Detrended Fluctuation Analysis

Detrended Fluctuation Analysis is a scaling analysis method that can be used to estimate long-term temporal correlation that have the power-law form.

Workings
We start off with having a bounded time series x_t, which we transform in the follow unbounded time series. This process is called detrending.


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

With $$\langle x\rangle$$ being the average of $$x_t.$$

Next we divide X_t into time windows, with each window containing L samples. We name these windows Y_j.

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


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

After that we create a log-log plot of $$L$$ against $$F(L)$$. A straight line on this log-log graph indicates statistical self-affinity expressed as $$F(L) \propto L^{\alpha}$$.

The alpha then says something about the self-correlations, and thus the degree of the power law.


 * $$\alpha$$: anti-correlated


 * $$\alpha \simeq 1/2$$: uncorrelated, white noise


 * $$\alpha>1/2$$: correlated


 * $$\alpha\simeq 1$$: 1/f-noise,


 * $$\alpha>1$$: non-stationary, unbounded


 * $$\alpha\simeq 3/2$$: Brownian noise

So for Self-Organized Criticality we want $$\alpha$$ to be between 0.5 and 1.