Flexible estimation of probability and cumulative density functions

Introduction: Probability density function (PDF) estimation is a very important issue in several interesting areas, such as blind signal processing and adaptive data processing. The estimation of the PDF or the cumulative density function (CDF) through use of an easy and fast method becomes a very important task. Several approaches exist [1] such as maximum likelihood estimation, kernel estimation, cluster analysis and the most important histogram method. An effective method is based on neural networks and provides a simple adaptation rule. An example can be found in [2] that is based on the information-theoretic approach proposed by Bell and Sejnowski [3]. In this Letter we propose the use of a single-input neuron, i.e. the nonlinear activation function, by adopting a flexible nonlinear function the shape of which can be changed during the learning process following the method shown by Solazzi et al. in [4]. This nonlinear function is implemented by a cubic spline function. Spline functions consist of a superposition of a certain number of cubic polynomial pieces, so their shape can be varied during the learning process. Some experimental results are shown in order to demonstrate the effectiveness of the proposed approach.