Detrended Fluctuation Analysis Calculator

Example Data Table

Formula Used

How to Use

1. Paste or upload your time series. Use time,value if you enable “First column is time”.
2. Choose missing-data handling. For gappy data, segmentation can be safer than interpolation.
3. Select detrending order (DFA-1 is common). Increase order for stronger polynomial trends.
4. Pick window sizes (auto log-spaced is recommended). Avoid very small windows for high orders.
5. Set fit range(s) where the log–log plot looks close to linear. Use range 2 to check crossovers.
6. Run DFA and inspect residuals + local slope for stability. Export CSV/PDF when satisfied.

Detrended Fluctuation Analysis Guide

Use this guide to pick settings, validate scaling, and report α responsibly for sensor, financial, or geophysical series and methods clearly.

1) What α measures in real signals

DFA estimates α from the slope of log10(F(n)) versus log10(n). In many physical signals, α≈0.5 suggests weak correlations, α>0.5 persistence, and α<0.5 anti-persistence. α well above 1 can reflect nonstationarity or remaining trends, so always check diagnostics.

2) Choosing window sizes that behave

Choose n across scales but keep enough windows per scale. A practical start is min n 8–16 and max n about N/4, where N is usable length. Use 12–25 log-spaced sizes to sample each decade evenly. If large n yields too few windows, reduce max.

3) Detrending order and drift removal

DFA-1 removes linear drift per window; DFA-2 removes quadratic curvature. Higher orders need larger windows to avoid overfitting. If you increase order, raise min n (for example, ≥4×(m+1)) and compare curves. Stable α across orders is a good sign.

4) Overlap and forward–backward segmentation

Overlapping windows reduce estimator noise by using more segments, especially for shorter series. Forward–backward segmentation adds windows from both ends, improving coverage when N is not a multiple of n. If F(n) looks jagged, try overlap with a modest step.

5) Fit range selection and crossovers

Fit α only where the log–log curve is close to linear. Very small n can show discretization and instrument effects, while very large n can be dominated by a few windows. Use the second fit range to quantify crossovers between short- and long-scale dynamics.

6) Residuals and local slope as quality checks

Residuals should scatter around zero without systematic curvature. The local slope plot estimates α over moving subsets; a flat plateau supports one exponent. Drift, periodic components, or mixed regimes typically bend the curve and create changing local slopes.

7) Handling missing values and uneven sampling

DFA assumes uniform sampling. If time steps vary, resample to a fixed Δt before analysis. For missing values, interpolation is reasonable for short gaps, while segmentation avoids fabricating long stretches. Report the method used because it can shift α at large scales.

8) Surrogate shuffles for correlation evidence

Shuffled surrogates keep the value distribution but remove correlations. If observed α differs strongly from the surrogate mean, that supports genuine temporal structure. Treat the z-score as screening, and confirm by varying fit ranges and detrending order to rule out artifacts.

FAQs

1) How much data do I need for reliable α?

More is better. Aim for several thousand points if possible, and ensure each fitted n has enough windows. Short series can work, but fit stability and confidence intervals will be weaker.

2) Should I always integrate the profile?

Standard DFA integrates the demeaned series to form the profile. Turning integration off can be useful for testing, but it changes the meaning of F(n). For typical DFA interpretation, keep integration enabled.

3) What detrending order should I start with?

DFA-1 is a common baseline. If your data shows curved drift, try DFA-2. Increase order cautiously and also increase the minimum window size, because higher orders need more points per window.

4) Why does α change when I adjust the fit range?

Scaling may not be uniform across all scales. Small n can be noisy, and large n may have too few windows. Use residuals and local slope to select a range where the curve is most linear and stable.

5) Is Theil–Sen better than OLS?

Theil–Sen is more robust to outlier points on the log–log curve. OLS provides standard errors and a classic R². If you suspect outliers or mixed regimes, Theil–Sen can be a good cross-check.

6) What does the surrogate shuffle test tell me?

It compares your α to α values from shuffled data. A large difference suggests correlations beyond randomness. It does not prove a specific model, and it can be fooled by nonstationary trends or preprocessing issues.

7) Can DFA handle unevenly sampled measurements?

Not directly. DFA assumes uniform sampling intervals. If your time steps vary, resample or interpolate onto a regular grid first, then run DFA. Always report the resampling method with your results.