.TH DFA 1 "31 July 2002" "DFA 4.2" "WFDB Applications Guide"
dfa \- detrended fluctuation analysis
\fBdfa\fR [ \fIoption\fR ... ]
The method of detrended fluctuation analysis (DFA) has proven useful in
revealing the extent of long-range correlations in seemingly irregular
Briefly, the time series to be analyzed is first integrated. Next,
the integrated time series is divided into boxes of equal length,
\fIn\fR. In each box of length \fIn\fR, a least squares line (or
polynomial curve of order \fIk\fR) is fit to the data (representing
the trend in that box). Next, we detrend the integrated time series
by subtracting the local trend in each box. The root-mean-square
fluctuation of this integrated and detrended time series is calculated
and denoted as \fIF(n)\fR.
This computation is repeated over all time scales (box sizes), from
\fIn = minbox\fR to \fIn = maxbox\fR, to characterize the relationship
between \fIF(n)\fR, the average fluctuation, and \fIn\fR, the box size.
Typically, \fIF(n)\fR will increase with box size \fIn\fR. A linear
relationship on a log-log plot indicates the presence of power law
(fractal) scaling. Under such conditions, the fluctuations can be
characterized by a scaling exponent, i.e., the slope of the line
relating \fIlog[F(n)]\fR to \fIlog[n]\fR.
This program performs detrended fluctuation analysis on a sequence of data
read from the standard input (which should contain a single column of numbers
in text format). The standard output contains two columns of numbers, which
are the base 10 logarithms of \fIn\fR and \fIF(n)\fR. Note that \fBdfa\fR
does \fInot\fR compute a scaling exponent; to do so, fit the output to
a line and measure its slope.
\fIOptions\fR may include:
Detrend the data using a polynomial of degree \fIk\fR (1: linear, 2: quadratic,
etc.). Default: \fIk\fR = 1 (linear detrending).
Print a usage summary and exit.
Do not integrate the input series. Use this option if the input series is
already integrated (for example, if it represents times of occurrence rather
Set the smallest box width. The default, and the minimum allowed value for
\fIminbox\fR, is \fI2k + 2\fR (where \fIk\fR is determined by the \fB-d\fR
option, see above).
Perform a sliding window DFA (measure the fluctuations using all possible
boxes at each box size). By default, fluctuations are measured using
non-overlapping boxes only. Using the \fB-s\fR option will make the
calculation much slower.
Set the largest box width. The default, and the maximum allowed value for
\fImaxbox\fR, is one-fourth the length of the input series.
.SH SEE ALSO
The DFA method was first proposed in Peng C-K, Buldyrev SV,
Havlin S, Simons M, Stanley HE, Goldberger AL. Mosaic organization of
DNA nucleotides. \fIPhys Rev E\fR 1994;\fB49\fR:1685-1689.
A detailed description of the algorithm and its application to
physiologic signals can be found in Peng C-K, Havlin S, Stanley HE,
Goldberger AL. Quantification of scaling exponents and crossover
phenomena in nonstationary heartbeat time series.
\fBdfa\fR is available as part of PhysioToolkit under the GPL (see
JE Mietus (joe at physionet dot org), C-K Peng, and GB Moody, based on C-K Peng's
original Fortran implementation.