Pearson correlation coefficient

We have gathered a lot of knowledge about Bivariate Normal Distributions [BVDs] and their contour ellipses in the math section of this blog. We can now analyze some secondary and funny properties of BVD contour and confidence ellipses. Among other things the variation of some key properties with the Pearson correlation coefficient ρ is of interest for data analysts. In… Read More »Properties of BVD confidence ellipses – I – constant limits and tangents in x- and y-direction during variation of the Pearson correlation coefficient