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

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 Bienaymé.