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half-axes

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

BVD contour ellipses

Bivariate Normal Distribution – Mahalanobis distance and contour ellipses

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