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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

Confidence ellipses for an approximate BVD

Bivariate Normal Distribution – integrated probability up to a given Mahalanobis distance, the Chi-squared distribution and confidence ellipses

In previous posts of this blog we have discussed the general form of the probability density function [pdf] of a Bivariate Normal Distribution [BVD]. In this post we consider the integral over a BVD’s pdf up to a defined value of the Mahalanobis Distance. A given value of the latter defines an elliptic contour line of constant probability density. With… Read More »Bivariate Normal Distribution – integrated probability up to a given Mahalanobis distance, the Chi-squared distribution and confidence ellipses

Ellipses constructed from elements of a matrix defining a quadratic form

Ellipses via matrix elements – II – numerical tests of formulas

During the last posts, I have discussed properties of ellipses and ways to (re-) construct them from elements of a symmetric, invertible and positive-definite (2×2)-matrix, which defines a quadratic form. In the context of Machine Learning we often have to determine confidence ellipses from elements of a numerically determined variance-covariance matrix of statistical bivariate vector-distributions. Formulas relating the geometric properties… Read More »Ellipses via matrix elements – II – numerical tests of formulas