Deciphering Dynamical Nonlinearities in Short Time Series Using Recurrent Neural Networks

Radhakrishnan Nagarajan

doi:10.1038/s41598-019-50625-y

Abstract

Surrogate testing techniques have been used widely to investigate the presence of dynamical nonlinearities, an essential ingredient of deterministic chaotic processes. Traditional surrogate testing subscribes to statistical hypothesis testing and investigates potential differences in discriminant statistics between the given empirical sample and its surrogate counterparts. The choice and estimation of the discriminant statistics can be challenging across short time series. Also, conclusion based on a single empirical sample is an inherent limitation. The present study proposes a recurrent neural network classification framework that uses the raw time series obviating the need for discriminant statistic while accommodating multiple time series realizations for enhanced generalizability of the findings. The results are demonstrated on short time series with lengths (L = 32, 64, 128) from continuous and discrete dynamical systems in chaotic regimes, nonlinear transform of linearly correlated noise and experimental data. Accuracy of the classifier is shown to be markedly higher than ≫50% for the processes in chaotic regimes whereas those of nonlinearly correlated noise were around ~50% similar to that of random guess from a one-sample binomial test. These results are promising and elucidate the usefulness of the proposed framework in identifying potential dynamical nonlinearities from short experimental time series.

Highlights

As in the case of single empirical sample, if the multiple time series realizations are sufficiently long it might be possible to statistically compare the distribution of discriminant statistic estimates on the given cohort to those estimated on its paired surrogate realizations addressing the null hypothesis that there is no significant difference in the discriminant estimates between the cohort and its surrogate counterpart, Fig. 1b
This is especially helpful across small lengths such as those discussed in the present study (L = 32, 64, 128) where estimation of discriminant statistics[37] can be challenging and unreliable. (b) It poses the classical statistical surrogate testing Fig. 1a,b, as a binary classification problem, Fig. 1c, using recurrent neural networks (RNN), Fig. 2, where the two classes of interest correspond to the multiple time series realizations from a given cohort and their corresponding Iterated Amplitude Adjusted Fourier Transform (IAAFT) surrogate counterparts
Traditional surrogate testing while helpful has inherent limitations. It subscribes to statistical hypothesis testing and investigates the separation of a chosen discriminant statistic or dynamical invariant between the given empirical sample and its surrogate counterpart

Summary

Introduction

While the choice of empirical sample can be attributed to implicit ergodic assumptions[36], generating long time series so as to enable robust estimation of dynamical invariants and discriminant statistics can be especially challenging in experimental settings as it demands controlling a number of factors Experimental time series such as those from physiological systems have been especially known to exhibit variations between subjects within a given disease group or cohort. As in the case of single empirical sample, if the multiple time series realizations are sufficiently long it might be possible to statistically compare the distribution of discriminant statistic estimates on the given cohort to those estimated on its paired surrogate realizations addressing the null hypothesis that there is no significant difference in the discriminant estimates between the cohort and its surrogate counterpart, Fig. 1b.

Methods

Results

Conclusion