Rescaled Range Method for Calculating Hurst Exponent

The calculation of Hurst exponent using the rescaled range method involves the following steps:

1. Perform rescaled range calculation on the data series to obtain a sequence of R/S values. This typically involves dividing the time series into sub-periods and calculating the range divided by standard deviation for each segment.

2. Compute the mean and standard deviation of the rescaled range values. The mean represents the average scaling behavior while the standard deviation measures the variability in the R/S values.

3. Normalize each rescaled range value by subtracting the mean and dividing by the standard deviation. This standardization process ensures consistent scaling across different data sets.

4. Accumulate the normalized values to derive the Hurst exponent. The Hurst exponent (H) is obtained through linear regression of log(R/S) against log(n), where n is the time scale, with H indicating the persistence (H>0.5) or anti-persistence (H<0.5) of the time series.

To implement this operation, the following modular subroutines are required:

- Rescaled range calculation subroutine: Handles data segmentation, range computation, and standard deviation calculation for each time window

- Mean and standard deviation computation subroutine: Statistical module that calculates central tendency and dispersion measures

- Numerical processing subroutine: Performs data normalization and standardization operations

- Hurst exponent accumulation subroutine: Implements the logarithmic regression and exponent extraction algorithm

By combining these subroutines, we can efficiently compute the Hurst exponent and conduct deeper analysis and research on time series data, particularly useful for financial market analysis, hydrological studies, and long-memory process identification.