zeonta.multifractal_dfa() — Width of the generalized-Hurst spectrum across fluctuation sizes (multifractality).
What it measures
dfa fits one scaling exponent to a return series, implicitly treating small and large fluctuations as scaling the same way. Kantelhardt et al. (2002) generalize DFA’s fluctuation function with a q-th-power average over boxes instead of a plain RMS, reusing the exact same per-box detrended fluctuations dfa computes: negative q weights small fluctuations more heavily, positive q weights large ones. A monofractal series (small and large fluctuations scale identically, e.g. plain fractional Brownian motion) has h(q) essentially constant across q; a genuinely multifractal series has h(q) vary with q, and this function reports that variation as a single number, the width of the generalized-Hurst spectrum between two chosen q extremes.
Formula
F_q(n) = {mean over boxes of [F^2(n,box)]^(q/2)}^(1/q) for q != 0, or exp(mean(ln[F^2(n,box)])/2) for q = 0; h(q) = slope of log(F_q(n)) vs log(n); MFDFA = h(q_min) - h(q_max)
Parameters
Required inputs: close
| Parameter | Default |
|---|---|
window |
100 |
q_min |
-5.0 |
q_max |
5.0 |
Returns
| Column |
|---|
MFDFA_100_-5.0_5.0 |
Usage
Examples run against the 300-bar OHLCV fixture in tests/data/ohlcv.csv, loaded as df. The output shown is the real output.
import pandas as pd
import zeonta
df = pd.read_csv('tests/data/ohlcv.csv', parse_dates=['date']).set_index('date')
zeonta.multifractal_dfa(df['close']).tail(3)
date
2024-10-25 1.318860
2024-10-26 0.871528
2024-10-27 1.168866
Name: MFDFA_100_-5.0_5.0, dtype: float64
Accessor form: df.zta.multifractal_dfa(...)
How to read it
Near 0 describes a monofractal series (dfa’s own single exponent already tells the whole story); larger describes a more strongly multifractal one, where small and large price swings genuinely follow different scaling laws. h(2) — the special case at q=2 this function does not expose on its own — reduces exactly to dfa’s own exponent, so dfa and this function are checking related but different things.
Pitfalls
q_min and q_max default to -5 and 5, the range most commonly scanned in the tutorial literature that has followed the original 2002 paper (e.g. Ihlen, 2012) — not something the paper itself mandates as the one correct choice. Like dfa, this divides each window into non-overlapping boxes counted from the start only, not also from the end as some MF-DFA implementations do, kept consistent with dfa’s own existing convention in this library.
Reference
Formula source: https://doi.org/10.1016/S0378-4371(02)01383-3