Abstract
This paper extends the kernel affine projection algorithm to a rich, flexible and cohesive taxonomy of fractional signal processing approach. The formulation of the algorithm is established on the inclusion of Riemann–Liouville fractional derivative to gradient-based stochastic Newton recursive method to minimize the cost function of the kernel affine projection algorithm. This approach extends the idea of fractional signal processing in reproducing kernel Hilbert space. The proposed algorithm is applied to the prediction of chaotic Lorenz time series and nonlinear channel equalization. Also the performance is validated in comparison with the least mean square algorithm, kernel least mean square algorithm, affine projection algorithm and kernel affine projection algorithm.
Highlights
Introduction to fractional derivativeFractional calculus is a widely used signal processing algorithm autoregressive (AR) systems identification
Kernel principal component analysis (KPCA) and kernel regression (Scholkopf et al 1997; Takeda et al 2007; Hardle and Vieu 1992) show desirable performance regarding classification in the complicated environment of statistical signal processing. These are batch mode methods and suffer the burden of high computational cost and memory usage. These issues are replaced by introducing the online kernel methods, such as kernel least mean square (KLMS) (Liu et al 2008), kernel affine projection algorithm (KAPA) (Liu and Principe 2008), kernel recursive least squares (KRLS) (Engel et al 2004; Liu et al 2015) and extended kernel recursive least squares (Ex-KRLS) (Liu et al 2009) algorithms
To remove the guesswork existing in tuning the step size parameter of the modified fractional least mean square algorithm (MFLMS) algorithm (Shoaib and Qureshi 2014b), a stochastic gradient-based method is introduced to adapt the step sizes of the MFLMS algorithm according to the mean square error and its application towards the prediction of Mackey glass as well as Lorenz time series
Summary
Kernel-based learning algorithms gained interest since the last few years. Mercer’s theorem is used in kernel-based learning algorithms to map the input data using some nonlinear kernel function to some higher dimensional feature space, known as reproducing kernel Hilbert space (RKHS), where the linear operations are performed on the input data. Kernel principal component analysis (KPCA) and kernel regression (Scholkopf et al 1997; Takeda et al 2007; Hardle and Vieu 1992) show desirable performance regarding classification in the complicated environment of statistical signal processing These are batch mode methods and suffer the burden of high computational cost and memory usage. To remove the guesswork existing in tuning the step size parameter of the MFLMS algorithm (Shoaib and Qureshi 2014b), a stochastic gradient-based method is introduced to adapt the step sizes of the MFLMS algorithm according to the mean square error and its application towards the prediction of Mackey glass as well as Lorenz time series. In this paper we introduced a mechanism that combined the adaptive fractional learning algorithms and online kernel-based filtering algorithms This idea greatly helps in improving the performance in solving nonlinear problems. The experimental results are discussed in “Experimental results” and “Conclusion and future work” sections comprises of conclusion along with future directions
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