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Fractal correlation in heterogeneous systems. (English) Zbl 0737.92005

The incentive to undertake the present work was the need to understand the heterogeneity of regional blood flows in the heart and lungs. (Other organs of course have heterogeneity in regional flows.) The fact that the relative dispersions of regional flows showed fractal characteristics indicates that there are correlations between neighbors. While the original analysis of relative dispersions at various volume element sizes does implicitly take into account that the subelements are neighbors, the fundamental idea of the approach is that by continuing to divide a region ever more finely one would reveal more and more of the underlying heterogeneity. Now the development of an algorithm for extended correlation allows the testing of the same data sets from a very different point of view, that of local correlation.
We investigate the relationship between the fractal measures of dispersion and correlation over long distances in signals whose dispersion is dependent on the size of the observed elements in a self- similar fashion. We show that the correlation depends on the fractal dimension \(D\) and that once \(D\) is known, the correlation between elements of sizes other than that for which \(D\) was determined is possible. We apply this idea of correlation between elements of different sizes to signals with power spectra that vary like \(1/f^{\beta}\), where \(f\) is the frequency and \(\beta\) is a positive constant \((\beta=1+2H)\).

MSC:

92C30 Physiology (general)
92-08 Computational methods for problems pertaining to biology
93E99 Stochastic systems and control
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