Adaptive step-size modified fractional least mean square algorithm for chaotic time series prediction

This paper presents an adaptive step-size modified fractional least mean square （AMFLMS） algorithm to deal with a nonlinear time series prediction. Here we incorporate adaptive gain parameters in the weight adaptation equation of the original MFLMS algorithm and also introduce a mechanism to adjust the order of the fractional derivative adaptively through a gradient-based approach. This approach permits an interesting achievement towards the performance of the filter in terms of handling nonlinear problems and it achieves less computational burden by avoiding the manual selection of adjustable parameters. We call this new algorithm the AMFLMS algorithm. The predictive performance for the nonlinear chaotic Mackey Glass and Lorenz time series was observed and evaluated using the classical LMS, Kernel LMS, MFLMS, and the AMFLMS filters. The simulation results for the Mackey glass time series, both without and with noise, confirm an improvement in terms of mean square error for the proposed algorithm. Its performance is also validated through the prediction of complex Lorenz series.     

A modified fractional least mean square algorithm for chaotic and nonstationary time series prediction

A method of modifying the architecture of fractional least mean square （FLMS） algorithm is presented to work with nonlinear time series prediction. Here we incorporate an adjustable gain parameter in the weight adaptation equation of the original FLMS algorithm and absorb the gamma function in the fractional step size parameter. This approach provides an interesting achievement in the performance of the filter in terms of handling the nonlinear problems with less computational burden by avoiding the evaluation of complex gamma function. We call this new algorithm as the modified fractional least mean square （MFLMS） algorithm. The predictive performance for the nonlinear Mackey glass chaotic time series is observed and evaluated using the classical LMS, FLMS, kernel LMS, and proposed MFLMS adaptive filters. The simulation results for the time series with and without noise confirm the superiority and improvement in the prediction capability of the proposed MFLMS predictor over its counterparts.