Detrended Fluctuation Analysis (DFA)

Computes the scaling exponent alpha of an activity time series using Detrended Fluctuation Analysis (Peng et al., 1994). DFA quantifies long-range temporal correlations: alpha approximately 0.5 indicates uncorrelated (white) noise, alpha approximately 1.0 indicates 1/f (pink) noise, and alpha approximately 1.5 indicates Brownian (random-walk / brown) noise. Healthy human activity fluctuations typically show alpha in the 0.9 to 1.0 range, with reductions reported in aging and Alzheimer's disease (Hu et al., 2009); those reference values were obtained with quadratic DFA-2 (detrend_order = 2), which can differ from the default linear DFA-1 on trend-carrying data.

Usage

fractal.dfa(
  x,
  scale_min = 4,
  scale_max = NULL,
  breakpoint_min = 90,
  detrend_order = 1L
)

Arguments

x

Numeric vector of activity counts (minute-level recommended). Internally analyzed on the longest continuous non-NA segment.

scale_min

Integer. Smallest window size (box length) in samples. Must be >= 4 so that a line can be detrended with residual degrees of freedom. Default 4.

scale_max

Integer or NULL. Largest window size in samples. If NULL (default) it is set to floor(N / 4) where N is the length of the analyzed segment, ensuring at least four windows at the largest scale. The top scales then have only about four windows and are somewhat undersampled; Hu et al. (2001) suggest a maximum nearer N / 10 for well-sampled fluctuations.

breakpoint_min

Numeric. Window-size boundary (in samples / minutes for minute-level data) separating the short-timescale exponent alpha1 (scales < breakpoint_min) from the long-timescale exponent alpha2 (scales >= breakpoint_min). Default 90.

detrend_order

Integer order of the within-window polynomial detrend: 1 (default) is linear DFA-1 (Peng et al. 1994); 2 is quadratic DFA-2, the convention Hu et al. (2009) used for activity data.

Value

A list with class "actiRhythm_dfa" containing:

alpha

Overall scaling exponent: slope of lm(log10(F) ~ log10(n)) across all scales.

alpha1

Short-timescale exponent (scales < breakpoint_min). NA if fewer than two qualifying scales.

alpha2

Long-timescale exponent (scales >= breakpoint_min). NA if fewer than two qualifying scales.

scales

Integer vector of window sizes n that were used.

fluctuations

Numeric vector of fluctuation magnitudes F(n) corresponding to scales.

n_used

Length of the analyzed (longest non-NA) segment.

breakpoint_min

The breakpoint value used.

On an unusable series (too short, all NA, or constant) the numeric scaling outputs are returned as NA with the same structure (never an error).

Details

Algorithm (integrated, non-overlapping DFA, polynomial detrend of order detrend_order):

1. Extract the longest continuous non-NA segment of x.

2. Integrate the mean-centred signal: y = cumsum(x - mean(x)).

3. For each window size n (log-spaced from scale_min to scale_max), split y into floor(N / n) non-overlapping windows of length n, fit and remove a least-squares polynomial of order detrend_order within each window, and pool the residuals. The fluctuation is F(n) = sqrt(mean(residuals^2)) over all pooled residuals.

4. The scaling exponent is the slope of log10(F(n)) regressed on log10(n).

Window sizes are unique integers chosen on a base-10 log grid, which gives approximately even spacing in the log-log fit and matches the convention used by reference implementations such as nonlinearTseries.

See also

multiscale.entropy, circadian.rhythm

Examples

# \donttest{
set.seed(1)
# White noise -> alpha near 0.5
fractal.dfa(rnorm(10000))$alpha
#> [1] 0.4953119
# Brown noise -> alpha near 1.5
fractal.dfa(cumsum(rnorm(10000)))$alpha
#> [1] 1.494906
# }