How DFA Works
Detrended Fluctuation Analysis measures long-range correlation in a time series.
The algorithm in 4 steps
- Profile: compute the cumulative sum of deviations from the mean
- Box: divide the profile into non-overlapping boxes of size s
- Detrend: fit and subtract a linear trend within each box
- Scale: measure the root-mean-square residual F(s) at each box size
The scaling exponent alpha is the slope of log F(s) vs log s.
What alpha means
| Alpha range | Interpretation |
|---|---|
| ~0.5 | White noise (uncorrelated) |
| 0.5 - 1.0 | Long-range correlated (healthy complexity) |
| ~1.0 | 1/f noise (pink noise) |
| > 1.0 | Non-stationary / trend-dominated |
Why it works for anomaly detection
Healthy complex systems maintain a characteristic alpha. When the system degrades, alpha shifts — often before any amplitude-based monitor fires. The structure changes first.
References
- Peng et al., “Mosaic organization of DNA nucleotide sequences,” Physical Review E 49(2), 1994.
- Peng et al., “Quantification of scaling exponents,” Chaos 5(1), 1995.
- Goldberger et al., “Fractal dynamics in physiology,” PNAS 99(suppl 1), 2002.