Enabling quaternion derivatives: the generalized HR calculus

Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis

Filtering and Tracking with Trinion-Valued Adaptive Algorithms

A new model for three-dimensional processes based on the trinion algebra is introduced for the first time. Compared with the pure quaternion model, the trinion model is more compact and computationally more efficient, while having similar or comparable performance in terms of adaptive linear filtering. Moreover, the trinion model can effectively represent the general relationship of state evolution in Kalman filtering, where the pure quaternion model fails. Simulations on real-world wind recordings and synthetic data sets are provided to demonstrate the potentials of this new modeling method

On gradient calculation in quaternion adaptive filtering

A novel way to calculate the gradient of real functions of quaternion variables, typical cost functions in quaternion signal processing, is proposed. This is achieved by revisiting quaternion involutions and by simplifying the existing ℍℝ derivatives. This has allowed us to express the class of quaternion least mean square (QLMS) algorithms in a more compact form while keeping the same generic form of LMS. Simulations in the prediction setting support the approach. © 2012 IEEE

On HR calculus, quaternion valued stochastic gradient, and adaptive three dimensional wind forecasting

Short term forecasting of wind field in the quaternion domain is addressed. This is achieved by casting the three components of wind speed (two horizontal and a vertical) into a pure quaternion and adding air temperature as a scalar component, to form the full quaternion. First, HR calculus is introduced in order to provide a unifying framework for the calculation of the derivatives of both analytic quaternion valued functions and real functions of quaternion variables, such as the standard cost function (error power). The analysis shows that the maximum change in the gradient is in the direction of the conjugate of the weight vector, conforming with the gradient calculation in the complex domain. For rigour, we also illustrate that the widely linear model is required in order to capture full second order information within three- and four-dimensional quaternion valued signals. The so established framework is used to illustrate a convenient way to derive the recently introduced quaternion least mean square (QLMS) and the widely linear QLMS (WL-QLMS). Simulations on short term prediction of real world wind signals support the approach. © 2010 IEEE

A class of quaternion valued affine projection algorithms

The strictly linear quaternion valued affine projection algorithm (QAPA) and its widely linear counterpart (WLQAPA) are introduced, in order to provide fast converging stochastic gradient learning in the quaternion domain, for the processing of both second order circular (proper) and second order noncircular (improper) signals. This is achieved based on the recent advances in augmented quaternion statistics, which employs all second order information available, together with the associated widely linear models and through performing rigorous gradient calculation (HR-calculus). Further, mean square error analysis is performed based on the energy conservation principle, which provides a theoretical justification for the WLQAPA offering enhanced steady state performance for quaternion noncircular (improper) signals, a typical case in real world scenarios. Simulations on benchmark circular and noncircular signals, and on noncircular real world 4D wind and 3D body motion data support the analysis

A Unifying Framework for the Analysis of Quaternion Valued Adaptive Filters

The recently proposed HR-calculus has enabled rigorous derivation of quaternion-valued adaptive filtering algorithms, and has also introduced several equivalent forms of the quaternion least mean square (QLMS). This work aims to address the uniqueness of the solutions to the stochastic gradient optimisation problems, and to provide a unified framework for the derivation and analysis of quaternion least mean square algorithms. In doing so, we assess and compare the properties of the adaptive algorithms in the context of their convergence and steady state performances. For generality, the convergence properties of both QLMS and its widely linear extension, the WL-QLMS are illuminated

On gradient calculation in quaternion adaptive filtering

A novel way to calculate the gradient of real functions of quaternion variables, typical cost functions in quaternion signal processing, is proposed. This is achieved by revisiting quaternion involutions and by simplifying the existing ℍℝ derivatives. This has allowed us to express the class of quaternion least mean square (QLMS) algorithms in a more compact form while keeping the same generic form of LMS. Simulations in the prediction setting support the approach. © 2012 IEEE

Blind extraction of improper quaternion sources

Blind extraction of quaternion-valued latent sources is addressed based on their local temporal properties. The extraction criterion is based on the minimum mean square widely linear prediction error, thus allowing for the extraction of both proper and improper quaternion sources. The use of the widely linear adaptive predictor is justified by the relationship between the mean square prediction error and the crosscorrelation and cross-pseudocorrelations of the source signals. Simulations on benchmark improper quaternion sources together with a real-world example of EEG artifact removal illustrate the usefulness of the proposed methodology. © 2011 IEEE