Hurst Coefficient Calculator
Understanding how a time series behaves — whether it trends, mean-reverts, or behaves randomly — is critical in areas like:
- Finance (e.g., stock price analysis)
- Geophysics
- Econometrics
- Hydrology
- Fractal analysis
The Hurst exponent or Hurst coefficient (H) measures the long-term memory or dependence in a dataset. First introduced by Harold Edwin Hurst, it quantifies whether a time series tends to regress to the mean or persist in a direction.
The Hurst Coefficient Calculator provides a quick estimation of this value using a simplified R/S method.
📊 Formula (Plain Text)
The simplified method uses:
H = log(R/S) / log(n)
Where:
- R/S = Rescaled range (range of cumulative deviations divided by standard deviation)
- n = Length of the time series
- H = Hurst exponent
This formula gives an estimate of the Hurst exponent, which falls between 0 and 1:
- H < 0.5: Mean-reverting (anti-persistent)
- H = 0.5: Random walk (Brownian motion)
- H > 0.5: Trending (persistent)
✅ How to Use the Calculator
- Enter the Sample Size (n)
The number of data points in your time series. - Enter the Rescaled Range (R/S)
You can calculate this from your dataset or estimate from a statistical tool. - Click “Calculate”
The calculator returns your Hurst coefficient.
🧮 Example Calculation
Let’s say:
- Sample size (n) = 100
- Rescaled range (R/S) = 5.23
Then:
- H = log(5.23) / log(100) ≈ 0.718
Since H > 0.5, this time series tends to trend, showing persistent behavior.
🔍 Interpretation of Hurst Exponent
| H Value | Interpretation | Behavior Type |
|---|---|---|
| < 0.5 | Anti-persistent | Mean-reverting |
| = 0.5 | Uncorrelated (random walk) | Purely random behavior |
| > 0.5 | Persistent | Trending / Long memory |
🧠 Applications of the Hurst Exponent
- Finance: Determine if a stock price series follows a random walk or trends.
- Weather Patterns: Detect climate persistence.
- Network Traffic: Analyze fractal patterns in data transfer rates.
- Hydrology: Originally developed to study Nile River flood trends.
- Cryptocurrency Analysis: Determine if crypto assets trend or revert.
❓ FAQs About Hurst Coefficient Calculator
1. What is the Hurst exponent?
A statistical measure indicating the tendency of a time series to trend, revert, or behave randomly.
2. What does H = 0.5 mean?
The data behaves like a random walk — no memory or trend.
3. Is this method precise?
It’s a simplified estimation. Advanced methods (e.g., DFA or R/S resampling) provide higher precision.
4. What is the rescaled range (R/S)?
The range of cumulative deviations from the mean divided by the standard deviation.
5. How do I calculate R/S?
You'll need to compute the cumulative deviation from the mean, find the range, and divide by the standard deviation of the series.
6. Can I use this for stock prices?
Yes — it’s commonly used to test if a financial asset trends or behaves randomly.
7. What if I get H > 1?
That’s invalid in most contexts — check your inputs. H should be between 0 and 1.
8. What does a low H (< 0.5) mean?
The series is mean-reverting. After going up, it’s likely to come down and vice versa.
9. Is this useful in crypto?
Yes — many traders analyze Hurst values for volatility and trend strength in crypto.
10. How does this relate to fractals?
The Hurst exponent is linked to fractal dimension — a higher H implies a smoother fractal path.
11. What does persistent behavior mean?
If a data point goes up, it’s likely the next point also goes up. The trend is more likely to continue.
12. Can H be exactly 1?
In theory, yes — but it usually indicates perfect predictability or a constant trend, which is rare.
13. What sample size is best?
Larger samples (n > 100) yield more stable and reliable estimates.
14. Is H used in machine learning?
It can be. H values are sometimes used as features for time series models or anomaly detection.
15. How does this relate to volatility?
Trending series (H > 0.5) may show lower volatility spikes but longer directional moves.
16. Can I use Excel to compute R/S?
Yes, though it’s manual. Scripts in Python, R, or MATLAB are faster for large datasets.
17. Is H affected by outliers?
Yes. Outliers can distort the range, affecting the R/S and thus the H value.
18. Is this only for time series?
Yes. Hurst exponent analysis requires data that is ordered in time.
19. Can I use this for rainfall, temperature, or wind patterns?
Definitely. It’s been widely applied in meteorological and climate sciences.
20. Is this tool academic or practical?
Both. Academics use it in research, while traders and engineers apply it for practical forecasting.
✅ Conclusion
The Hurst Coefficient Calculator offers a quick and accessible way to estimate the memory characteristics of a time series. Whether you're analyzing stock market behavior, weather patterns, or data stream reliability, the Hurst exponent provides deep insight into how predictable or random your system is.