R: Detrended Fluctuation Analysis

dfa {fractalRegression}

R Documentation

Detrended Fluctuation Analysis

Description

Fast function for computing detrended fluctuation analysis (DFA), a widely used method for estimating long-range temporal correlations in time series data. DFA is also a form of mono-fractal analysis that indicates the degree of self-similarity across temporal scales.

Usage

dfa(x, order, verbose, scales, scale_ratio = 2)

Arguments

`x`	A real valued vector (i.e., time series data) to be analyzed.
`order`	An integer indicating the polynomial order used for detrending the local windows (e.g, 1 = linear, 2 = quadratic, etc.). There is not a pre-determined limit on the order of the polynomial order but the user should avoid using a large polynomial on small windows. This can result in overfitting and non-meaningful estimates.
`verbose`	If the value of verbose = 1, then a list object is returned that includes: `log_scales` the log of all included scales, `log_rms` the log root mean square error (RMS) per scale, and `alpha` the overall `\alpha` estimate. If the value of verbose = 0, then a list containing only 'alpha' will be returned.
`scales`	An integer valued vector indicating the scales one wishes to resolve in the analysis. Best practice is to use scales which are evenly spaced in the logarithmic domain e.g., `scales = 2^(4:(N/4))`, where N is the length of the time series. Other, logarithmic bases may also be used to give finer resolution of scales while maintaining ~= spacing in the log domain e.g, `scales = unique(floor(1.1^(30:(N/4))))`. Note that fractional bases may produce duplicate values after the necessary floor function.
`scale_ratio`	A scaling factor by which successive window sizes were created. The default is 2 but should be addressed according to how scales were generated for example using `logscale(16, 100, 1.1)`, where 1.1 is the scale ratio.

Details

Details of the algorithm are specified in detail in Peng et al. (1994) and visualized nicely in Kelty-Stephen et al. (2016). The output of the algorithm is an \alpha (alpha) estimate which is a generalization of the Hurst Exponent. Conventional interpretation of \alpha is:

\alpha < 0.5 = anti-correlated
\alpha ~= 0.5 = uncorrelated, white noise
\alpha > 0.5 = temporally correlated
\alpha ~= 1 = 1/f-noise, pink noise
\alpha > 1 = non-stationary and unbounded
\alpha ~= 1.5 = fractional brownian motion

We recommend a few points of consideration here in using this function. One is to be sure to verify there are not cross-over points in the logScale- logFluctuation plots (Peng et al., 1995; Perakakis et al ., 2009). Cross- over points (or a visible change in the slope as a function of of scale) indicate that a mono-fractal characterization does not sufficiently characterize the data. If cross-over points are evident, we recommend proceeding to using the mfdfa() to estimate the multi-fractal fluctuation dynamics across scales.

While it is common to use only linear detrending with DFA, it is important to inspect the trends in the data to determine if it would be more appropriate to use a higher order polynomial for detrending, and/or compare the DFA output for different polynomial orders (see Kantelhardt et al., 2001).

General recommendations for choosing the min and max scale are an sc_min = 10 and sc_max = (N/4), where N is the number of observations. See Eke et al. (2002) and Gulich and Zunino (2014) for additional considerations.

Value

The object returned can take the following forms:

If the value of verbose = 1, then a list object is returned that includes: log_scales the log of all included scales, log_rms the log root mean square error (RMS) per scale, and alpha the overall \alpha estimate.
If the value of verbose = 0, then a list containing only 'alpha' the estimated scaling exponent \alpha will be returned.

References

Eke, A., Herman, P., Kocsis, L., & Kozak, L. R. (2002). Fractal characterization of complexity in temporal physiological signals. Physiological measurement, 23(1), R1-R38.

Gulich, D., & Zunino, L. (2014). A criterion for the determination of optimal scaling ranges in DFA and MF-DFA. Physica A: Statistical Mechanics and its Applications, 397, 17-30.

Kantelhardt, J. W., Koscielny-Bunde, E., Rego, H. H., Havlin, S., & Bunde, A. (2001). Detecting long-range correlations with detrended fluctuation analysis. Physica A: Statistical Mechanics and its Applications, 295(3-4), 441-454.

Kelty-Stephen, D. G., Stirling, L. A., & Lipsitz, L. A. (2016). Multifractal temporal correlations in circle-tracing behaviors are associated with the executive function of rule-switching assessed by the Trail Making Test. Psychological assessment, 28(2), 171-180.

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.

Perakakis, P., Taylor, M., Martinez-Nieto, E., Revithi, I., & Vila, J. (2009). Breathing frequency bias in fractal analysis of heart rate variability. Biological psychology, 82(1), 82-88.

Examples

noise <- rnorm(5000)

scales <- c(16,32,64,128,256,512,1024)

dfa.noise.out <- dfa(
    x = noise, 
    order = 1, 
    verbose = 1, 
    scales = scales,
    scale_ratio = 2)

pink.noise <- fgn_sim(n = 5000, H = 0.9)

dfa.pink.out <- dfa(
    x = pink.noise, 
    order = 1, 
    verbose = 1, 
    scales = scales, 
    scale_ratio = 2)

anticorr.noise <- fgn_sim(n = 5000, H = 0.25)

dfa.anticorr.out <- dfa(
    x = anticorr.noise, 
    order = 1, 
    verbose = 1, 
    scales = scales, 
    scale_ratio = 2)

[Package fractalRegression version 1.2 Index]