Quaternion-Valued Nonlinear Adaptive Filtering

Abstract

Abstract-A class of nonlinear quaternion-valued adaptive filtering algorithms is proposed based on locally analytic nonlinear activation functions. To circumvent the stringent standard analyticity conditions which are prohibitive to the development of nonlinear adaptive quaternion-valued estimation models, we use the fact that stochastic gradient learning algorithms require only local analyticity at the operating point in the estimation space. It is shown that the quaternion-valued exponential function is locally analytic, and, since local analyticity extends to polynomials, products, and ratios, we show that a class of transcendental nonlinear functions can serve as activation functions in nonlinear and neural adaptive models. This provides a unifying framework for the derivation of gradient-based learning algorithms in the quaternion domain, and the derived algorithms are shown to have the same generic form as their real-and complex-valued counterparts. To make such models second-order optimal for the generality of quaternion signals (both circular and noncircular), we use recent developments in augmented quaternion statistics to introduce widely linear versions of the proposed nonlinear adaptive quaternion valued filters. This allows full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances to cater rigorously for second-order noncircularity (improperness), and the corresponding power mismatch in the signal components. Simulations over a range of circular and noncircular synthetic processes and a real world 3-D noncircular wind signal support the approach.Index Terms-Augmented quaternion statistics, H-circularity, nonlinear adaptive filtering, quaternion least mean square, widely linear modeling, widely linear quaternion least mean square, wind prediction.

Design of Quaternion-Neural-Network-Based Self-Tuning Control Systems

“…Convergence analysis of the quaternion neural network is important and several studies have been conducted. (7,13,16) In this study, the difference in the cost function ∆J (k) is derived as…”

“…(1,2) Quaternion neural networks have been demonstrated to perform better than real-number neural networks, because the former can cope with multidimensional issues more efficiently by employing quaternions directly. Many studies have successfully used quaternion neural networks in applications requiring multidimensional signal processing, for example, colour image processing, (3,4) signal processing, (5,6) filtering, (7) inverse problems, (8) and classification problems. (9) In previous studies, we presented robot control applications of quaternion neural networks to solve forward and inverse kinematics of a robot manipulator.…”

The quaternion kernel least squares

“…The optimal least squares solution of the kernel regression problem in (9) has the same form as its real-valued counterpart in (5). These two forms are equivalent for realvalued kernels of quaternion variable.…”

“…The properties of the quaternion normed division algebra provide physically meaningful representation and more accurate rotation and orientation modelling of 3D objects than real vectors. Thus, quaternion valued algorithms exhibit more degrees of freedom than their real-valued counterparts, and are particularly suited for rotation and orientation applications as well as unified 3D and 4D modelling [9,10]. Our aim is to show that kernel regression also benefits from the enhanced dimensionality of quaternion valued kernels, and their inherent ability to both represent inter-dependence between signal components and to model highly coupled multidimensional data features.…”

Remarks on robot controller application of Clifford multi-layer neural networks

“…In artificial neural networks, the neural architectures based on some Clifford subalgebras, for example, complex numbers and quaternions, are believed to solve the multi-inputs/outputs problems efficiently because Clifford algebra is a powerful framework to represent data in the number of different dimensions. Complex multilayer neural networks using back-propagation algorithm have been introduced [1] [2] and the characteristics of quaternion multi-layer neural networks have also been investigated in [3] [4] [5] [6] [7]. These works are the recent contribution to this field.…”