Hurst Exponent
Written by Maria Perica, 3/22/24
The Hurst exponent is a measure of trends within time series over time, or the long-term memory/self-similarity of a time series. The value of the Hurst exponent can tell you how predictable future values of a time series are based on past values, i.e. does the time-series follow a mean-reverting pattern over time, does it cluster in a certain direction or trend in a certain direction, or is it random and relatively unpredictable?
There are a number of methods to calculate the Hurst exponent or estimate Hurst parameters. The first method proposed historically is the rescaled range (or R/S). While the R/S is a very commonly used method and superior to some methods for calculating Hurst, it is sensitive to non-stationarities, and thus may not be the best choice for potentially non-stationary time series.
A potentially better method for non-stationary data is detrended fluctuation analysis (or DFA). DFA allows for quantification of self-similarity, like R/S, but is more robust to non-stationary data. It has been suggested that fMRI BOLD data is likely non-stationary, and thus DFA may be a better way of detecting true trends in BOLD time courses as compared to R/S. When doing a DFA analysis, you obtain an 'alpha' parameter. Higher values of the DFA alpha parameter suggest more correlation over different scales, while lower values suggest less correlation.
Some paper examples using DFA with BOLD data: https://www.sciencedirect.com/science/article/pii/S1053811920307102?via%3Dihub, https://onlinelibrary.wiley.com/doi/full/10.1002/hbm.25030
It has been suggested that the Hurst exponent can be a measure of E/I: https://elifesciences.org/articles/55684, https://www.nature.com/articles/s41467-023-41686-9
Higher values of Hurst have been suggested to reflect relatively more inhibition compared to excitation. Therefore, more neural inhibition may be associated with more correlation in a BOLD time series.