# Detrended fluctuation analysis

In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing time series 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, 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.

Peng et al. introduced DFA in 1994 in a paper that has been cited over 2000 times as of 2013[1] and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities.

## Calculation

Given a bounded time series $x_t$, $t \in \mathbb{N}$, integration or summation first converts this into an unbounded process $X_t$:

$X_t=\sum_{i=1}^t (x_i-\langle x_i\rangle)$

$X_t$ is called cumulative sum or profile. This process converts, for example, an i.i.d. white noise process into a random walk.

Next, $X_t$ is divided into time windows $Y_j$ of length $L$ samples, and a local least squares straight-line fit (the local trend) is calculated by minimising the squared error $E^2$ with respect to the slope and intercept parameters $a, b$:

$E^2 = \sum_{j = 1}^L \left( Y_j - j a - b \right)^2.$

Trends of higher order, can be removed by higher order DFA, where the linear function $j a +b$ is replaced by a polynomial of order $n$.[2] Next, the root-mean-square deviation from the trend, the fluctuation, is calculated over every window at every time scale:

$F( L ) = \left[ \frac{1}{L}\sum_{j = 1}^L \left( Y_j - j a - b \right)^2 \right]^{\frac{1}{2}}.$

This detrending followed by fluctuation measurement process is repeated over the whole signal at a range of different window sizes $L$, and a log-log graph of $L$ against $F(L)$ is constructed.

A straight line on this log-log graph indicates statistical self-affinity expressed as $F(L) \propto L^{\alpha}$. The scaling exponent $\alpha$ is calculated as the slope of a straight line fit to the log-log graph of $L$ against $F(L)$ using least-squares. This exponent is a generalization of the Hurst exponent. Because the expected displacement in an uncorrelated random walk of length L grows like $\sqrt{L}$, an exponent of $\tfrac{1}{2}$ would correspond to uncorrelated white noise. When the exponent is between 0 and 1, the result is Fractional Brownian motion, with the precise value giving information about the series self-correlations:

• $\alpha<1/2$: anti-correlated
• $\alpha \simeq 1/2$: uncorrelated, white noise
• $\alpha>1/2$: correlated
• $\alpha\simeq 1$: 1/f-noise, pink noise
• $\alpha>1$: non-stationary, unbounded
• $\alpha\simeq 3/2$: Brownian noise

There are different orders of DFA. In the described case, linear fits ($n=1$) are applied to the profile, thus it is called DFA1. In general, DFA$n$, uses polynomial fits of order $n$. Due to the summation (integration) from $x_i$ to $X_t$, linear trends in the mean of the profile represent constant trends in the initial sequence, and DFA1 only removes such constant trends (steps) in the $x_i$. In general DFA of order $n$ removes (polynomial) trends of order $n-1$. For linear trends in the mean of $x_i$ at least DFA2 is needed. The Hurst R/S analysis removes constants trends in the original sequence and thus, in its detrending it is equivalent to DFA1. The DFA method was applied to many systems; e.g., DNA sequences,[3][4] neuronal oscillations,[5] speech pathology detection,[6] and heartbeat fluctuation in different sleep stages.[7] Effect of trends on DFA were studied in[8] and relation to the power spectrum method is presented in.[9]

Since in the fluctuation function $F(L)$ the square(root) is used, DFA measures the scaling-behavior of the second moment-fluctuations, this means $\alpha=\alpha(2)$. The multifractal generalization (MF-DFA)[10] uses a variable moment $q$ and provides $\alpha(q)$. Kantelhardt et al. intended this scaling exponent as a generalization of the classical Hurst exponent. The classical Hurst exponent corresponds to the second moment for stationary cases $H=\alpha(2)$ and to the second moment minus 1 for nonstationary cases $H=\alpha(2)-1$.[5][10]

## Relations to other methods

In the case of power-law decaying auto-correlations, the correlation function decays with an exponent $\gamma$: $C(L)\sim L^{-\gamma}\!\$. In addition the power spectrum decays as $P(f)\sim f^{-\beta}\!\$. The three exponent are related by:[3]

• $\gamma=2-2\alpha$
• $\beta=2\alpha-1$ and
• $\gamma=1-\beta$.

The relations can be derived using the Wiener–Khinchin theorem.

Thus, $\alpha$ is tied to the slope of the power spectrum $\beta$ and is used to describe the color of noise by this relationship: $\alpha = (\beta+1)/2$.

For fractional Gaussian noise (FGN), we have $\beta \in [-1,1]$, and thus $\alpha = [0,1]$, and $\beta = 2H-1$, where $H$ is the Hurst exponent. $\alpha$ for FGN is equal to $H$.

For fractional Brownian motion (FBM), we have $\beta \in [1,3]$, and thus $\alpha = [1,2]$, and $\beta = 2H+1$, where $H$ is the Hurst exponent. $\alpha$ for FBM is equal to $H+1$. In this context, FBM is the cumulative sum or the integral of FGN, thus, the exponents of their power spectra differ by 2.

## Pitfalls in interpretation

As with most methods that depend upon line fitting, it is always possible to find a number $\alpha$ by the DFA method, but this does not necessarily imply that the time series is self-similar. Self-similarity requires that the points on the log-log graph are sufficiently collinear across a very wide range of window sizes $L$.

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. Therefore, the DFA scaling exponent $\alpha$ is not a fractal dimension sharing all the desirable properties of the Hausdorff dimension, for example, although in certain special cases it can be shown to be related to the box-counting dimension for the graph of a time series.