On The Cost Functions And Update Methods Of Adaptive Filtering Algorithms

As an important part of statistical signal processing,an adaptive filter(AF)has been widely used and developed in the fields of signal processing and positioning.For stationary data,in the process of signal processing,the system parameters can be adjusted according to the change of input signal,adaptively,and the parameters of noise and signal need not to be estimated in advance.For non-stationary data,based on the state space model,the AF based on Kalman filter has good filtering performance for state estimation.In addition,as an emerging machine learning method,the extreme learning machine(ELM)has been widely used in many fields such as signal processing and machine learning.From the perspective of learning efficiency,ELM has the characteristics of fast learning and good performance.Recently,AFs are developed from three aspects including the optimization criterion,filter structure and weight update methods.Among them,the optimization criterion in AF mainly includes the minimum mean square error(MMSE),maximum correntropy criterion(MCC),generalized maximum correntropy criterion(GMCC)and so on.The MMSE criterion can achieve the best filtering performance in the presence of Gaussian noise.While the MCC based on the Gaussian kernel function can effectively suppress the impact of non-Gaussian noise such as impulse noise due to the included high-order statistical characteristics of the error.However,the Gaussian kernel is not always the best choice in correntropy.Recently,based on the generalized Gaussian density(GGD)function,a generalized correntropy is proposed.The traditional correntropy with the Gaussian kernel is a special form of the generalized correntropy.Similar to the MCC,based on the generalized correntropy,the generalized maximum corretropy criterion(GMCC)can be used as an optimization criterion in estimation-related problems.For the study of filter structure,it mainly includes the feedforward and feedback structures.The traditional filtering algorithm mainly uses the feedforward structure.For the weight update method,the traditional adaptive filtering algorithms mainly use the Newton's descent method and gradient descent method.As a generalized derivative form,the q derivative and fractional-order derivative have been more and more applied in adaptive filtering.Therefore,based on the optimal criterion and the update method,this thesis proposes three kinds of methods for improving filtering accuracy and robustness.(1)In order to obtain higher filtering accuracy and faster convergence speed of the filter,the q derivative is combined with affine projection algorithm(APA),generating a q-affine projection algorithm(q-APA).Since the q derivative calculates the secant of the cost function and the traditional derivative calculates the tangent,the q derivative can avoid the local optimal solution of the cost function compared with the traditional derivative.However,the adaptive filtering algorithm based on the q derivative with constant value cannot improve the filtering accuracy and convergence rate of the filter,simultaneously.Therefore,based on q-APA,the variable q-APA is developed.Since V-q-APA can adjust the q value during the weight update process adaptively,V-q-APA can improve the filtering accuracy and convergence rate of the filter,simultaneously.(2)Since the MCC can combat larger outliers in the presence of non-Gaussian noise,the MCC-based adaptive filtering algorithm shows a better robustness against impulsive interferences.Compared with the traditional derivative,the system described by the fractional-order derivative is closer to the actual system.Therefore,based on the MCC criterion,the fractional-order maximum correntropy criterion(FMCC)algorithm is proposed.Compared with the MCC algorithm,the FMCC algorithm proposed in this thesis can achieve better prediction accuracy and faster convergence at the cost of increasing computational complexity.(3)For improving the robustness and filtering accuracy of the filter,the GMCC is applied into the online sequential extreme learning machine(OS-ELM),generating the online sequential extreme learning machine based on the generalized maximum correntropy criterion(OS-ELM-GMCC).As a generalized form of MCC,OS-ELM-GMCC can achieve better filtering performance by selecting appropriate parameters.