In
stochastic processes,
chaos theory
Chaos theory is an interdisciplinary area of Scientific method, scientific study and branch of mathematics. It focuses on underlying patterns and Deterministic system, deterministic Scientific law, laws of dynamical systems that are highly sens ...
and
time series analysis
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
, detrended fluctuation analysis (DFA) is a method for determining the statistical
self-affinity of a signal. It is useful for analysing
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
that appear to be
long-memory processes (diverging
correlation time, e.g. power-law decaying
autocorrelation function) or
1/f noise.
The obtained exponent is similar to the
Hurst exponent
The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst expo ...
, except that DFA may also be applied to signals whose underlying statistics (such as mean and variance) or dynamics are
non-stationary (changing with time). It is related to measures based upon spectral techniques such as
autocorrelation and
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
.
Peng et al. introduced DFA in 1994 in a paper that has been cited over 3,000 times as of 2022 and represents an extension of the (ordinary)
fluctuation analysis (FA), which is affected by non-stationarities.
Systematic studies of the advantages and limitations of the DFA method were performed by PCh Ivanov et al. in a series of papers focusing on the effects of different types of nonstationarities in real-world signals: (1) types of trends; (2) random outliers/spikes, noisy segments, signals composed of parts with different correlation; (3) nonlinear filters; (4) missing data; (5) signal coarse-graining procedures and comparing DFA performance with moving average techniques (cumulative citations > 4,000).�
Datasetsgenerated to test DFA are available on PhysioNet.
Definition
Algorithm
Given: a
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
.
Compute its average value
.
Sum it into a process
. This is the cumulative sum, or profile, of the original time series. For example, the profile of an
i.i.d. white noise is a standard
random walk
In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space.
An elementary example of a rand ...
.
Select a set
of integers, such that
, the smallest
, the largest
, and the sequence is roughly distributed evenly in log-scale:
. In other words, it is approximately a
geometric progression.
For each
, divide the sequence
into consecutive segments of length
. Within each segment, compute the
least squares straight-line fit (the local trend). Let
be the resulting piecewise-linear fit.
Compute the
root-mean-square deviation from the local trend (local fluctuation):
And their root-mean-square is the total fluctuation:
:
(If
is not divisible by
, then one can either discard the remainder of the sequence, or repeat the procedure on the reversed sequence, then take their root-mean-square.)
Make the
log-log plot .
Interpretation
A straight line of slope
on the log-log plot indicates a statistical
self-affinity of form
. Since
monotonically increases with
, we always have
.
The scaling exponent
is a generalization of the
Hurst exponent
The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst expo ...
, with the precise value giving information about the series self-correlations:
*
: anti-correlated
*
: uncorrelated,
white noise
*
: correlated
*
: 1/f-noise,
pink noise
*
: non-stationary, unbounded
*
:
Brownian noise
Because the expected displacement in an
uncorrelated random walk of length N grows like
, an exponent of
would correspond to uncorrelated white noise. When the exponent is between 0 and 1, the result is
fractional Gaussian noise.
Pitfalls in interpretation
Though the DFA algorithm always produces a positive number
for any time series, it does not necessarily imply that the time series is self-similar.
Self-similarity requires the log-log graph to be sufficiently linear over a wide range of
. Furthermore, a combination of techniques including
maximum likelihood estimation (MLE), rather than least-squares has been shown to better approximate the scaling, or power-law, exponent.
Also, there are many scaling exponent-like quantities that can be measured for a self-similar time series, including the divider dimension and
Hurst exponent
The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst expo ...
. Therefore, the DFA scaling exponent
is not a
fractal dimension
In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured.
It ...
, and does not have certain desirable properties that the
Hausdorff dimension has, though in certain special cases it is related to the
box-counting dimension for the graph of a time series.
Generalizations
Generalization to polynomial trends (higher order DFA)
The standard DFA algorithm given above removes a linear trend in each segment. If we remove a degree-n polynomial trend in each segment, it is called DFAn, or higher order DFA.
Since
is a cumulative sum of
, a linear trend in
is a constant trend in
, which is a constant trend in
(visible as short sections of "flat plateaus"). In this regard, DFA1 removes the mean from segments of the time series
before quantifying the fluctuation.
Similarly, a degree n trend in
is a degree (n-1) trend in
. For example, DFA1 removes linear trends from segments of the time series
before quantifying the fluctuation, DFA1 removes parabolic trends from
, and so on.
The Hurst
R/S analysis removes constant trends in the original sequence and thus, in its detrending it is equivalent to DFA1.
Generalization to different moments (multifractal DFA)
DFA can be generalized by computing
then making the log-log plot of
, If there is a strong linearity in the plot of
, then that slope is
.
DFA is the special case where
.
Multifractal systems scale as a function
. Essentially, the scaling exponents need not be independent of the scale of the system. In particular, DFA measures the scaling-behavior of the second moment-fluctuations.
Kantelhardt et al. intended this scaling exponent as a generalization of the classical Hurst exponent. The classical Hurst exponent corresponds to
for stationary cases, and
for nonstationary cases.
Applications
The DFA method has been applied to many systems, e.g. DNA sequences;
heartbeat dynamics in sleep and wake, sleep stages, rest and exercise, and across circadian phases; locomotor gate and wrist dynamics, neuronal oscillations,
speech pathology detection, and animal behavior pattern analysis.
Relations to other methods, for specific types of signal
For signals with power-law-decaying autocorrelation
In the case of power-law decaying auto-correlations, the
correlation function decays with an exponent
:
.
In addition the
power spectrum
In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of Power (physics), power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be ...
decays as
.
The three exponents are related by:
*
*
and
*
.
The relations can be derived using the
Wiener–Khinchin theorem. The relation of DFA to the power spectrum method has been well studied.
Thus,
is tied to the slope of the power spectrum
and is used to describe the
color of noise by this relationship:
.
For fractional Gaussian noise
For
fractional Gaussian noise (FGN), we have
, and thus