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quadratic form

Compute confidence ellipses – II – equivalence of Schelp’s basic construction method for confidence ellipse with other approaches

In the 1st post of this series, I have motivated a simple method for constructing confidence ellipses for assumedly Bivariate Normal Distributions [BVD] or at least approximate BVDs. A reader has asked me, whether one can prove more rigidly that the proposed method of C. Schelp is equivalent to other BVD-based methods. Well, in this post we show that the… Read More »Compute confidence ellipses – II – equivalence of Schelp’s basic construction method for confidence ellipse with other approaches

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

Surfaces of n-dimensional ellipsoids – I – quadratic form and matrix equation

Multidimensional ellipsoids are mathematically interesting figures per se. But there is a reason why they sometimes also appear in the context of Machine Learning experiments. One reason is that Multivariate Normal Distributions [MND] often describe the statistical distributions of properties which characterize natural objects we investigate by ML-methods. And the locations of constant probability density of MNDs are surfaces of… Read More »Surfaces of n-dimensional ellipsoids – I – quadratic form and matrix equation