Katz Fractal Dimension

Computes Katz's fractal dimension (KFD) of a one-dimensional signal (Katz, 1988) – a waveform-complexity measure derived from the total length of the signal curve relative to its extent. Let \(L\) be the sum of the absolute successive differences (the curve length), \(a\) their mean, and \(d\) the maximum absolute distance from the first sample; then \(\mathrm{KFD} = \log_{10}(L/a) / \log_{10}(d/a)\). A smoother, more regular signal has a lower KFD; a more convoluted (noise-like) waveform a higher one.

Usage

katzFD(x)

Arguments

x: A numeric vector (the time series).

Value

A single numeric value, Katz's fractal dimension.

Details

This uses the amplitude-only convention adopted by the de-facto reference implementations (Python antropy, NeuroKit2, pyeeg, mne-features), in which both the successive distances and the extent are measured in the signal's amplitude units alone; it reproduces antropy.katz_fd and NeuroKit2's fractal_katz bit-for-bit. (Katz's original definition uses the Euclidean distance in the sample-index/amplitude plane, which mixes the two axes' units, is scale-dependent, and is not comparable across tools; the amplitude-only form is what the field uses and what makes KFD reproducible.)

References

Katz, M. J. (1988). "Fractals and the analysis of waveforms." Computers in Biology and Medicine, 18(3), 145–156. doi:10.1016/0010-4825(88)90041-8

Examples

katzFD(sin(seq(0, 20 * pi, length.out = 500)))  # low (smooth oscillation)
#> [1] 2.461042
set.seed(1); katzFD(rnorm(500))                 # higher (noise-like)
#> [1] 4.653128