Fractional-integer-order echo state network for time series prediction

In this paper, a new echo state network with fractional -order reservoir and integer -order reservoir in series configuration, called fractional -integer -order ESN (FIO-ESN), is proposed for time series prediction. Firstly, considering the infinite memory of fractional -order reservoir, the feature information of input signals will be amplified through the fractional -order reservoir, and then the magnified feature information can be extracted twice by using the integer -order reservoir with very large input weights. Secondly, the magnitude of the fractional -order reservoir state is increased through the integer -order reservoir, and then the output weight can be computed in a reasonable range. Thirdly, in order to realize the stable application of the FIO-ESN, a sufficient stability criterion for the FIO-ESN is given by using an LMI approach. Fourthly, in order to reduce the dependence of the prediction accuracy of the FIO-ESN on the fractional -integer -order reservoir parameters, an optimization algorithm based on gradient descent is given. Finally, two numerical simulation examples and one real -world example are used for demonstrating the feasibility of stability criterion and the learning performance of the FIO-ESN.