Evaluating Temporal Correlations in Time Series Using Permutation Entropy, Ordinal Probabilities and Machine Learning.

Bruno R R Boaretto,Thiago L Prado,Sergio R Lopes,Kalel L Rossi,Roberto C Budzinski,Cristina Masoller

doi:10.3390/e23081025

Abstract

Time series analysis comprises a wide repertoire of methods for extracting information from data sets. Despite great advances in time series analysis, identifying and quantifying the strength of nonlinear temporal correlations remain a challenge. We have recently proposed a new method based on training a machine learning algorithm to predict the temporal correlation parameter, , of flicker noise (FN) time series. The algorithm is trained using as input features the probabilities of ordinal patterns computed from FN time series, , generated with different values of . Then, the ordinal probabilities computed from the time series of interest, , are used as input features to the trained algorithm and that returns a value, , that contains meaningful information about the temporal correlations present in . We have also shown that the difference, , of the permutation entropy (PE) of the time series of interest, , and the PE of a FN time series generated with , , allows the identification of the underlying determinism in . Here, we apply our methodology to different datasets and analyze how and correlate with well-known quantifiers of chaos and complexity. We also discuss the limitations for identifying determinism in highly chaotic time series and in periodic time series contaminated by noise. The open source algorithm is available on Github.

Highlights

Thanks to huge advances in data science and computing power, a wide repertoire of time series analysis methods [1,2,3,4,5,6,7] are available for the quantitative characterization of time series and are routinely used in all fields of science and technology, social sciences, economy and finance, etc
We address the following questions: Can we distinguish a highly chaotic time series from a stochastic one? Can we identify a periodic signal hidden by noise? In addition, to gain insight into the information encapsulated by αe and Ω, we contrast them with well known quantifiers of chaos and complexity: the maximum Lyapunov exponent and the ordinal-based statistical complexity measure [25]
In both panels the black line represents flicker noise (FN) signals generated with different values of α, which are perfectly recovered by the artificial neural network (ANN)

Summary

Introduction

Thanks to huge advances in data science and computing power, a wide repertoire of time series analysis methods [1,2,3,4,5,6,7] are available for the quantitative characterization of time series and are routinely used in all fields of science and technology, social sciences, economy and finance, etc. As any algorithm will return, at least, a number (i.e., a “feature” that encapsulates some property of the time series), in order to interpret the information in the obtained features and to assess the performance of different algorithms, appropriate surrogates [8] or a “reference model” (where the systems that generates the data are known) need to be used. A FN time series, xαFN (t), is characterized by a power spectrum

Methods

Results

Conclusion