Non-parametric analysis of serial dependence in time series using ordinal patterns

Statistics and contrasts of order patterns in univariate time series

Order patterns apply well to many fields, because of minimal stationarity assumptions. Here, we fix the methodology of patterns of length 3 by introducing an orthogonal system of four pattern contrasts, that is, weighted differences of pattern frequencies. These contrasts are statistically independent and turn up as eigenvectors of a covariance matrix both in the independence model and the random walk model. The most important contrast is the turning rate. It can be used to evaluate sleep depth directly from EEG (electroencephalographic brain data). The paper discusses fluctuations of permutation entropy, statistical tests, and the need of new models for noises like EEG.

Permutation group entropy: A new route to complexity for real-valued processes

This is a review of group entropy and its application to permutation complexity. Specifically, we revisit a new approach to the notion of complexity in the time series analysis based on both permutation entropy and group entropy. As a result, the permutation entropy rate can be extended from deterministic dynamics to random processes. More generally, our approach provides a unified framework to discuss chaotic and random behaviors.

Adaptation of Partial Mutual Information from Mixed Embedding to Discrete-Valued Time Series

A causality analysis aims at estimating the interactions of the observed variables and subsequently the connectivity structure of the observed dynamical system or stochastic process. The partial mutual information from mixed embedding (PMIME) is found appropriate for the causality analysis of continuous-valued time series, even of high dimension, as it applies a dimension reduction by selecting the most relevant lag variables of all the observed variables to the response, using conditional mutual information (CMI). The presence of lag components of the driving variable in this vector implies a direct causal (driving-response) effect. In this study, the PMIME is appropriately adapted to discrete-valued multivariate time series, called the discrete PMIME (DPMIME). An appropriate estimation of the discrete probability distributions and CMI for discrete variables is implemented in the DPMIME. Further, the asymptotic distribution of the estimated CMI is derived, allowing for a parametric significance test for the CMI in the DPMIME, whereas for the PMIME, there is no parametric test for the CMI and the test is performed using resampling. Monte Carlo simulations are performed using different generating systems of discrete-valued time series. The simulation suggests that the parametric significance test for the CMI in the progressive algorithm of the DPMIME is compared favorably to the corresponding resampling significance test, and the accuracy of the DPMIME in the estimation of direct causality converges with the time-series length to the accuracy of the PMIME. Further, the DPMIME is used to investigate whether the global financial crisis has an effect on the causality network of the financial world market.

Non-parametric tests for serial dependence in time series based on asymptotic implementations of ordinal-pattern statistics

Ordinal patterns can be used to construct non-parametric hypothesis tests that aim to discover (possibly non-linear) serial dependence in a real-valued time series. We derive the asymptotic distribution of the vector of sample frequencies of ordinal patterns and that of various corresponding tests statistics such that the targeted tests for serial dependence are easily implemented based on asymptotic approximations. Simulations are used to check the finite-sample performance of these tests as well as their power properties with respect to various alternative scenarios. The application and interpretation of the tests in practice are illustrated by an environmental data example.

Assessing serial dependence in ordinal patterns processes using chi-squared tests with application to EEG data analysis

We extend Elsinger's work on chi-squared tests for independence using ordinal patterns and investigate the general class of m-dependent ordinal patterns processes, to which belong ordinal patterns processes derived from random walk, white noise, and moving average processes. We describe chi-squared asymptotically distributed statistics for such processes that take into account necessary constraints on ordinal patterns probabilities and propose a test for m-dependence, with which we are able to quantify the range of serial dependence in a process. We apply the test to epilepsy electroencephalography time series data and observe shorter m-dependence associated with seizures, suggesting that the range of serial dependence decreases during those events.