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covariance matrix

3-dim projections of 4-dim MVN

Covariance matrix of a cut-off Multivariate Normal Distribution – II – integrals over volume and surface of an n-dimensional sphere

In the 1st post of this series, we have posed the following problem: Take the probability density of a Multivariate Normal Distribution [MVN], but set it to zero at Mahalanobis distances bigger than a finite Mahalanobis distance D. Take a respective volume region enclosed by a contour surface of constant probability density, for a Mahalanobis distance dm ≤ D. We… Read More »Covariance matrix of a cut-off Multivariate Normal Distribution – II – integrals over volume and surface of an n-dimensional sphere

Cut-off BVN limited to an ellipsoidal core

Covariance matrix of a cut-off Multivariate Normal Distribution – I – intricate integrals with exponentials over the volumes and surfaces of n-dimensional ellipsoids?

In Machine Learning and statistics one sometimes has to work with a data sample whose underlying probability distribution approximates a Multivariate Normal Distribution [MVN] – but only within the ellipsoidal region of a central core. The core’s surface is assumed to reflect a contour surface of the MVN and would therefore be given by a finite Mahalanobis distance D. I.e.,… Read More »Covariance matrix of a cut-off Multivariate Normal Distribution – I – intricate integrals with exponentials over the volumes and surfaces of n-dimensional ellipsoids?

Iterative method to compute the covariance-matrix of normal MVN-like inner cores of multivariate distributions with strongly asymmetric outer layers – I

In other posts in this blog (see [1] to [3]), I have discussed multiple methods to calculate and construct confidence ellipses of “Bivariate Normal Distributions” [BVNs]. BVNs are the marginal distributions of “Multivariate Normal Distributions” [MVNs] in e.g. n dimensions ( n > 2). Therefore, two-dimensional confidence ellipses appear as projections of n-dimensional concentric confidence ellipsoids of MVNs onto (2-dim) coordinate planes. The properties of the confidence ellipsoids, which also give us contours of the probability density, are defined by the variance-covariance matrix Σ of the MVN. This post discusses a method to compute the confidence ellipsoids and ellipses for an inner MVN-like core of an otherwise largely asymmetric distribution, which in its overall shape and structure deviates strongly from a MVN.

Read More »Iterative method to compute the covariance-matrix of normal MVN-like inner cores of multivariate distributions with strongly asymmetric outer layers – I
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