# 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 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). Datasets generated to test DFA are available on PhysioNet.

## Definition
### Algorithm

Given: a time series $x_{1},x_{2},...,x_{N}$.
Compute its average value $\langle x\rangle ={\frac {1}{N}}\sum _{t=1}^{N}x_{t}$.
Sum it into a process $X_{t}=\sum _{i=1}^{t}(x_{i}-\langle x\rangle )$. 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.
Select a set $T=\{n_{1},...,n_{k}\}$ of integers, such that $n_{1}<n_{2}<\cdots <n_{k}$, the smallest $n_{1}\approx 4$, the largest $n_{k}\approx N$, and the sequence is roughly distributed evenly in log-scale: $\log(n_{2})-\log(n_{1})\approx \log(n_{3})-\log(n_{2})\approx \cdots$. In other words, it is approximately a geometric progression.
For each $n\in T$, divide the sequence $X_{t}$ into consecutive segments of length $n$. Within each segment, compute the least squares straight-line fit (the local trend). Let $Y_{1,n},Y_{2,n},...,Y_{N,n}$ be the resulting piecewise-linear fit.
Compute the root-mean-square deviation from the local trend (local fluctuation):$F(n,i)={\sqrt {{\frac {1}{n}}\sum _{t=in+1}^{in+n}\left(X_{t}-Y_{t,n}\right)^{2}}}.$And their root-mean-square is the total fluctuation:
    $F(n)={\sqrt {{\frac {1}{N/n}}\sum _{i=1}^{N/n}F(n,i)^{2}}}.$
(If $N$ is not divisible by $n$, 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 $\log n-\log F(n)$.

### Interpretation

A straight line of slope $\alpha$ on the log-log plot indicates a statistical self-affinity of form $F(n)\propto n^{\alpha }$. Since $F(n)$ monotonically increases with $n$, we always have $\alpha >0$.
The scaling exponent $\alpha$ is a generalization of the Hurst exponent, 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
Because the expected displacement in an uncorrelated random walk of length N grows like ${\sqrt {N}}$, an exponent of ${\tfrac {1}{2}}$ 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 $\alpha$ 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 $n$. 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. Therefore, the DFA scaling exponent $\alpha$ is not a fractal dimension, 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 $X_{t}$ is a cumulative sum of $x_{t}-\langle x\rangle$, a linear trend in $X_{t}$ is a constant trend in $x_{t}-\langle x\rangle$, which is a constant trend in $x_{t}$ (visible as short sections of "flat plateaus"). In this regard, DFA1 removes the mean from segments of the time series $x_{t}$ before quantifying the fluctuation.
Similarly, a degree n trend in $X_{t}$ is a degree (n-1) trend in $x_{t}$. For example, DFA1 removes linear trends from segments of the time series $x_{t}$ before quantifying the fluctuation, DFA1 removes parabolic trends from $x_{t}$, 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$F_{q}(n)=\left({\frac {1}{N/n}}\sum _{i=1}^{N/n}F(n,i)^{q}\right)^{1/q}.$then making the log-log plot of $\log n-\log F_{q}(n)$, If there is a strong linearity in the plot of $\log n-\log F_{q}(n)$, then that slope is $\alpha (q)$. DFA is the special case where $q=2$.
Multifractal systems scale as a function $F_{q}(n)\propto n^{\alpha (q)}$. 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 $H=\alpha (2)$ for stationary cases, and $H=\alpha (2)-1$ 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 $\gamma$: $C(L)\sim L^{-\gamma }\!\$. In addition the power spectrum decays as $P(f)\sim f^{-\beta }\!\$. The three exponents are related by:
* $\gamma =2-2\alpha$
* $\beta =2\alpha -1$ and
* $\gamma =1-\beta$.
The relations can be derived using the Wiener–Khinchin theorem. The relation of DFA to the power spectrum method has been well studied.
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

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

### For fractional Brownian motion

For fractional Brownian motion (FBM), we have $\beta \in [1,3]$, and thus $\alpha \in [1,2]$, 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.