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The method of detrended fluctuation analysis (DFA) has proven useful in revealing the extent of long-range correlations in seemingly irregular time series.

Briefly, the time series to be analyzed is first integrated. Next, the
integrated time series is divided into boxes of equal length, *n*. In each
box of length *n*, a least squares line (or polynomial curve of order *k*)
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 *F(n)*
.

This computation is repeated over all
time scales (box sizes), from *n = minbox* to *n = maxbox*, to characterize
the relationship between *F(n)*
, the average fluctuation, and *n*, the box
size. Typically, *F(n)*
will increase with box size *n*. 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 *log[F(n)
]* to *log[n]*.

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 *n* and *F(n)*
. Note that **dfa** does *not* compute
a scaling exponent; to do so, fit the output to a line and measure its
slope.

*Options* may include:

**-d***k*- Detrend the data using a polynomial of degree
*k*(1: linear, 2: quadratic, etc.). Default:*k*= 1 (linear detrending). **-h**- Print a usage summary and exit.
**-i**- 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 than intervals).
**-l***minbox*- Set the
smallest box width. The default, and the minimum allowed value for
*minbox*, is*2k + 2*(where*k*is determined by the**-d**option, see above). **-s**- 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
**-s**option will make the calculation much slower. **-u***maxbox*- Set the largest box width. The default, and the maximum allowed
value for
*maxbox*, is one-fourth the length of the input series.

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. *Phys
Rev E* 1994;**49**: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. *Chaos* 1995;**5**:82-87.

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Updated 8 March 2019