Iterative Reservoir Computing Networks for Reconstructing Irregular Time Series

Time series data with missing entries are ubiquitous in a broad spectrum of practical and clinical applications, from climatology and cell biology to personalized medicine. This undesired structure arising either due to undesired artifacts (e.g., noise) or by design (e.g., asynchronous or aperiodic sampling in distributed sensors) results in irregularity in the temporal dimension and forms a bottleneck in data mining. Although extensive data science approaches have been proposed to address learning problems involving irregular data, the emphasis was largely placed on filling in the missing entries via interpolation and binning, or the methods were tailored to specific data analytic tasks. In this article, we develop a reservoir computing (RC)-based iterative learning method for recovering missing data in irregular time series generated by dynamical systems and networks. In particular, we formulate this learning task as a fixed-point iterative learning problem and develop a training procedure using an RC network (RCN). We find that when the irregular time series has "sufficient" samples to train an RCN within a tolerant training error then the missing samples in the time series can be recovered systematically. We also derive sufficient conditions with respect to the choices of the reservoir parameters that guarantee the convergence of the iterative procedure. We present several numerical experiments to demonstrate the efficacy of the developed iterative RCN approach. Specifically, we illustrate the capability of our approach to recover missing data in irregular time series generated by chaotic Rössler and Kuramoto-Sivashinsky (KS) systems. Finally, we also report the results of incorporating our approach in an irregular medical data classification task.