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Reservoir Computing (RC) is a type of recursive neural network (RNN), andthere can be no doubt that the RC will be more and more widely used forbuilding future prediction models for time-series data, with low training cost,high speed and high computational power. However, research into themathematical structure of RC neural networks has only recently begun. Bollt(2021) clarified the necessity of the autoregressive (AR) model for gaining theinsight into the mathematical structure of RC neural networks, and indicatedthat the Wold decomposition theorem is the milestone for understanding ofthese. Keeping this celebrated result in mind, in this paper, we clarify hiddenstructures of input and recurrent weight matrices in RC neural networks, andshow that such structures attain perfect prediction for the AR type of timeseries data.

Mathematical structure of perfect predictive reservoir computing for autoregressive type of time series data