Kernel Filter Algorithm Based On Quaternion

The development of sensor technology enables researchers to obtain high-dimensional data,and the data obtained by actual sensors are mostly non-linear data.In order to cope with the existing situation,more suitable filtering algorithms are needed,and quaternion kernel filtering algorithm can handle high-dimensional nonlinear data well.As the basis of quaternion filtering,the research of quaternion gradient is also the emphasis in the field of quaternion filtering.Therefore,in this thesis,the quaternion kernel filtering algorithm including quaternion gradient is studied in depth.The contributions of this thesis are as follows.This thesis firstly introduces quaternion-related theories,including quaternion algebras and quaternion gradient update rules,as well as the composition of quaternion signals and signal analysis and statistical theory of quaternion signals.The statistical methods and classification of quaternion signals are also given in this thesis.At the same time,this thesis also summarizes the research basis of quaternion kernel filtering,that is,the composition of the quaternion kernel and the usage of the kernel method,which lays a theoretical foundation for the subsequent research on quaternion-based kernel filtering algorithms.Then this thesis introduces the GHR-based quaternion gradient in detail,analyzes the disadvantages of traditional HR derivatives in quaternion filtering applications and the reasons why it is difficult for generalization.Besides,this thesis gives the properties of the GHR quaternion gradient.This thesis combines the quaternion involution with the HR derivative and applies them to the derive the quaternion gradient.A new gradient based on the quaternion involution is proposed,which greatly reduces the difficulty of calculating the quaternion derivative.This thesis also gives some classical derivative calculations based on quaternion involution,and the results are identical to the derivative results based on GHR.Finally,this thesis proposes two new quaternion linear filtering algorithms which are quaternion least mean square(LMS-QI)algorithm and quaternion recursive least squares(QRLS)algorithm,and two new quaternion kernel filtering algorithm,which are quaternion kernel least mean square(KLMS-QI)algorithm and quaternion kernel recursive least squares(KRLS-QI)algorithm.The forms of the new algorithms proposed in this thesis all have the same form as the algorithms corresponding to real and complex domain.In this thesis,the correctness of LMS-QI and QRLS is verified by comparing with the corresponding four-dimensional real algorithms.Experiments show that when using a real kernel,the KLMS-QI algorithm has the best performance comparing with other quaternion kernel least mean square algorithms.When using a quaternion kernel,the performance if similar to those of other quaternion kernel least mean square algorithms;when the parameter,the KRLS-QI algorithm has significantly better performance than other algorithms.When testing the algorithms,this thesis uses synthetic proper data,improper data,and real collected EEG data for experiments.