Turn noisy signals into a clear Hurst estimate fast with robust options. Compare scaling across windows, validate fit quality, and download results as files.
Example input values and the kind of output you should expect.
| Index | Signal value | Comment |
|---|---|---|
| 1 | 0.12 | Small fluctuation around a baseline |
| 2 | 0.18 | Noise with mild persistence |
| 3 | 0.27 | Short burst typical in physical signals |
| 4 | 0.31 | Trend-like behavior may bias naive methods |
| 5 | 0.29 | Local reversal can indicate anti-persistence |
Split the signal into windows of length s. For each window, compute the cumulative deviation
Y(k)=∑(xᵢ−x̄), the range R=max(Y)−min(Y), and the standard deviation S.
The statistic R/S scales as:
R/S(s) ∝ sᴴ → fit slope of ln(R/S) vs ln(s).
Demean the series and integrate to a profile y(k)=∑(xᵢ−x̄). In each window size s,
fit a polynomial trend (order 1 or 2) and compute the RMS fluctuation F(s).
F(s) ∝ sᴴ → fit slope of ln(F) vs ln(s).
Form non-overlapping block means of size m and compute Var(block mean).
For long-memory increment processes, the scaling behaves approximately as:
Var(m) ∝ m^(2H−2) → compute H=(slope+2)/2 from log–log fit.
Many measurements are not memoryless. Turbulent velocity traces, plasma fluctuations, seismic noise, and surface roughness profiles can carry correlations across wide scales. The Hurst exponent condenses that behavior into one statistic, helping you compare runs, identify regime shifts, and quantify complexity using consistent settings. In practice, it complements spectra and autocorrelation by emphasizing scaling.
H near 0.50 suggests weak long‑range correlation within the fitted window range. Values above about 0.60 imply persistence, where increases tend to follow increases, while values below about 0.40 suggest anti‑persistence and frequent reversals. Estimates near 0.2 or 0.9 often require careful artifact checks.
Rescaled range (R/S) computes the range of cumulative deviations divided by the window standard deviation and fits ln(R/S) versus ln(s). DFA integrates the demeaned series, detrends each window, then fits ln(F) versus ln(s). Variance‑time estimates H from block‑mean variance scaling as a fast cross‑check.
A stable slope needs multiple window sizes and enough segments per window. Start with 10–14 scales and keep the maximum window at or below N/4 so at least four segments contribute. For uncertainty near ±0.05, 200–1000 samples often outperform short records, especially when noise is heavy‑tailed.
Sensor drift, heating, or background changes can mimic persistence. DFA reduces this bias by detrending locally; order 2 can help when curvature remains. If your signal is strictly positive and spans large ranges, log returns can stabilize variance. Avoid log returns when values cross or touch zero.
The log–log regression provides R² and an approximate 95% confidence interval from slope uncertainty. High R² suggests cleaner scaling, but also inspect the scaling table for monotonic behavior across adjacent windows. If only two scales dominate, widen the scale set or shorten the maximum window to improve robustness. Refit after adjustments and confirm the slope remains stable.
Record the method, preprocessing choice, minimum and maximum windows, and the number of scales used. Report H with its interval and R², then export CSV to archive the scaling table alongside raw data. Attach the PDF summary to lab notes so another analyst can reproduce the estimate exactly. Include units, sampling rate, and any filtering to avoid hidden processing differences later.
Avoid interpreting H from extremely short series, strongly periodic signals, or heavily quantized measurements without additional checks. If methods disagree, treat that as information: trends, regime changes, or mixed processes may be present. With transparent window choices and exported diagnostics, H becomes a comparable descriptor of dynamics.
It indicates roughly uncorrelated increments at the tested scales, similar to white noise or a memoryless process. Real data can still show short-scale structure outside the fitted range.
DFA is typically more stable when slow drifts exist, because it detrends inside each window. Start with DFA order 1, then compare order 2 if curvature remains.
Low R² often means the signal does not follow a clean power-law scaling across selected windows. Adjust window limits, increase data length, or try another estimator.
More is better. Around 100–500 points usually improves stability, while very short series can produce wide uncertainty and unstable slopes, especially at large windows.
No. Log returns require strictly positive values. They are useful for multiplicative growth signals and amplitude changes, but they are inappropriate for data that cross zero.
Try min window 8, max window near N/4, and 10–14 scales. Ensure each scale has at least two segments, and preferably more, to reduce noise.
They respond differently to nonstationarity, trends, and finite samples. Comparing methods is a useful diagnostic; consistent estimates across methods increase confidence in the result.
Measure persistence, reveal correlations, and understand complex dynamics better.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.