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Fractal Objects and Self-Similar Processes

Before describing the metrics we use to quantitatively characterize the fractal properties of heart rate and gait dynamics, we first review the meaning of the term fractal. The concept of a fractal is most often associated with geometrical objects satisfying two criteria: self-similarity and fractional dimensionality. Self-similarity means that an object is composed of sub-units and sub-sub-units on multiple levels that (statistically) resemble the structure of the whole object [7]. Mathematically, this property should hold on all scales. However, in the real world, there are necessarily lower and upper bounds over which such self-similar behavior applies. The second criterion for a fractal object is that it have a fractional dimension. This requirement distinguishes fractals from Euclidean objects, which have integer dimensions. As a simple example, a solid cube is self-similar since it can be divided into sub-units of 8 smaller solid cubes that resemble the large cube, and so on. However, the cube (despite its self-similarity) is not a fractal because it has an (=3) dimension. (Click here for a hands-on experiment about fractal curves.)

The concept of a fractal structure, which lacks a characteristic length scale, can be extended to the analysis of complex temporal processes. However, a challenge in detecting and quantifying self-similar scaling in complex time series is the following: Although time series are usually plotted on a 2-dimensional surface, a time series actually involves two different physical variables. For example, in Figure 1, the horizontal axis represents ``time,'' while the vertical axis represents the value of the variable that changes over time (in this case, heart rate). These two axes have independent physical units, minutes and beats/minute, respectively. (Even in cases where the two axes of a time series have the same units, their intrinsic physical meaning is still different.) This situation is different from that of geometrical curves (such as coastlines and mountain ranges) embedded in a 2-dimensional plane, where both axes represent the same physical variable. To determine if a 2-dimensional curve is self-similar, we can do the following test: (i) take a subset of the object and rescale it to the same size of the original object, using the same magnification factor for both its width and height; and then (ii) compare the statistical properties of the rescaled object with the original object. In contrast, to properly compare a subset of a time series with the original data set, we need two magnification factors (along the horizontal and vertical axes), since these two axes represent different physical variables.

To put the above discussion into mathematical terms: A time-dependent process (or time series) is self-similar if
where tex2html_wrap_inline997 means that the statistical properties of both sides of the equation are identical. In other words, a self-similar process, y(t), with a parameter tex2html_wrap_inline1001 has the identical probability distribution as a properly rescaled process, tex2html_wrap_inline1003, i.e., a time series which has been rescaled on the x-axis by a factor a (tex2html_wrap_inline1007) and on the y-axis by a factor of tex2html_wrap_inline1009 (tex2html_wrap_inline1011). The exponent tex2html_wrap_inline1001 is called the self-similarity parameter.

In practice, however, it is impossible to determine whether two processes are statistically identical, because this strict criterion requires their having identical distribution functions (including not just the mean and variance, but all higher moments as well). Therefore, one usually approximates this equality with a weaker criterion by examining only the means and variances (first and second moments) of the distribution functions for both sides of Eq. 1.

Figure: Illustration of the concept of self-similarity for a simulated random walk. (a) Two observation windows, with time scales tex2html_wrap_inline1015 and tex2html_wrap_inline1017, are shown for a self-similar time series y(t). (b) Magnification of the smaller window with time scale tex2html_wrap_inline1015. Note that the fluctuations in (a) and (b) look similar provided that two different magnification factors, tex2html_wrap_inline1023 and tex2html_wrap_inline1025, are applied on the horizontal and vertical scales, respectively. (c) The probability distribution, P(y), of the variable y for the two windows in (a), where tex2html_wrap_inline1031 and tex2html_wrap_inline1033 indicate the standard deviations for these two distribution functions. (d) Log-log plot of the characteristic scales of fluctuations, s, versus the window sizes, n.

Figure 2a shows an example of a self-similar time series. We note that with the appropriate choice of scaling factors on the x- and y-axis, the rescaled time series (Fig. 2b) resembles the original time series (Fig. 2a). The self-similarity parameter tex2html_wrap_inline1001 defined in Eq. 1 can be calculated by a simple relation
where tex2html_wrap_inline1023 and tex2html_wrap_inline1025 are the appropriate magnification factors along the horizontal and vertical direction, respectively.

In practice, we usually do not know the value of the tex2html_wrap_inline1001 exponent in advance. Instead, we face the challenge of extracting this scaling exponent (if one does exist) from a given time series. To this end it is necessary to study the time series on observation windows with different sizes and adopt the weak criterion of self-similarity defined above to calculate the exponent tex2html_wrap_inline1001. The basic idea is illustrated in Fig. 2. Two observation windows (Fig. 2a), window 1 with horizontal size tex2html_wrap_inline1015 and window 2 with horizontal size tex2html_wrap_inline1017, were arbitrarily selected to demonstrate the procedure. The goal is to find the correct magnification factors such that we can rescale window 1 to resemble window 2. It is straightforward to determine the magnification factor along the horizontal direction, tex2html_wrap_inline1061. But for the magnification factor along the vertical direction, tex2html_wrap_inline1025, we need to determine the vertical characteristic scales of windows 1 and 2. One way to do this is by examining the probability distributions (histograms) of the variable y for these two observation windows (Fig. 2c). A reasonable estimate of the characteristic scales for the vertical heights, i.e., the typical fluctuations of y, can be defined by using the standard deviations of these two histograms, denoted as tex2html_wrap_inline1031 and tex2html_wrap_inline1033, respectively. Thus, we have tex2html_wrap_inline1073. Substituting tex2html_wrap_inline1023 and tex2html_wrap_inline1025 into Eq. 2, we obtain
This relation is simply the slope of the line that joins these two points, (tex2html_wrap_inline1015, tex2html_wrap_inline1031) and (tex2html_wrap_inline1017, tex2html_wrap_inline1033), on a log-log plot (Fig. 2d).

In analyzing ``real-world'' time series, we perform the above calculations using the following procedures: (1) For any given size of observation window, the time series is divided into subsets of independent windows of the same size. To obtain a more reliable estimation of the characteristic fluctuation at this window size, we average over all individual values of s obtained from these subsets. (2) We then repeat these calculations, not just for two window sizes (as illustrated above), but for many different window sizes. The exponent tex2html_wrap_inline1001 is estimated by fitting a line on the log-log plot of s versus n across the relevant range of scales.

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Next: Mapping Real-World Time Series Up: Fractal Analysis Methods Previous: Fractal Analysis Methods