XJ 1 A22Acos2nf 27f2

where X = 2p - 1 = exp(-l/£). These analytical results give excellent agreement with the numerical data. One way to estimate the correlation length is to measure a limit of S{f) for f—¥ 0. This quantity can be applied to detect a characteristic patch size in the DNA sequence (see sections "Alternation of Nucleotide Frequencies" and "Models of Long Range Anti-Correlations"). Another, more accurate method60 is to plot the inverse power spectrum 1 IS(f) versus/2 (Fig. 7B) and to measure the slope of this graph for/"2 —> 0. Indeed, according to Eq.(29), this slope is equal to 2<ljn2. These two methods give consistent results for exponentially decaying correlations, but technically speaking they measure two different properties of the power spectrum. In fact, the latter method gives the so called Debye persistence length Jf - j; Cirjp'dr, which is not the same as correlation length E,, but is proportional to E, for exponentially decreasing correlations, C(r) - exp(-r/<j|).

Figure 8A shows the power spectrum for a two-dimensional Ising model on a L X L = 210 X 210 square lattice computed averaging power spectra for L horizontal rows each consisting of N= L = 210 points. The figure shows a remarkable straight line indicating long range power law correlations. However, the slope of the line ¡3 = 0.86 corresponds to /= 0.14 which is almost two times smaller than the theoretical exact value /= 1) = 0.25. The discrepancy shows that the power spectrum analysis of a finite system may often give inaccurate values of the correlation exponents.

Figure 8B shows a log-log plot of the power spectrum for a two-dimensional anti-ferromagnetic Ising model, plotted versus 1/2 —f. The analysis in the previous section shows that since 1/2 is the frequency of the anti-ferromagnetic correlations, the power spectrum must have a power-law singularity in this point. Indeed, the graph gives an approximately straight line with slope -/3 = -0.84 similar to the case of ferromagnetic interactions.

Figure 8. A) Power spectrum of the 210 X 210 Ising model at the critical point. The slope of the straight line gives ¡5 = 0.86. B) Power spectrum of the 2 10x210anti -ferromagnetic Ising model at the critical point plotted versus \l2-f. The slope of the straight line gives ft = 0.84.

Detrended Fluctuation Analysis (DFA)

A somewhat more intuitive way to study correlations was proposed in the studies of the fluctuations of environmental records by Hurst in 1964.7 This method is especially useful for short records. The idea is based on comparison of the behavior of the standard deviation of the record averaged over increasing periods of time with the analogous behavior for an uncorrelated record. According to the law of large numbers, the standard deviation of the averaged uncorrelated time series must decrease as the square root of the number of measurements in the averaging interval. This method naturally emerges when the goal is to determine an average value of a quantity (e.g., magnetization in the Ising model, or concentration of a certain nucleotide type in a DNA sequence) obtained in many successive measurements and to asses an error bar of this averaged value. Since the average is equal to the sum divided by the number of measurements, the same analysis can be performed in terms of the sum. In addition to its analytical merits, this method provides a useful graphical description of a time series which otherwise is difficult to see due to high frequency fluctuations.

A variant of Hurst analysis was developed in reference 64 under the name of detrended fluctuation analysis (DFA). The DFA method comprises the following steps:

1. For a numerical sequence s{k), k = 1,2,...L compute a running sum:

which can be represented graphically as a one dimensional landscape, (see Fig. 9A).

2. For any sliding observation box of length r which includes r + 1 values y(k),y(k + 1 ),...y(k + r) define a linear functionyj,(x) = a/, +b& which provides the least square fit for these values, i.e., a/, and b^ are such that the sum of r + 1 squares

n=k has a minimal possible value ^2,nin (r). Note that bk has the meaning of the average value (s(k)) for this observation box, which is the local trend of the values^). For a non-stationary sequence, the local average values (s(k)) can change with time. Since these trends are subtracted in each observation box, this analysis is called detrended. Note that Fk min (1) = 0, so it is a trivial value which can be excluded from the analysis.

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