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ellipse

Properties of BVD confidence ellipses – II – dependency of 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 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

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

Cholesky decomposition of an ellipse-defining symmetric matrix

An ellipse can be defined via a symmetric, invertible and positive-definite (2×2)-matrix Aq. Such a matrix provides a quadratic form which in turn correlates the components of position vectors to points on an ellipse. This post shows that a Cholesky decomposition of the inverse of Aq provides a method to create an ellipse from a simple set of vectors which… Read More »Cholesky decomposition of an ellipse-defining symmetric matrix