For a self-similar process with , the fluctuations grow with the window size in a power-law way. Therefore, the fluctuations on large observation windows are exponentially larger than those of smaller windows. As a result, the time series is unbounded. However, for most physiologic time series of interest, such as heart rate and gait, are bounded--they cannot have arbitrarily large amplitudes no matter how long the data set is. This practical restriction causes further complications for our analyses. Consider the case of the heart rate time series shown in Fig. 3a. If we zoom in on a subset of the time series, we notice an apparently self-similar pattern. To visualize this self-similarity, we do not need to rescale the y-axis ()--only rescaling the x-axis is needed. Therefore, according to Eq. 3, the self-similarity parameter is 0--not an informative result. Consider another example where we randomize the sequential order of the original heart rate time series generating a completely uncorrelated ``control'' time series (Fig. 3b)--white noise. The white noise data set also has a self-similarity parameter of 0. However, it is obvious that the patterns in Figs. 3a and b are quite different. An immediate problem, therefore, is how to distinguish the trivial parameter 0 in the latter case of uncorrelated noise, from the non-trivial parameter 0 computed for the original data.
Figure: A cardiac inter-heartbeat interval (inverse of heart rate) time series is shown in (A) and a randomized control is shown in (B). Successive magnifications of the sub-sets show that both time series are self-similar with a trivial exponent (i.e., ), albeit the patterns are very different in (A) and (B).
Physicists and mathematicians have developed an innovative solution for this central problem in time series analysis [8, 9]. The ``trick'' is to study the fractal properties of the accumulated (integrated) time series, rather than those of the original signals [7, 10]. One well-known physical example with relevance to biological time series is the dynamics of Brownian motion. In this case, the random force (noise) acting on particles is bounded, similar to physiologic time series. However, the trajectory (an integration of all previous forces) of the Brownian particle is not bounded and exhibits fractal properties that can be quantified by a self-similarity parameter. When we apply fractal scaling analysis to the integrated time series of Figs. 3a and b, the self-similarity parameters are indeed different in these two cases, providing meaningful distinctions between the original and the randomized control data sets. The details of this analysis will be discussed in the next section.
In summary, mapping the original bounded time series to an integrated signal is a crucial step in fractal time series analysis. In the rest of this chapter, therefore, we apply fractal analysis techniques after integration of the original time series.