Statistics and modelling of order patterns in univariate time series

TL;DR

This paper develops a statistical framework for analyzing order patterns in univariate time series, introducing orthogonal contrasts and applying them to EEG sleep data, permutation entropy, and stationary processes.

Contribution

It introduces an orthogonal system of pattern contrasts for length 3, enabling independent statistical analysis and applications to EEG and permutation entropy.

Findings

Orthogonal pattern contrasts are statistically independent.

Turning rate effectively evaluates sleep depth from EEG.

Permutation entropy fluctuations are analyzed with new models.

Abstract

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. 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 turning rate. It can be used to evaluate sleep depth directly from EEG data. The paper discusses fluctuations of permutation entropy, statistical tests, and the need of new models for noises like EEG. We show how ordinal stationary processes can be constructed without any numerical values. An order by coin-tossing is a natural example. Every partially stationary probability measure on patterns of length $m$ can be extended to a stationary measure on patterns of infinite length.