Calculate the Hurst coefficient, range of cumulative deviations, standard deviation, or window size from the other three inputs in one step.
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Hurst Coefficient Formula
The calculator uses a single-scale rescaled range approximation for the Hurst coefficient:
Rearranged forms are used when you leave a different field blank:
- H = Hurst coefficient, unitless
- R = range of cumulative deviations
- S = standard deviation
- T = window size, measured as the number of observations
To calculate H, enter R, S, and T. The calculator divides R by S, takes the logarithm of that ratio, then divides by the logarithm of T.
To calculate R, enter H, S, and T. The calculator raises T to the power of H, then multiplies by S.
To calculate S, enter H, R, and T. The calculator divides R by T raised to the power of H.
To calculate T, enter H, R, and S. The calculator raises the ratio R/S to the power of 1/H.
R and S must use compatible units because the ratio R/S needs to be unitless. If you select length units, the calculator converts R and S to common units before calculating.
Interpreting Hurst Coefficient Values
| H value | Common interpretation | What it suggests about the series |
|---|---|---|
| H < 0.5 | Anti-persistent | Increases are more likely to be followed by decreases, and decreases by increases. |
| H = 0.5 | Random walk-like | The series has little evidence of long-term dependence. |
| H > 0.5 | Persistent | Increases tend to be followed by increases, and decreases by decreases. |
| H near 1 | Strong persistence | The series may show strong trend-following behavior over the tested scale. |
Input Requirements and Unit Notes
| Input | Requirement | Note |
|---|---|---|
| Hurst coefficient, H | Unitless | Cannot be zero when solving for T. |
| Range, R | Positive value | Use the same type of unit as S, such as meters with meters or feet with feet. |
| Standard deviation, S | Positive value | S cannot be zero because R/S is used in the formula. |
| Window size, T | Number of observations | T should be greater than 1 when calculating H because log(T) is in the denominator. |
Example Calculations
Example 1: Calculate the Hurst coefficient
Suppose the range of cumulative deviations is 12, the standard deviation is 3, and the window size is 16 observations.
The Hurst coefficient is approximately 0.5000.
Example 2: Calculate the range of cumulative deviations
Suppose H = 0.7, S = 2.5, and T = 20 observations.
The range of cumulative deviations is approximately 20.3545.
FAQ
What does the Hurst coefficient measure?
The Hurst coefficient measures long-term dependence or memory in a time series. A value below 0.5 suggests mean-reverting or anti-persistent behavior. A value near 0.5 suggests random walk-like behavior. A value above 0.5 suggests persistent behavior, where movements tend to continue in the same direction.
Why does the calculator use R divided by S?
The rescaled range method compares the range of cumulative deviations, R, with the standard deviation, S. Dividing R by S normalizes the range so it can be compared across different scales or units. The Hurst coefficient is then estimated from how that rescaled range changes with the window size T.
Is this the same as a full Hurst exponent estimate?
This calculator uses a single-scale approximation: H ≈ log(R/S) / log(T). A full rescaled range analysis usually calculates R/S across several window sizes and estimates H from the slope of a log-log regression. The single-scale version is useful for quick checks, but it may be less stable than a multi-scale estimate.