Detrended fluctuation analysis

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In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining a fractal scaling exponent, a measure of the statistical self-similarity of a signal. It is useful for analysing time series that appear to be 1/f noise or long-memory processes.

It is similar to the Hurst exponent, except that it may also be applied to signals whose underlying statistics (such as mean and variance) or dynamics are non-stationary (changing with time). It can be shown to be related to measures based upon spectral techniques such as autocorrelation and Fourier transform.

1 Calculation
2 Pitfalls in interpretation
3 References
4 External links

[edit] Calculation

Given a bounded time series $x t$ , $t \in \mathbb{N}$ , integration first converts this into an unbounded process $X t$ :

$X_t=\sum_{i=1}^t x_i$

This integration process converts, for example, an i.i.d. white noise process into a self-similar random walk.

Next, $X t$ is divided into time windows 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 $a, b$ :

$E^2 = \sum_{i = 1}^L \left( X_i - ai - b \right)^2.$

Next, the root-mean-square deviation from the trend, the fluctuation, is calculated over every window at every time scale:

$F( L ) = \left[ \frac{1}{L}\sum_{i = 1}^L \left( X_i - ai - b \right)^2 \right]^{\frac{1}{2}}.$

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 graph indicates statistical self-similarity expressed as $F(L) \propto L^{\alpha}$ . The scaling exponent $α$ is calculated as the slope of a straight line fit to the log-log graph of $L$ against $F (L)$ using least-squares.

[edit] Pitfalls in interpretation

As with most methods that depend upon line fitting, it is always possible to find a number $α$ 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 $α$ is not a fractal dimension sharing the all the desireable 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.

[edit] References

Hu, K. et al., (2001) Effect of trends on detrended fluctuation analysis, Phys Rev E, 64(1) 011114

Little M., McSharry P. et al., (2006) Nonlinear, Biophysically-Informed Speech Pathology Detection, IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. 2 II-1080-II-1083

[edit] External links

Systems Analysis, Modelling and Prediction (SAMP), University of Oxford FastDFA MATLAB code for rapidly calculating the DFA scaling exponent on very large datasets.
Physionet A good overview of DFA and C code to calculate it.

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Categories: Stochastic processes | Time series analysis