Using missing ordinal patterns to detect nonlinearity in time series data

Abstract

The number of missing ordinal patterns (NMP) is the number of ordinal patterns that do not appear in a series after it has been symbolized using the Bandt and Pompe methodology. In this paper, the NMP is demonstrated as a test for nonlinearity using a surrogate framework in order to see if the NMP for a series is statistically different from the NMP of iterative amplitude adjusted Fourier transform (IAAFT) surrogates. It is found that the NMP works well as a test statistic for nonlinearity, even in the cases of very short time series. Both model and experimental time series are used to demonstrate the efficacy of the NMP as a test for nonlinearity.

Decreased electrocortical temporal complexity distinguishes sleep from wakefulness

“…In contrast, the field of non-linear dynamics has developed measures and models that account for the complexity of the systems and their emerging interactions. [10][11][12][13] These properties are fundamental for the characterization of the thalamo-cortical function and for the emergence of consciousness. 14 A general approach to study time-signals is the characterization of their randomness; for example, by means of the Shannon entropy (SE), 15 which measures the average unpredictability of a signal.…”

“…In contrast to other methods, PeEn is a time-series complexity measure that is simple to implement, is robust to noise and short time-series, and works for arbitrary data sets. 13,16,17,[20][21][22][23][24][25][26] In particular, it has been shown that PeEn applied to EEG signals captures different states associated with the level of consciousness, both during anesthesia [26][27][28][29] and sleep. 30,31 Hence, in order to study the thalamo-cortical function during W and sleep, PeEn is a practical and reliable method, where results can be understood from primary principles, and can be related to the signal characteristics.…”

Is human atrial fibrillation stochastic or deterministic?—Insights from missing ordinal patterns and causal entropy-complexity plane analysis

“…The presence of a higher number of MOPs in the time series under examination compared to IAAFT surrogate data distinguishes deterministic time series from correlated stochastic time series. 41 IAAFT surrogate data are generated such that the surrogate data have the same power spectrum, autocorrelations, and probability distribution with the original time series. As a result, the derived surrogates have the same probability distribution and power spectrum with potential high order correlations being randomized.…”

“…44 To overcome this limitation, Kulp et al used IAAFT surrogate data and the BP MOP paradigm, to detect non-linear determinism in both theoretical and experimental deterministic time series and distinguish them from correlated stochastic time series. 41 There are no studies to date evaluating the ability of the MOP paradigm (as a stand-alone method or in a surrogate data framework) in distinguishing determinism from nonlinear stochastic processes.…”

“…61 The IAAFT surrogate data have been used with the BP MOP paradigm successfully, in both theoretical and experimental settings. 41 We used the causal entropy-complexity plane of the BP symbolization of the AA time series and the fBm to differentiate deterministic time series from highly correlated noise. 17,38 In addition, we used rigorous statistical analysis to quantify our findings using a relatively large sample size of AF episodes.…”