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 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
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
I continue with my posts on Bivariate Normal Distributions [BVDs]. In this post we consider the exponent of a BVD’s probability density function [pdf]. This function is governed by a central matrix Σ-1, the inverse of the variance-covariance matrix of the BVD’s random vector. We define the so called Mahalanobis distance dm for BVD vectors. A constant value of the… Read More »Bivariate Normal Distribution – Mahalanobis distance and contour ellipses