## Zr lrlifi

It can be shown, that for a long enough sequence L —» of uncorrected values s{k) (i.e., C(r) = 0 for r > 1) with finite mean and variance C(0), we must have ifl(r) -> (r+ 3)C(0)/ 15. Thus the graph of Fo{r) for such a sequence on a log-log plot is a straight line with slope the a = 1/2 if plotted versus r + 3. Any deviation from the straight line behavior indicates the presence of correlations or anti-correlations. It can be also shown that for a sequence with long range power law correlations C(r) - for 0 < y < 1, the detrended fluctuation also grows as a power law Fpir) - r°- as r —> =*>, where a = 1-7/2 > 1/2, is called the Hurst exponent of the time series.

A Relation between DFA and Power Spectrum

There are many different ways to subtract local trends in Eq. (31).65 One can subtract polynomials of various powers or linear combinations of sines and cosines of certain frequency instead of linear functions. All these different types of DFA have certain advantages and disadvantages. One way to subtract local trends is first to subtract a global trend and plot a sequence yD(k) =y(k) - ky{L)IL. Next, compute a discrete Fourier transform with N = L of this function yif) = F'jd and subtract from the function yi)(k) a low frequency approximation yr(k) = ULl,lf]<l/r~y{f)exp(-2mJk),

(see Fig. 9B). A visual comparison of Figures 9A and 9B, suggests that these two procedures of subtracting local trends are equivalent. Thus we can define a Fourier detrended fluctuation as

According to Eq. (28), the residuals in the right hand side of Eq. (34) are equal to the high frequency part of the inverse Fourier transform:

The Fourier basis vectors are mutually orthogonal, i.e., X>=i exp(2jciqklL)exp{—27liqklL) = L8pv where Spq = 1 ifp = q and <5^ = 0, otherwise. Thus, according to the ¿-dimensional analogy of the Pythagorean theorem, the square of the vector yo(k) — yXk) is equal to the sum of the squares of its orthogonal components and therefore,

The latter sum is nothing but the sum of all the high frequency components of the power spectrum Sy(j) of the integrated signal.

Equation (35) allows us to derive the relation (33) between the exponents a and J. Indeed, in continuum limit, this sum corresponds to the integral Jy=1/ rSyif)df- |y=1/ rS{f)f~2df, where S(f) is the power spectrum of the original, non-integrated sequence s(x) and the factor/" 2 comes from the fact that the Fourier transform of the integrated sequence is proportional to the Fourier transform of the original sequence divided by/. As we see above (26), in case of power law correlations with exponent J, we have S(f) - f^1. Thus

If we assume that Fop(r) - Fo(r) = r" as visual inspection of Figure 9 suggests, we have a= 1 -y/2.

Figure 10A shows linear DFA and Fourier DFA for a one-dimensional Ising model on a double logarithmic plot. These two methods are graphically introduced in Figure 9. One can see a sharp transition from the correlated behavior for r~ E, with slope air) > 1 to an uncorrelated behavior for r » E, with slope a(r) ~ 1/2. The change of the slope can be also studied by plotting the local slope <x(r) versus r (Fig. 10B). This graph shows that Fourier DFA can detect the correlation length more accurately than the linear DFA.

Figure 11 shows analogous plots for the two-dimensional Ising model with long range correlations y= 1/4. One can see again that the Fourier DFA is more accurate in finding the correct value of the exponent a = 1 — yf2 = 0.875 than linear DFA.

In summary, we introduce three methods to study correlations: autocorrelation function C(r), power spectrum S(f), and DFA or Hurst analysis Fo(r). For a signal with long range power law correlations y< 1, all three quantities behave as power law:

Figure 10. A) Linear detrended fluctuation (O) and Fourier detrended fluctuation (□) of the one dimensional Ising model for (T= 0.6, L = 216). The slopes of linear fits give local values of a = 1.24 (thin line) and a = 1.38 (bold line) for small r~ <!; = 14 and a = 0.42 (thin line), a = 0.47 (bold line) for an uncorrelated regime r » B) The slope a(r) of the detrended fluctuations as function of r. Note that Fourier DFA gives a strong maximum at r = 2£ while linear DFA shows monotonic decay of a.

Figure 10. A) Linear detrended fluctuation (O) and Fourier detrended fluctuation (□) of the one dimensional Ising model for (T= 0.6, L = 216). The slopes of linear fits give local values of a = 1.24 (thin line) and a = 1.38 (bold line) for small r~ <!; = 14 and a = 0.42 (thin line), a = 0.47 (bold line) for an uncorrelated regime r » B) The slope a(r) of the detrended fluctuations as function of r. Note that Fourier DFA gives a strong maximum at r = 2£ while linear DFA shows monotonic decay of a.

Figure 11. Linear detrended fluctuation (O) and Fourier detrended fluctuation (□) of the two dimensional Ising model for (T= Tt.L = 210). The slopes of linear fits give local values of a = 0.95 (thin line) and a = 0.88 (bold line) for small r < L. The steep jump in Fourier DFA at L = 210, indicates quasi-periodicity with period L = 210 due to the "bagel" geometry of the model.

Figure 11. Linear detrended fluctuation (O) and Fourier detrended fluctuation (□) of the two dimensional Ising model for (T= Tt.L = 210). The slopes of linear fits give local values of a = 0.95 (thin line) and a = 0.88 (bold line) for small r < L. The steep jump in Fourier DFA at L = 210, indicates quasi-periodicity with period L = 210 due to the "bagel" geometry of the model.

where the exponents a, ¡3, and /are related via the following linear relations:

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