Performance Bounds of Quaternion Estimators

Estimation of Forearm Motion Based on EMG Using Quaternion Neural Network

Quaternions are useful for representing data in three-dimensional space, and the quaternion neural network is effective for learning data in this context. On the other hand, estimating biological motion based on myopotential can be performed directly using electromyogram (EMG) signals as the computer interface. The trajectory of human forearm movement within the three-dimensional space can provide important information. In this study, the relationship between the myopotential of the upper arm muscles and the forearm motion was estimated and investigated using a quaternion neural network.

Multiple-Model Adaptive Estimation for 3-D and 4-D Signals: A Widely Linear Quaternion Approach

Quaternion state estimation techniques have been used in various applications, yet they are only suitable for dynamical systems represented by a single known model. In order to deal with model uncertainty, this paper proposes a class of widely linear quaternion multiple-model adaptive estimation (WL-QMMAE) algorithms based on widely linear quaternion Kalman filters and Bayesian inference. The augmented second-order quaternion statistics is employed to capture complete second-order statistical information in improper quaternion signals. Within the WL-QMMAE framework, a widely linear quaternion interacting multiple-model algorithm is proposed to track time-variant model uncertainty, while a widely linear quaternion static multiple-model algorithm is proposed for time-invariant model uncertainty. A performance analysis of the proposed algorithms shows that, as expected, the WL-QMMAE reduces to semiwidely linear QMMAE for [Formula: see text]-improper signals and further reduces to strictly linear QMMAE for proper signals. Simulation results indicate that for improper signals, the proposed WL-QMMAE algorithms exhibit an enhanced performance over their strictly linear counterparts. The effectiveness of the proposed recursive performance analysis algorithm is also validated.

Decomposition of Rotor Hopfield Neural Networks Using Complex Numbers

A complex-valued Hopfield neural network (CHNN) is a multistate model of a Hopfield neural network. It has the disadvantage of low noise tolerance. Meanwhile, a symmetric CHNN (SCHNN) is a modification of a CHNN that improves noise tolerance. Furthermore, a rotor Hopfield neural network (RHNN) is an extension of a CHNN. It has twice the storage capacity of CHNNs and SCHNNs, and much better noise tolerance than CHNNs, although it requires twice many connection parameters. In this brief, we investigate the relations between CHNN, SCHNN, and RHNN; an RHNN is uniquely decomposed into a CHNN and SCHNN. In addition, the Hebbian learning rule for RHNNs is decomposed into those for CHNNs and SCHNNs.

A Robust Diffusion Estimation Algorithm with Self-Adjusting Step-Size in WSNs

In wireless sensor networks (WSNs), each sensor node can estimate the global parameter from the local data in a distributed manner. This paper proposed a robust diffusion estimation algorithm based on a minimum error entropy criterion with a self-adjusting step-size, which are referred to as the diffusion MEE-SAS (DMEE-SAS) algorithm. The DMEE-SAS algorithm has a fast speed of convergence and is robust against non-Gaussian noise in the measurements. The detailed performance analysis of the DMEE-SAS algorithm is performed. By combining the DMEE-SAS algorithm with the diffusion minimum error entropy (DMEE) algorithm, an Improving DMEE-SAS algorithm is proposed for a non-stationary environment where tracking is very important. The Improving DMEE-SAS algorithm can avoid insensitivity of the DMEE-SAS algorithm due to the small effective step-size near the optimal estimator and obtain a fast convergence speed. Numerical simulations are given to verify the effectiveness and advantages of these proposed algorithms.

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions typically require the calculation of the gradient and Hessian. However, real functions of quaternion variables are essentially nonanalytic, which are prohibitive to the development of quaternion-valued learning systems. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized Hamilton-real (GHR) calculus, thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field, rather than using the isomorphism with the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the novel quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are also introduced, opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms. The proposed GHR calculus is shown to yield the same generic algorithm forms as the corresponding real- and complex-valued algorithms. Advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks.