Skip to content

eremo

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

Ellipses determined from a matrix mediating a quadratic form

Ellipses via matrix elements – I – basic derivations and formulas

Ellipses are specific two-dimensional geometrical objects. They are of interest in many contexts – e.g. in physics, engineering and in cryptography. However, they also appear in statistics. For example, in the form of elliptic contour lines of Bivariate Normal Distributions [BVDs] and as elliptic contours of the projections of Multivariate Normal Distributions [MVDs] onto coordinate planes. Approximate BVDs/MVDs in turn… Read More »Ellipses via matrix elements – I – basic derivations and formulas

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