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## Relationship Between Self-Similarity and Auto-Correlation Functions

The self-similarity parameter of an integrated time series is related to the more familiar auto-correlation function, , of the original (non-integrated) signal. Briefly:

• For white noise where the value at one instant is completely uncorrelated with any previous values, the integrated value, y(k), corresponds to a random walk and therefore [7, 17]. The auto-correlation function, , is 0 for any (time-lag) not equal to 0.
• Many natural phenomena are characterized by short-term correlations with a characteristic time scale, , and an autocorrelation function, that decays exponentially, [i.e., ]. The initial slope of vs. may be different from 0.5, but will approach 0.5 for large window sizes.
• An greater than 0.5 and less than or equal to 1.0 indicates persistent long-range power-law correlations, i.e., . The relation between and is . Note also that the power spectrum, S(f), of the original (non-integrated) signal is also of a power-law form, i.e., . Because the power spectrum density is simply the Fourier transform of the autocorrelation function, . The case of is a special one which has interested physicists and biologists for many years--it corresponds to 1/f noise ( ).
• When , power-law anti-correlations are present such that large values are more likely to be followed by small values and vice versa .
• When , correlations exist but cease to be of a power-law form; indicates brown noise, the integration of white noise.

The exponent can also be viewed as an indicator of the ``roughness'' of the original time series: the larger the value of , the smoother the time series. In this context, 1/f noise can be interpreted as a compromise or ``trade-off'' between the complete unpredictability of white noise (very rough ``landscape'') and the much smoother landscape of Brownian noise .

In the next sections, we apply these scaling analyses to the output of two complex integrated neural control systems, namely those regulating human heart rate and gait dynamics in health and disease.   Next: Fractal Dynamics of Human Up: Fractal Analysis Methods Previous: Detrended Fluctuation Analysis (DFA)