Abstract
Information dynamics provides a broad set of measures for characterizing how a dynamical system stores, processes, and transmits information. While estimators for these measures are commonly used in applications, the statistical properties of these estimators for finite time series are not well understood. In particular, the precision of a given estimate is generally unknown. We develop confidence intervals for generic information-dynamic parameters using a bootstrap procedure. The bootstrap procedure uses an echo state network, a particular instance of a reservoir computer, as a simulator to generate bootstrap samples from a given time series. We perform a Monte Carlo analysis to investigate the performance of the bootstrap confidence intervals in terms of their coverage and expected lengths with two model systems and compare their performance to a simulator based on the random analog predictor. We find that our bootstrap procedure generates confidence intervals with nominal, or near nominal, coverage of the information-dynamic measures, with smaller expected length than the random analog predictor-based confidence intervals. Finally, we demonstrate the applicability of the confidence intervals for characterizing the information dynamics of a time series of sunspot counts.
Highlights
Information dynamics provides a set of tools for the analysis of systems that evolve in time
We have developed a method for constructing bootstrap con dence intervals for information-dynamic properties of a stochastic dynamical system using an echo state network as a simulator for use in the bootstrap procedure
The echo state network is an especially useful simulator in this setting, since it is both straightforward to t to the observed time series, requiring only a regularized linear regression, and e cient to simulate from, requiring only a linear update to the system state passed through a single activation function at each time step
Summary
Information dynamics provides a set of tools for the analysis of systems that evolve in time. Echo state networks are an ideal candidate to use as a simulator for bootstrapping the sampling distribution of the estimator of an information-dynamic measure for several reasons They are computationally inexpensive to t to a time series, with a cost that scales quadratically in the length of the time series. The value of chosen by this model selection criterion will be used for the estimation of both the entropy rate and active information storage, since both measures are related to the amount of the past necessary to optimally predict the next-step future. Where ρt,k is the distance to the kth nearest neighbor of (Xt− :t, Xt) under the in nity norm, and NS(t; ρt,k) is the number of sample points within a distance ρt,k of the appropriate subcomponent of (Xt− :t, Xt) in the space S.6,46,47 Both the entropy rate and active information storage estimators require a choice of k for the k-nearest neighbor density estimator implicit in their construction. This is a conservative choice and will result in the widest possible con dence intervals
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