The last months I have seldom written posts in this blog. The reason is that I am occupied with a book about “The geometry of Multivariate Normal Distributions”. Which will cover a lot more topics than the ones discussed in this blog so far. One of those topics is the use of quadratic form functions as graph functions to produce… Read More »Quadratic form functions as graph functions – I – level sets and gradients
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 MVN-like inner cores of multivariate distributions with strongly asymmetric outer layers – IIn previous mathematical posts of this blog, we have studied some core properties of Bivariate Normal Distributions [BVDs]. During the rather mathematical tour de force we have come across various methods to construct and plot confidence ellipses for a given confidence level and respective Mahalanobis distance from the distribution’s center. We have also covered the mathematical derivation of the methods.… Read More »How to compute confidence ellipses – III – 4 alternative construction methods
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 »How to compute confidence ellipses – II – equivalence of Schelp’s basic construction method for confidence ellipse with other approaches