The quaternion minimum error entropy algorithm with fiducial point for nonlinear adaptive systems

Abstract

In this paper, we develop a kernel adaptive filter for quaternion domain data, based on information theoretic learning cost function which could be useful for quaternion based kernel applications of nonlinear filtering. The new algorithm is based on error entropy function with Fiducial point and is referred to as the quaternion minimum error entropy with fiducial point (QKMEEF) algorithm. In our previous work we developed quaternion kernel adaptive filter based on minimum error entropy referred to as the quaternion KMEE (QKMEE) algorithm [1]. Since entropy does not change with the mean of the distribution, the algorithm may converge to a set of optimal weights without having zero mean error. Traditionally, to make the zero mean output error, the output during testing session was biased with the mean of errors of training session. However, for non-symmetric or heavy tails error PDF the estimation of error mean is problematic. The minimum error entropy criterion, minimizes Renyi’s quadratic entropy of the error between the filter output and desired response or indirectly maximizing the error information potential. Here, the approach is applied to quaternions. Adaptive filtering in quaterion domain intrinsically incorporates component-wise real valued cross-correlation or the coupling within the dimensions of the quaternion input. We apply generalized Hamilton-real (GHR) calculus that is applicable to Hilbert space for evaluating the cost function gradient to develop the Quaternion Minimum Error Entropy Algorithm with Fiducial points. Simulation results are used to show the behavior of the new algorithm (QKMEEF) when signal is non-Gaussian in presence of unimodal noise versus bi-modal noise distributions. Simulation results also show that the new algorithm Quat-KMEEF can track and predict the 4-Dimensional non-stationary process signals where there are correlations between components better than quadruple real-valued KMEEF and Quat-KLMS algorithms.