Multifractal Spectrum via the Chhabra-Jensen Method

Estimates the multifractal spectrum f(alpha) and generalised dimension spectrum D(q) of a strictly positive time series using the direct box-counting method of Chhabra & Jensen (1989). Clean-room C++ reimplementation from the published algorithm (the reference MATLAB implementation consulted, ChhabraJensen_Yuj_w0.m, co-authored by L. Franca, had no license header; see inst/COPYRIGHTS). The moments core was validated against a Python transliteration of that reference on synthetic test data (exact match to displayed precision).

Usage

chhabra_jensen(
  x,
  q_values = c(seq(-10, -0.1, by = 0.1), seq(1, 10, by = 0.1)),
  scales = NULL
)

Arguments

x: Numeric vector, strictly positive (treat it as a measure — e.g. apply a sigmoid transform first if your data can be negative). length(x) must be evenly divisible by 2^scale for every value in scales, i.e. dyadic lengths (powers of two) work best.
q_values: Numeric vector of multifractal orders. Per the original author's guidance, values strictly between 0 and 1 (exclusive) are numerically unstable for this method and best avoided – a warning is issued if any are supplied. Default skips that range: c(seq(-10, -0.1, by = 0.1), seq(1, 10, by = 0.1)).
scales: Integer vector of box-counting scale exponents; the box size at each scale is 2^scale. Default 1:floor(log2(length(x)) - 2), which keeps at least 4 points per box at the coarsest scale.

Value

A list with:

alpha: Singularity strength alpha(q).
falpha: Multifractal spectrum f(alpha(q)).
Dq: Generalised dimension spectrum D(q).
r_squared_alpha, r_squared_falpha, r_squared_Dq: R-squared of the linear fit underlying each of the above, per q – inspect these before trusting a given q value's estimate.
q: The q_values used.
mu_scale, Ma, Mf, Md: The underlying regression inputs, included for completeness.

References

Chhabra A, Jensen RV. Direct determination of the f(alpha) singularity spectrum. Phys Rev Lett 1989;62:1327-1330.

Franca LGS, Miranda JGV, Leite M, Sharma NK, Walker MC, Lemieux L, Wang Y. Fractal and multifractal properties of electrographic recordings of human brain activity: toward its use as a signal feature for machine learning in clinical applications. Front Physiol 2018;9:1767. Compares MF-DFA, MF-DMA, and Chhabra-Jensen on simulated and human intracranial EEG data and finds Chhabra-Jensen the most stable/reliable of the three – the basis for implementing this method (rather than MF-DFA) alongside mfdma() in Rtractor.

Examples

set.seed(1)
x <- abs(rnorm(1024)) + 0.01
res <- chhabra_jensen(x, scales = 1:6)
plot(res$alpha, res$falpha, type = "b")