Widely Linear Quaternion-Valued Least-Mean Kurtosis Algorithm

A widely linear quaternion-valued least-mean kurtosis (WL-QLMK) algorithm is introduced for adaptive filtering of quaternion-valued circular and noncircular signals. In the design, kurtosis-based cost function is first defined in the quaternion domain by integrating the widely linear model, and augmented statistics, and then minimized using the recently developed generalized Hamilton-real (GHR) calculus. In this way, the novel WL-QLMK algorithm is obtained for training quaternion-valued adaptive filter structures. Furthermore, its steady-state performance is theoretically analyzed to determine the bounds of the step size, which provides a theoretical justification for simulations. The simulation results over both benchmark system identification scenarios, and one-step-ahead predictions of real-world 4D pathological resting tremors show that the proposed WL-QLMK algorithm, by virtue of its newly defined cost function, significantly enhances the performance compared to the recently developed quaternion-valued algorithms, especially for noncircular signals.

[1]  Nobuyuki Matsui,et al.  Quaternion neural network with geometrical operators , 2004, J. Intell. Fuzzy Syst..

[2]  Badong Chen,et al.  Kernel Least Mean Kurtosis Based Online Chaotic Time Series Prediction , 2013 .

[3]  Dongpo Xu,et al.  Enabling quaternion derivatives: the generalized HR calculus , 2014, Royal Society Open Science.

[4]  Danilo P. Mandic,et al.  Cost-effective quaternion minimum mean square error estimation: From widely linear to four-channel processing , 2017, Signal Process..

[5]  Cameron N. Riviere,et al.  Modeling and canceling tremor in human-machine interfaces , 1996 .

[6]  Xiangyang Wang,et al.  A Novel Least-Mean Kurtosis Adaptive Filtering Algorithm Based on Geometric Algebra , 2019, IEEE Access.

[7]  Anthony G. Constantinides,et al.  Least-mean kurtosis: a novel higher-order statistics based adaptive filtering algorithm , 1994 .

[8]  José Carlos M. Bermudez,et al.  Stochastic analysis of the Least Mean Kurtosis algorithm for Gaussian inputs , 2016, Digit. Signal Process..

[9]  Danilo P. Mandic,et al.  Performance analysis of the deficient length augmented CLMS algorithm for second order noncircular complex signals , 2018, Signal Process..

[10]  Engin Cemal Menguc.  Novel quaternion-valued least-mean kurtosis adaptive filtering algorithm based on the GHR calculus , 2018, IET Signal Process..

[11]  Thomas K. Paul,et al.  A Kernel Adaptive Algorithm for Quaternion-Valued Inputs , 2015, IEEE Transactions on Neural Networks and Learning Systems.

[12]  Danilo P. Mandic,et al.  Performance Analysis of Quaternion-Valued Adaptive Filters in Nonstationary Environments , 2018, IEEE Transactions on Signal Processing.

[13]  Danilo P. Mandic,et al.  The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes , 2009, IEEE Transactions on Signal Processing.

[14]  Danilo P. Mandic,et al.  Quaternion-Valued Echo State Networks , 2015, IEEE Transactions on Neural Networks and Learning Systems.

[15]  Tyseer Aboulnasr,et al.  A robust variable step-size LMS-type algorithm: analysis and simulations , 1997, IEEE Trans. Signal Process..

[16]  Jack B. Kuipers,et al.  Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality , 2002 .

[17]  Yi Li,et al.  Adaptive Beamforming for Vector-Sensor Arrays Based on a Reweighted Zero-Attracting Quaternion-Valued LMS Algorithm , 2016, IEEE Transactions on Circuits and Systems II: Express Briefs.

[18]  Haiquan Zhao,et al.  Widely linear least mean kurtosis‐based frequency estimation of three‐phase power system , 2020 .

[19]  Danilo P. Mandic,et al.  A perspective on CLMS as a deficient length augmented CLMS: Dealing with second order noncircularity , 2018, Signal Process..

[20]  Rajnikant V. Patel,et al.  WAKE: Wavelet Decomposition Coupled with Adaptive Kalman Filtering for Pathological Tremor Extraction , 2017, Biomed. Signal Process. Control..

[21]  Huazhong Shu,et al.  Removing Gaussian noise for colour images by quaternion representation and optimisation of weights in non-local means filter , 2014, IET Image Process..

[22]  Sheng Zhang,et al.  New Steady-State Analysis Results of Variable Step-Size LMS Algorithm With Different Noise Distributions , 2014, IEEE Signal Processing Letters.

[23]  Jessica Koehler,et al.  Advanced Digital Signal Processing And Noise Reduction , 2016 .

[24]  Daniel Pérez Palomar,et al.  Quaternion ICA From Second-Order Statistics , 2011, IEEE Transactions on Signal Processing.

[25]  Danilo P. Mandic,et al.  Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms , 2016, IEEE Transactions on Neural Networks and Learning Systems.

[26]  Paulo S. R. Diniz,et al.  New Trinion and Quaternion Set-Membership Affine Projection Algorithms , 2017, IEEE Transactions on Circuits and Systems II: Express Briefs.

[27]  Ghasem Alipoor,et al.  LMSK: a robust higher-order gradient-based adaptive algorithm , 2019, IET Signal Process..

[28]  Tao Chen,et al.  Quaternion least mean kurtosis algorithm for adaptive filtering of 3D and 4D signal processes , 2017, 2017 20th International Conference on Information Fusion (Fusion).

[29]  Danilo P. Mandic,et al.  A Quaternion Widely Linear Adaptive Filter , 2010, IEEE Transactions on Signal Processing.

[30]  Yili Xia,et al.  Robust frequency estimation of unbalanced power system using a phase angle error based least mean kurtosis algorithm , 2019 .

[31]  Danilo P. Mandic,et al.  Performance Analysis of Deficient Length Quaternion Least Mean Square Adaptive Filters , 2020, IEEE Transactions on Signal Processing.

[32]  Kazuyuki Aihara,et al.  Quaternion-valued short term joint forecasting of three-dimensional wind and atmospheric parameters , 2011 .

[33]  Jeng-Kuang Hwang,et al.  A Gradient-Based Variable Step-Size Scheme for Kurtosis of Estimated Error , 2010, IEEE Signal Processing Letters.

[34]  Engin Cemal Mengüç,et al.  An Augmented Complex-Valued Least-Mean Kurtosis Algorithm for the Filtering of Noncircular Signals , 2018, IEEE Transactions on Signal Processing.

[35]  Nurettin Acır,et al.  Kurtosis-Based CRTRL Algorithms for Fully Connected Recurrent Neural Networks , 2018, IEEE Transactions on Neural Networks and Learning Systems.

[36]  Francesco Grigoli,et al.  Optimal reorientation of geophysical sensors: A quaternion-based analytical solution , 2015 .

[37]  Noureddine Manamanni,et al.  Complementary Observer for Body Segments Motion Capturing by Inertial and Magnetic Sensors , 2014, IEEE/ASME Transactions on Mechatronics.