Probabilistic Models Based on the Pi-Sigmoid Distribution

Mixture models constitute a popular type of probabilistic neural networks which model the density of a dataset using a convex combination of statistical distributions, with the Gaussian distribution being the one most commonly used. In this work we propose a new probability density function, called the Π-sigmoid, from its ability to form the shape of the letter “Π” by appropriately combining two sigmoid functions. We demonstrate its modeling properties and the different shapes that can take for particular values of its parameters. We then present the Π-sigmoid mixture model and propose a maximum likelihood estimation method to estimate the parameters of such a mixture model using the Generalized Expectation Maximization algorithm. We assess the performance of the proposed method using synthetic datasets and also on image segmentation and illustrate its superiority over Gaussian mixture models.