Flat-topped Probability Density Functions for Mixture Models

This paper investigates probability density functions (PDFs) that are continuous everywhere, nearly uniform around the mode of distribution, and adaptable to a variety of distribution shapes ranging from bell-shaped to rectangular. From the viewpoint of computational tractability, the PDF based on the Fermi-Dirac or logistic function is advantageous in estimating its shape parameters. The most appropriate PDF for n-variate distribution is of the form: pleft(xright)proptoleft[coshleft(left[left(x-mright)^mathsfTboldsymbolSigma-1left(x-mright)right]n/2right)+coshleft(rnright)right]-1 where x,m∈ℝn, boldsymbolSigma is an n× n positive definite matrix, and r>0 is a shape parameter. The flat-topped PDFs can be used as a component of mixture models in machine learning to improve goodness of fit and make a model as simple as possible.