Evaluating Temporal Correlations in Time Series Using Permutation Entropy, Ordinal Probabilities and Machine Learning - Bruno R. R. Boaretto; Roberto C. Budzinski; Kalel L. Rossi; Thiago L. Prado; Sergio R. Lopes; Cristina Masoller (Entropy, 2021)
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, <semantics>α</semantics>, of flicker noise (FN) time series. The algorithm is trained using as input features the probabilities of ordinal patterns computed from FN time series, <semantics><msubsup>xαFN</msubsup>(t)</semantics>, generated with different values of <semantics>α</semantics>. Then, the ordinal probabilities computed from the time series of interest, <semantics>x(t)</semantics>, are used as input features to the trained algorithm and that returns a value, <semantics>αe</semantics>, that contains meaningful information about the temporal correlations present in <semantics>x(t)</semantics>. We have also shown that the difference, <semantics>Ω</semantics>, of the permutation entropy (PE) of the time series of interest, <semantics>x(t)</semantics>, and the PE of a FN time series generated with <semantics>α=αe</semantics>, <semantics><msubsup>xαeFN</msubsup>(t)</semantics>, allows the identification of the underlying determinism in <semantics>x(t)</semantics>. Here, we apply our methodology to different datasets and analyze how <semantics>αe</semantics> and <semantics>Ω</semantics> 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.