The DFA package provides tools to perform a detrended fluctuation analysis (DFA) and estimates the scaling exponent from the results. DFA is used to characterize long memory dependence in stochastic fractal time series.
To install the package:
pkg> add https://github.com/abcsds/DFA.jl
We'll perform a DFA and estimates the scaling exponent for a random time series.
using DFA
x = rand(10000)
<!-- n, Fn = dfa(x) -->
Fn = dfa(x)
You can also specify the following key arguments:
- order: the order of the polynomial fit. Default:
1. - overlap: the overlap of blocks in partitioning the time data expressed as a fraction in [
0,1). A positive overlap will slow down the calculations slightly with the (possible)
effect of generating less biased results. Default:
0. - boxmax: an integer denoting the maximum block size to use in partitioning the data. Default:
div(length(x), 2). - boxmin: an integer denoting the minimum block size to use in partitioning the data. Default:
2*(order+1). - boxratio: the ratio of successive boxes. This argument is used as an input to the logScale
function. Default:
2.
To perform a DFA on x with boxmax=1000, boxmin=4, boxratio=1.2, overlap=0.5:
scales, fluc = dfa(x, boxmax=1000, boxmin=4, boxratio=1.2, overlap=0.5)
To estimates the scaling exponent:
intercept, α = polyfit(log10.(scales), log10.(fluc)) # α is scaling exponent
To plot F(n):
using plots
scatter(scales, fluc, "o")
To plot F(n) with fitted line:
log_scales = log10.(scales)
plot(log_scales, log10.(fluc), "o", log_scales, α.*log_scales.+intercept)
- Peng C-K, Buldyrev SV, Havlin S, Simons M, Stanley HE, and Goldberger AL (1994), Mosaic organization of DNA nucleotides, Physical Review E, 49, 1685–1689.
- Peng C-K, Havlin S, Stanley HE, and Goldberger AL (1995), Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series, Chaos, 5, 82–87.
- Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, Mark RG, Mietus JE, Moody GB, Peng C-K, Stanley HE (2000, June 13), PhysioBank, PhysioToolkit, and Physionet: Components of a New Research Resource for Complex Physiologic Signals, Circulation, 101(23), e215- e220.
- Goldsmith, R. L., Bigger, J. T., Steinman, R. C., & Fleiss, J. L. (1992). Comparison of 24-hour parasympathetic activity in endurance-trained and untrained young men. Journal of the American College of Cardiology, 20(3), 552–558. https://doi.org/10.1016/0735-1097(92)90007-A