Acknowledgements

illustration, each of these figures shows the logarithmic scaling plots produced by applying the corresponding fractal analysis technique to the specific pair of fBm traces illustrated in Figure 18. That is, each fractal analysis technique under consideration quantifies the fractal characteristic of the input trace by determining the slope of a best-fit line to a log–log scaling plot; Figures 21–24 provide examples of

In each of Figures 21–24, the vertical dashed lines indicate the cutoffs between which the scaling plot is fitted with a straight line whose slope is measured to determine Hout. For both traces in each of these figures, the coarsescale analysis cutoff corresponds to the location of the line labeled "1/5 of trace." The fine-scale analysis cutoff for the raw trace (red points) corresponds to the location of the line labeled "5 points" (corresponding to the data in Figure 17), while the fine-scale analysis cutoff for the filtered trace (blue points) may be chosen as 10 data points (corresponding to the data in Figure 19) or 20 data points (corresponding to the data in Figure 20), as

Contrasting the trends displayed in Figures 19 and 20 with those displayed in Figures 16 and 17 highlights the inherent challenge in assessing the fractal properties of time-series structures that suffer from limited total length and/or limited resolution/spectral content. Indeed, accommodating the impact of a minimum feature size that is significantly in excess of the trace's resolution limit generally necessitates restricting a fractal analysis to length scales larger still than even this observed minimum feature size. This in turn often restricts an analysis of scaling properties to a consideration of relatively few orders of magnitude in length. For example, performing a fractal analysis of a 512-point Fourier filtered trace using analysis cutoffs corresponding to 10 data points and 1/5 of the trace length corresponds to an analysis of the scaling behavior over barely more than one order of magnitude in length scale; attempting to increase the accuracy of the measurement by raising the fine-scale cutoff to 20 data points further reduces the scaling range to

Moreover, Figures 21–24 demonstrate the difficulty in identifying an appropriate fine-scale cutoff for fractal analysis of a time-series trace, even when the minimum feature size found in the trace is easily identifiable and/or well-defined. The examples of Figures 21–24 further highlight an important distinction between the application of fractal analysis techniques to spatial and time-series fractals. In the case of spatial fractals, it often is reasonable to expect to observe fractal scaling behavior between the length scales corresponding to physical constraints (and in particular at length scales sufficiently far from these cutoffs). By contrast, and as seen in Figures 21–24, the effect of imposing (or observing) a finite minimum feature size on a time-series trace is evident at all scales, not just at those smaller than the minimum observed period. Accordingly, and as further illustrated in Figures 21–24, this effect may impact the slope of a best-fit line to a logarithmic scaling plot (and, hence, the measured fractal dimension) even when this slope is evaluated between cutoffs that are expected to compensate for the fine-scale limi-

In light of these results, one must take care when applying these analysis techniques to data sets limited in length or spectral content, as it may be difficult to make a compelling argument for the empirical presence of fractal behavior when examining such a narrow range of length scales. Nevertheless, it is instructive to

represented by respective dashed vertical lines in Figures 21–24.

these logarithmic scaling plots.

Fractal Analysis

8. Conclusions

0.71 orders of magnitude.

tation.

28

The authors wish to thank Drs. Adam Micolich, Rick Montgomery, Billy Scannell, and Matthew Fairbanks for fruitful discussions. Generous support for this work was provided by the WM Keck Foundation.
