Multifractal Detrending Moving Average (MFDMA)

Estimates the multifractal scaling properties of a time series using the detrending moving average algorithm (Gu & Zhou 2010). Clean-room C++ reimplementation from the published algorithm (the reference MATLAB implementation consulted, MFDMA_1D.m, had no license header; see inst/COPYRIGHTS). The segment-fluctuation core was validated against a Python transliteration of that reference on synthetic test data (exact match to displayed precision).

Usage

mfdma(
  x,
  n_min = 10L,
  n_max = NULL,
  n_scales = 30L,
  theta = 0,
  q = seq(-4, 4, by = 0.1)
)

Arguments

x

Numeric vector. The time series to analyse.

n_min, n_max

Integer. Lower/upper bound of the segment size n. Following the reference implementation's guidance: n_min around 10; n_max around 10% of length(x). Defaults: n_min = 10, n_max = round(length(x) / 10).

n_scales

Integer. Number of segment sizes to evaluate (log-spaced between n_min and n_max). Default 30.

theta

Numeric in [0, 1]. Position of the moving-average window: 0 (default, recommended) = backward MFDMA, 0.5 = centered, 1 = forward.

q

Numeric vector. Multifractal orders to evaluate. Default seq(-4, 4, by = 0.1).

Value

A list with:

n

Segment sizes evaluated.

Fq

Matrix of the q-th order fluctuation function (segment size x q).

tau

Multifractal scaling exponent tau(q).

alpha

Singularity strength alpha(q) (trimmed at both ends by the local-slope smoothing window; shorter than q).

f

Multifractal spectrum f(alpha).

q

The q values corresponding to alpha/f (trimmed to match).

References

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

Examples

set.seed(1)
x <- rnorm(4000)
res <- mfdma(x, n_min = 10, n_max = 400, n_scales = 20)
plot(res$alpha, res$f, type = "b")