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covariance matrix

Iterative method to compute the covariance-matrix of MVN-like inner cores of multivariate distributions with strongly asymmetric outer layers – I

In other posts in this blog (see [1] to [3]), I have discussed multiple methods to calculate and construct confidence ellipses of “Bivariate Normal Distributions” [BVNs]. BVNs are the marginal distributions of “Multivariate Normal Distributions” [MVNs] in e.g. n dimensions ( n > 2). Therefore, two-dimensional confidence ellipses appear as projections of n-dimensional concentric confidence ellipsoids of MVNs onto (2-dim) coordinate planes. The properties of the confidence ellipsoids, which also give us contours of the probability density, are defined by the variance-covariance matrix Σ of the MVN. This post discusses a method to compute the confidence ellipsoids and ellipses for an inner MVN-like core of an otherwise largely asymmetric distribution, which in its overall shape and structure deviates strongly from a MVN.

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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

Confidence ellipses based on covariance matrix

How to compute confidence ellipses – I – simple method based on the Pearson correlation coefficient

This post was motivated by a publication of Carsten Schelp [1]. Actually, a long time ago. I used his results in 2021, when I had to plot confidence ellipses during the analysis of statistical (multivariate) vector distributions produced a Machine Learning algorithm. So, all acknowledgements belong to Schelp’s work. However, his ideas have also triggered some of my own efforts… Read More »How to compute confidence ellipses – I – simple method based on the Pearson correlation coefficient

Contour ellipses from Cholesky decomp ot various covariance matrices

Bivariate Normal Distributions – parameterization of contour ellipses in terms of the Mahalanobis distance and an angle

In my last post about Bivariate Normal Distributions [BVD] I have discussed why contour lines of a BVD’s probability density function [pdf] are concentric ellipses. These contour ellipses are defined by constant values of the so called Mahalanobis distance. In addition, I have discussed a method to create these ellipses from values of the elements of the BVD’s (variance-) covariance… Read More »Bivariate Normal Distributions – parameterization of contour ellipses in terms of the Mahalanobis distance and an angle