The quaternion kernel least squares

The quaternion kernel least squares algorithm (QKLS) is introduced as a generic kernel framework for the estimation of multivariate quaternion valued signals. This is achieved based on the concepts of quaternion inner product and quaternion positive definiteness, allowing us to define quaternion kernel regression. Next, the least squares solution is derived using the recently introduced Hℝ calculus. We also show that QKLS is a generic extension of standard kernel least squares, and their equivalence is established for real valued kernels. The superiority of the quaternion-valued linear kernel with respect to its real-valued counterpart is illustrated for both synthetic and real-world prediction applications, in terms of accuracy and robustness to overfitting.standard kernel least squares,quaternion-valued linear kernelreal-world prediction applications,real-world 3D inertial body sensor signals.synthetic autoregressive processes

[1]  Anthony Widjaja,et al.  Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond , 2003, IEEE Transactions on Neural Networks.

[2]  Daniel Pérez Palomar,et al.  Maximum likelihood ICA of quaternion Gaussian vectors , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[3]  Danilo P. Mandic,et al.  Quaternion-Valued Nonlinear Adaptive Filtering , 2011, IEEE Transactions on Neural Networks.

[4]  Alexander J. Smola,et al.  Online learning with kernels , 2001, IEEE Transactions on Signal Processing.

[5]  N. Jacobson,et al.  Basic Algebra I , 1976 .

[6]  Alistair Shilton,et al.  Mercer's theorem for quaternionic kernels , 2007 .

[7]  Alexander J. Smola,et al.  Support Vector Regression Machines , 1996, NIPS.

[8]  Miguel Lázaro-Gredilla,et al.  Kernel Recursive Least-Squares Tracker for Time-Varying Regression , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[9]  Alexander Gammerman,et al.  Ridge Regression Learning Algorithm in Dual Variables , 1998, ICML.

[10]  Danilo P. Mandic,et al.  Applications of complex augmented kernels to wind profile prediction , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[11]  Sun-Yuan Kung,et al.  Kernel Approaches to Unsupervised and Supervised Machine Learning , 2009, PCM.

[12]  Weifeng Liu,et al.  Kernel Adaptive Filtering: A Comprehensive Introduction , 2010 .

[13]  Sergios Theodoridis,et al.  The Augmented Complex Kernel LMS , 2011, IEEE Transactions on Signal Processing.

[14]  Paul Honeine,et al.  Online Prediction of Time Series Data With Kernels , 2009, IEEE Transactions on Signal Processing.

[15]  Gavin C. Cawley,et al.  Heteroscedastic kernel ridge regression , 2004, Neurocomputing.

[16]  Danilo P. Mandic,et al.  A novel augmented complex valued kernel LMS , 2012, 2012 IEEE 7th Sensor Array and Multichannel Signal Processing Workshop (SAM).

[17]  N. Aronszajn.  Theory of Reproducing Kernels. , 1950 .

[18]  Weifeng Liu,et al.  Kernel Adaptive Filtering , 2010 .

[19]  公庄 庸三.  Basic Algebra = 代数学入門 , 2002 .

[20]  A. Friedman.  Foundations of modern analysis , 1970 .

[21]  Tokunbo Ogunfunmi,et al.  An alternative kernel adaptive filtering algorithm for quaternion-valued data , 2012, Proceedings of The 2012 Asia Pacific Signal and Information Processing Association Annual Summit and Conference.

[22]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.

[23]  Alistair Shilton,et al.  Quaternionic and complex-valued Support Vector Regression for Equalization and Function Approximation , 2007, 2007 International Joint Conference on Neural Networks.

[24]  Danilo P. Mandic,et al.  A Quaternion Gradient Operator and Its Applications , 2011, IEEE Signal Processing Letters.