Order patterns, their variation and change points in financial time series and Brownian motion

TL;DR

This paper investigates order patterns and permutation entropy in financial time series and Brownian motion, highlighting stable pattern frequencies at small lags and identifying key parameters for change point detection.

Contribution

It introduces a new analysis of order patterns in financial data and Brownian motion, emphasizing the roles of turning rate and up-down balance for change point detection.

Findings

Pattern frequencies remain constant for small lags in financial data.

Turning rate is effective for change point detection in EEG data.

Up-down balance is most effective for change point detection in financial data.

Abstract

Order patterns and permutation entropy have become useful tools for studying biomedical, geophysical or climate time series. Here we study day-to-day market data, and Brownian motion which is a good model for their order patterns. A crucial point is that for small lags (1 up to 6 days), pattern frequencies in financial data remain essentially constant. The two most important order parameters of a time series are turning rate and up-down balance. For change points in EEG brain data, turning rate is excellent while for financial data, up-down balance seems the best. The fit of Brownian motion with respect to these parameters is tested, providing a new version of a forgotten test by Bienaym'e.