Statistical Modeling of Univariate Unimodal Data Using Π-Sigmoid Mixture Models
Abstract: Unimodal datasets demonstrate the primary characteristic of grouping behavior around a single peak. Assessing data unimodality means estimating whether the data has been generated by an arbitrary unimodal distribution like Gaussian, Student’s t, Triangular etc. Existing statistical tests for assessing unimodality of univariate data do not provide a unimodal statistical model. Thus, one has to assume a specific parametric form of the density (e.g. Gaussian) and estimate its parameters through maximum likelihood. A notable exception is UU-test which provides such a model in the form of a mixture of uniform distributions. In this work, we rely of \(\varPi \)-sigmoid mixture models to improve the UU-test solution for unimodal datasets. Such mixture models employ the \(\varPi \)-sigmoid distribution, which is flexible enough to model arbitrarily shaped unimodal distributions, from bell-shaped (e.g. Gaussian) to uniform. We present a training methodology that results in a final mixture with fewer components and superior modeling of the underlying unimodal data distribution. Comparative experimental results on various synthetic and real univariate datasets indicate the effectiveness of our method.
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