Working Paper Article Version 1 This version is not peer-reviewed

# Small Order Patterns in Big Time Series: A Practical Guide

Version 1 : Received: 30 April 2019 / Approved: 5 May 2019 / Online: 5 May 2019 (12:50:33 CEST)

A peer-reviewed article of this Preprint also exists.

Bandt, C. Small Order Patterns in Big Time Series: A Practical Guide. Entropy 2019, 21, 613. Bandt, C. Small Order Patterns in Big Time Series: A Practical Guide. Entropy 2019, 21, 613.

Journal reference: Entropy 2019, 21, 613
DOI: 10.3390/e21060613

## Abstract

The study of order patterns of three equally spaced values $x_t,x_{t+d},x_{t+2d}$ in a time series is a powerful tool. The lag $d$ is changed in a wide range so that differences of frequencies of order patterns become autocorrelation functions. Similar to a spectrogram in speech analysis, four ordinal autocorrelation functions are used to visualize big data series, as for instance heart and brain activity over many hours. The method applies to real data without preprocessing, outliers and missing data do not matter. On the theoretical side, we study properties of order correlation functions and show that the four autocorrelation functions are orthogonal in a certain sense. An analysis of variance of a modified permutation entropy can be performed with four variance components associated with the functions.

## Subject Areas

permutation entropy; autocorrelation; time series; order pattern