Program DFluC

Program DFluC is a MATLAB program for Detrended Fluctuation Analysis (DFA).

As compared to other DFA software, Program DFluC is implemented with special attention to edge cases such as partly unobserved and/or irregularly sampled data.

Prolegomenon

DFA is a method of estimating the Hurst exponent, which is used to quantify self-similarity and long-range dependence of a 1-dimensional process.

[Note. For convenience this documentation does not distinguish between a proper process and an array of (observed and possibly some unobserved or "missing") values sampled from that process. The process/array is denoted by y below and also referred to as "data". ("Signal" is another common term but is not used in this documentation.)]

DFA consists of the following steps:

1. Divide data into non-overlapping segments or "boxes" of a certain length.
2. Subtract local polynomial trends fit within each box and calculate the root mean square of the residuals (root-mean-square fluctuation).
3. Repeat the previous steps for different box sizes.
4. Estimate the power-law relationship between root-mean-square fluctuation and box size by means of regression.

When dealing with partly unobserved or irregularly sampled data, Program DFluC fits local polynomial trends to data points in situ and avoids "stitching" or "warping" data, in order not to create artificial jumps or alter the autocorrelation structures. You can of course "stitch" and "warp" data yourself, if you want to, by removing all unobserved entries and running Program DFluC without providing the actual sample points. You would then need to interpret the result with caution.

User Guide

You should be able to run Program DFluC in any reasonably recent version of MATLAB. To estimate the Hurst exponent H of a regularly sampled (i.e., sampled at constant intervals) random walk-like process y, you can simply execute

H = dfa(y)

in MATLAB. Program DFluC allows NaN entries in y and interprets them as unobserved (missing) values.

Important Note. Given a noise-like process, it is common to estimate the Hurst exponent of its (random walk-like) integrated process. You can do so with Program DFluC by executing

H = dfa(cumsum(y - mean(y)))

in cases where y is noise-like and regularly sampled without missing values. The cumulative sum cumsum(y - mean(y)) is often called the "profile" in the literature. Many other DFA implementations automatically apply cumulative sum to any data whatsoever. However, this trick only makes sense for regularly sampled data without missing values. If this is not the case, then you need more clever methods to derive the integrated process (a topic that is beyond the scope of this documentation). For regularly sampled data with missing values, a simple workaround is to concatenate the observed parts and then take the cumulative sum of the resulting "stitched" process. The effect of such a procedure on DFA depends on many factors and has been documented in detail by others [Phys. Rev. E 81: 031101]. You should critically evaluate whether it is appropriate for your data.

Program DFluC can be run with up to 5 additional input arguments:

H = dfa(y, x, detrend_order, [xstart xend], box_sizes, plotting)

(setting any of the additional arguments to [] means using the default for that argument), where

• x is an array indicating the sample points (where the values of y are sampled). This is helpful if y is irregularly sampled, or if you want to specify [xstart xend]. If x is not provided, then y is assumed to be regularly sampled.

• detrend_order specifies the degree of polynomials to be used for local detrending. Default is 2.

• [xstart xend] defines the portion of data for which the Hurst exponent is to be estimated. xstart and xend are assumed to have the same unit as x. If [xstart xend] is not provided, then the Hurst exponent will be estimated for the entire range of x.

• box_sizes is an array indicating several box sizes to be used for local polynomial detrending. Each entry is assumed to have the same unit as x. If box_sizes is not provided, then 50 automatically determined box sizes will be used.

  You should choose box sizes such that each box covers a sufficient number of data points, while at least a few boxes are available (i.e., not too small, not too big).

• plotting is true if a plot of log(root-mean-square fluctuation) versus log(box size) is to be shown, false otherwise. The slope of the least-squares fit line is the estimated Hurst exponent H. Default is false.

  If you could see an apparent crossover in this plot, then it's better to re-estimate the Hurst exponent for small and large box sizes separately. Crossovers may be due to finite data length, noise/artifact contamination, or intrinsic properties of the process (a complete survey of the reasons for crossovers is beyond the scope of this documentation) and thus warrant careful study (e.g., by checking whether crossovers depend on detrend_order).

The Hurst exponent is a dimensionless measure and does not depend on the units of x and y as long as you make sure that x, [xstart xend] and box_sizes are expressed in the same unit if you set them manually. To reduce rounding errors, it is best if you choose a unit that renders x, [xstart xend] and box_sizes integer-valued.

License, Disclaimer, and Reference

Program DFluC is intended to be an academic software program. Permission to use, copy, modify, and distribute the software and its documentation for not-for-profit purposes is granted to any person obtaining a copy of the source code, provided that this permission notice appear in all copies. For other uses, please contact the author (Y. Wei).

THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.

If you publish results obtained with Program DFluC, you may include the following citation:

Colombo, Wei, Ramautar, Linkenkaer-Hansen, Tagliazucchi, Van Someren. More severe insomnia complaints in people with stronger long-range temporal correlations in wake resting-state EEG. Frontiers in Physiology 7 (2016). DOI: 10.3389/fphys.2016.00576