Multifractal methods: MFDMA vs Chhabra-Jensen

Two estimators, one underlying idea

Rtractor ships two multifractal spectrum estimators: mfdma() (Gu & Zhou 2010) and chhabra_jensen() (Chhabra & Jensen 1989). Both try to answer the same question – does this time series need more than one scaling exponent to describe it, and if so, what’s the full spectrum of exponents present? – but they get there differently.

mfdma() is a multifractal generalisation of DFA: it detrends the series against a moving average at a range of scales, then looks at how the q-th order fluctuation function scales with segment size. chhabra_jensen() instead treats the series as a measure and does direct box-counting, avoiding the Legendre transform that mfdma() (and MF-DFA) rely on.

That difference matters in practice. França et al. (2018) benchmarked MF-DFA, MF-DMA, and Chhabra-Jensen against each other on both simulated and human intracranial EEG data, and found Chhabra-Jensen the most stable of the three across repeated epochs of the same underlying signal – the reason it’s implemented here alongside mfdma(), rather than reaching for MF-DFA. See ?chhabra_jensen for the full reference.

Ground truth, not just plausible-looking data

It’s easy to run a multifractal estimator on rnorm() and see some output, but that doesn’t tell you whether the estimator is actually detecting multifractality correctly – white noise is close to monofractal, so a badly-behaved estimator could still produce a narrow-looking spectrum purely by chance.

pmodel() solves this by generating a p-model binomial cascade (Meneveau & Sreenivasan 1987) whose multifractal properties are directly controlled by its p parameter: values near 0.5 are essentially monofractal, and values far from 0.5 are strongly multifractal. This gives both estimators a signal with known ground truth to be checked against, rather than just a signal that happens to look complex.

y_calm   <- pmodel(8192, p = 0.48, seed = 1)  # near-monofractal
y_multi  <- pmodel(8192, p = 0.15, seed = 1)  # multifractal
y_strong <- pmodel(8192, p = 0.05, seed = 1)  # strongly multifractal

Both estimators expect the raw p-model output directly – mfdma() and chhabra_jensen() each do their own internal integration/box-counting, so don’t difference or log-transform it first.

MFDMA on the ground-truth series

mf_calm   <- mfdma(y_calm,   n_min = 10, n_max = 400, n_scales = 25)
mf_multi  <- mfdma(y_multi,  n_min = 10, n_max = 400, n_scales = 25)
mf_strong <- mfdma(y_strong, n_min = 10, n_max = 400, n_scales = 25)

widths_mfdma <- c(
  calm   = diff(range(mf_calm$alpha)),
  multi  = diff(range(mf_multi$alpha)),
  strong = diff(range(mf_strong$alpha))
)
widths_mfdma
#>       calm      multi     strong 
#> 0.01884055 2.51858115 4.29434619

The estimated spectrum width increases monotonically as p moves away from 0.5, exactly as the underlying cascade predicts – mfdma() is genuinely picking up the multifractal structure, not just returning a wide spectrum regardless of input.

Chhabra-Jensen on the same series

cj_calm   <- chhabra_jensen(y_calm,   scales = 1:11)
cj_multi  <- chhabra_jensen(y_multi,  scales = 1:11)
cj_strong <- chhabra_jensen(y_strong, scales = 1:11)

widths_cj <- c(
  calm   = diff(range(cj_calm$alpha)),
  multi  = diff(range(cj_multi$alpha)),
  strong = diff(range(cj_strong$alpha))
)
widths_cj
#>       calm      multi     strong 
#> 0.04389655 2.50250019 4.24792751

Same pattern, and the two estimators land within a few percent of each other on the same data – independent cross-validation, not just each method being internally consistent with itself.

plot(
  mf_multi$alpha, mf_multi$f, type = "b", col = rtractor_palette("core")[["steel_blue"]],
  xlab = expression(alpha), ylab = expression(f(alpha)),
  main = "p = 0.15: MFDMA vs Chhabra-Jensen",
  ylim = c(0, 1.05)
)
lines(cj_multi$alpha, cj_multi$falpha, type = "b", col = rtractor_palette("core")[["coral"]])
legend(
  "bottomright", legend = c("MFDMA", "Chhabra-Jensen"),
  col = rtractor_palette("core")[c("steel_blue", "coral")], lty = 1, pch = 1, bty = "n"
)

chhabra_jensen()’s spectrum also has a theoretical constraint worth knowing about: the peak height, max(falpha), should sit close to 1 – the fractal dimension of the one-dimensional support the measure lives on. That’s a useful sanity check independent of the p-model comparison above:

max(cj_multi$falpha)
#> [1] 0.9945943

Which one should I use?

• If you want the more stable estimate across repeated epochs of the same signal, and your series can be made strictly positive (or you’re willing to apply a sigmoid transform), reach for chhabra_jensen() first – see França et al. (2018) for why.
• If your series already has a natural DFA-style workflow around it, or you need mfdma()’s cross-package parity with other MF-DFA-family tools, mfdma() is the more familiar entry point.
• Either way, check the R-squared / fit diagnostics each function returns (r_squared_alpha, r_squared_falpha, r_squared_Dq for chhabra_jensen()) before trusting an individual q value’s estimate, especially near the edges of the q range.

References

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

Gu GF, Zhou WX. Detrending moving average algorithm for multifractals. Phys Rev E 2010;82:011136.

Meneveau C, Sreenivasan KR. Simple multifractal cascade model for fully developed turbulence. Phys Rev Lett 1987;59:1424-1427.

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.