## Abstract

Many methods exist for quantifying the fractal characteristics of a structure via a fractal dimension. As a traditional example, a fractal dimension of a spatial fractal structure may be quantified via a box-counting fractal analysis that probes a manner in which the structure fills space. However, such spatial analyses generally are not well-suited for the analysis of so-called “time-series” fractals, which may exhibit exact or statistical self-affinity but which inherently lack well-defined spatial characteristics. In this chapter, we introduce and investigate a variety of fractal analysis techniques directed to time-series structures. We investigate the fidelity of such techniques by applying each technique to sets of computer-generated time-series data sets with well-defined fractal characteristics. Additionally, we investigate the inherent challenges in quantifying fractal characteristics (and indeed of verifying the presence of such fractal characteristics) in time-series traces modeled to resemble physical data sets.

### Keywords

- fractal
- spatial fractal
- time-series fractal
- fractal analysis
- fractal dimension
- self-similarity
- self-affinity
- topological dimension
- embedding dimension
- similarity dimension
- box-counting dimension
- covering dimension
- variational box-counting
- Hurst exponent
- variance method
- Dubuc variation method
- adaptive fractal analysis
- power-law noise
- Brownian motion
- fractional Brownian motion

## 1. Introduction

In this chapter, we explore a species of fractals known as “time-series” fractals. Such structures generally may be conceived (and visualized) as functions of independent variables whose plots exhibit shapes and patterns that are evocative of the more familiar spatial fractals. However, lacking well-defined spatial characteristics, time-series fractals call for analytical tools that depart from those of the world of spatial fractals. To lay the foundation for a discussion of such analytical tools, we begin with an overview of fractal structures and traditional fractal analysis techniques. We then introduce time-series fractals and investigate the unique analytical tools necessitated by such structures. Finally, we investigate the relative fidelity of these analytical tools, as well as the shortcomings inherent in performing fractal analysis on time-series fractals of limited length and/or fine-scale detail.

## 2. Motivating the fractal dimension

Mathematician Benoit B. Mandelbrot often is credited with introducing the notion of a fractional, or fractal, dimension in his 1967 paper, “How long is the coast of Britain?” [1]. In fact, however, the curious nature of coastline measurements had been discussed by Lewis Fry Richardson 6 years prior in the *General Systems Yearbook* [2]. Richardson, a pacifist and mathematician, sought to investigate the hypothesis that the likelihood that war would erupt between a pair of neighboring nations is related to the length of the nations’ shared border. As Richardson and Mandelbrot note, such a hypothesis is difficult to evaluate, since individual records of the length of Britain’s west coast varied by up to a factor of three. Indeed, as the precision of such measurements increases—that is, by decreasing the length of the “ruler” used to trace the profile—the measured total length appears to increase as well. This quality reflects the fact that the outline of the British coastline is an example of a “self-similar” structure—that is, a structure that exhibits the same statistical qualities, or even the exact details, across a wide range of length scales. In light of this apparent fundamental indeterminacy, Mandelbrot posits that familiar geometrical metrics such as length are inadequate for describing the complexity found in nature.

Recognizing Richardson’s prior investigations, Mandelbrot notes that Richardson had indeed produced an empirical relation between a measured coast length ^{1}, dimension to quantify the nature of such shapes.

Following Mandelbrot’s example, to generalize the concept of a geometrical dimension, we may begin by examining the scaling behavior of such trivially self-similar objects as a line, a square, and a cube. For example, consider a line segment of length *similarity dimension* of the structure in question.

Applying the concept of a similarity dimension to less trivial shapes is straightforward in the case of exactly self-similar structures, such as structures that are constructed via iteration of a generating pattern. As an example, consider the Koch curve, illustrated in Figure 2. The Koch curve is constructed as follows: Beginning with a line segment of unity length, replace the middle third of the segment with an equilateral triangle whose base has a length of 1/3 and overlies the original line segment, then remove this overlapping base segment. The resulting figure thus consists of four line segments, each of which has a length of 1/3. Iterating this process for each new line segment yields a sequence of figures that exhibit increasingly fine structure, with the limiting state of this series exhibiting exact self-similarity, in the sense that a nontrivial subset of the shape is exactly identical to the whole. This exact self-similarity is illustrated in Figure 2, which shows that the full Koch curve may be described as being formed from four exact copies of itself, each scaled down by a factor of 1/3. Thus, we can apply the above relation to find that the Koch curve has a similarity dimension of

The similarity dimension described above represents but one example of a plurality of dimensions that can be defined and calculated for a given figure. Indeed, the utility of the similarity dimension is limited by the fact that it applies only to figures that exhibit exact self-similarity; by contrast, the complexity witnessed in natural systems such as coastlines generally exhibits self-similarity only in the statistical sense. As an example, Figure 3 illustrates a structure that exhibits statistical self-similarity. Specifically, Figure 3 illustrates an example of a modified Koch curve formed by randomizing the orientations of the line segments as the structure is generated.

As a tool for quantifying the nature of such fractal structures that do not exhibit exact self-similarity, we now turn to the (roughly self-explanatory) “box-counting dimension,” also known as the “covering dimension.” Given a structure that extends in two dimensions^{2}, the box-counting dimension may be determined as follows: First, superimpose a square grid with individual boxes of size *D*. Such a plot is generally known as a scaling plot.

The box-counting method also may be described in more geometrically intuitive terms. For example, and as shown in Figure 4, one may observe that the set of all occupied boxes at a given length scale

While the box-counting method of estimating fractal dimension is conceptually straightforward, some care must be taken to preserve the utility of the method. For example, one must select an appropriate range of box sizes

Such conditions necessitate careful determination of the appropriate range of length scales over which to assess fractal scaling behavior. This determination may be made empirically, such as by observing the range of length scales over which the scaling plot is sufficiently linear. Alternatively, this determination may be made by convention, such as may be based on statistical arguments. In practice, it is generally not known *a priori* whether a structure under consideration should even be expected to *be* a fractal, and hence whether it should be expected to produce a scaling plot with a linear trend between cutoffs defined by appropriate physical and/or measurement limitations. Accordingly, it is preferred to adopt conventions with some degree of universality and that do not presuppose the existence of the fractal scaling behavior under investigation. More specifically, it is common to adopt the following conventions, noting that the ranges may be bounded by physical and/or measurement limitations. The coarse-scale analysis cutoff generally corresponds to a limit of the range of length scales measured, which in turn generally is related to the coarse-scale size of the structure itself. This limit is conventionally set at

As a further consideration in optimizing the performance of the box-counting method, one must select the position and orientation of the box grid relative to the structure in question. To the extent that the box-counting method seeks to probe an inherent quality of a structure, the observed fractal dimension should not be affected by a spatial translation or rotation of the grid with respect to the structure, since the structure itself has no preferred orientation. However, consider the case shown in Figure 5, in which the box-counting method is applied to a fractal profile. In the box-counting scheme discussed above, all boxes that contain any portion of the structure under examination are counted toward the total; applying this to the structure of Figure 5, we find that 35 boxes are filled using this box size

## 3. Time-series fractal structures

The fractal structures discussed above generally represent examples of spatial fractal structures—that is, structures with spatial extent and whose fractal characteristics are embodied in their spatial form. However, many observable structures and phenomena exhibit fractal behavior while lacking spatial form. Another important class of structures to which fractal analysis may be directed is that of “time-series” structures—that is, structures that may be represented as a single-valued function of a single independent variable. As suggested by their name, a time-series structure may refer to some variable quantity—say, stock market prices, or atmospheric pressure—that fluctuates in time, but for the purposes of this work we intend for the term to refer to any data set or plot consisting of a dependent variable that may be represented as a single-valued function of an independent variable.

As with the spatial structures considered above, a time-series structure may exhibit fractal scaling properties in either a statistical or an exact sense, which may be quantified using the formalism of fractal dimensions. Unfortunately, the box-counting methods described above for measuring a fractal dimension are ill-suited to time-series structures. Simply put, this limitation arises from the fact that box-counting methods assess the fractal dimension of shapes that extend in space, while the spatial “shape” of a time-series structure is inherently undefined. That is, since the two axes of a plot representing a time-series data set generally represent variables with distinct units, the geometric aspect ratio of such a plot is fundamentally undefined.

As an example, consider the data set displayed in Figure 6, which plots the daily closing price of a certain technology stock over a period of roughly 16 years. Specifically, Figure 6 illustrates three representations of the same data set, with the respective *y*-axis of each illustration scaled by a distinct factor. In qualitative terms, one may be tempted to conclude that the data in the top panel appear the most linear and that the data in the bottom panel appear the most space-filling. Accordingly, given that a box-counting fractal analysis technique essentially assesses the space-filling properties of a structure, applying a box-counting analysis to each plot would yield distinct results for each plot.

The difficulty here lies in the fact that a box-counting fractal analysis necessarily treats a figure as a spatial entity whose orthogonal dimensions have the same units. By contrast, a time-series trace such as the one displayed in Figure 6 lacks this property, but may still exhibit fractal characteristics in the form of either statistical or exact self-affinity. As discussed above, exact and statistical self-similarity describe structures whose precise details or statistical properties (respectively) are repeated as its orthogonal dimensions are rescaled by a similar factor. By contrast, exact and statistical self-affinity refer to structures whose precise details or statistical properties (respectively) are repeated as its two orthogonal dimensions are resized by independent quantities [4].

Due to the incommensurability of the orthogonal axes defining a time-series trace, such structures cannot exhibit self-similarity, only self-affinity. As an example, Figure 7 displays the data set shown in Figure 6 alongside a subset of the data set. When this subset is appropriately rescaled in each of the *x*- and *y*-axis, the resulting plot shares the general statistical properties of the original trace, and hence exhibits statistical self-affinity.

It also is possible, albeit less common, for a time-series trace to exhibit *exact* self-affinity. As an example, Figure 8 illustrates three experimentally measured data sets in which rescaling the *x*- and *y*-axes of the traces by carefully chosen factors produces structures that share the characteristics of the original traces [5].

## 4. Fractal analysis of time-series traces: beyond box-counting

As discussed above, when applying a box-counting method to a time-series structure, the measured scaling properties of the structure will depend on the aspect ratio with which the data are presented, which is in turn an arbitrary choice. Accordingly, applying a box-counting method to a time-series trace will return a fractal dimension that is essentially arbitrary. Thus, it is necessary to develop fractal analysis techniques that are insensitive to such artificial geometric parameters. In the following, we survey a sampling of such techniques proposed in the literature.

Returning to the example of Figure 5, above, this figure in fact illustrates the variational box-counting method as applied to fractal profile in the form of a time-series fractal. Indeed, fractal analyses of such time-series fractal structures have traditionally been performed using the variational box-counting method [6, 7], which does offer performance improvements over the traditional fixed-grid box-counting method. Nonetheless, the variational box-counting method still suffers from a fatal flaw. To see why this is so, consider the plots shown in Figure 9.

Figure 9 illustrates the stock price data of Figures 6 and 7 represented in two plots with the price axes respectively scaled by two different factors, as well as a visualization of a variational box-count method applied at a “length” scale

Of course, the fundamental issue is that the concept of an *x*- and *y*-axes such that the domain and range of the plot each run from 0 to 1, and the structure may be analyzed via a box-counting analysis that utilizes a square grid that just circumscribes the trace. While such a normalization convention may provide a consistent method for investigating the relative scaling properties among a set of related time-series traces, the absolute values of the dimensions produced by such analyses would remain essentially arbitrary.

Developing a fractal analysis technique that is appropriate for time-series structures generally amounts to taking one of two approaches: (1) to treat the time-series structure as a geometric figure without a well-defined aspect ratio, or (2) to treat the time-series structure as an ordered record of a process that exhibits a quantifiable degree of randomness. Following the latter approach, Harold Edwin Hurst introduced a formalism for quantifying the nature of self-affine time-series structures in a 1951 paper on the long-term storage capacity of water reservoirs [8].

In Ref. [8], Hurst introduces the concept of the “Hurst exponent”

The Hurst exponent of a data set may be calculated by examining the scaling properties of a “rescaled range” of the data, as follows. Consider a data set *R*/*S*) statistic is then defined as:

where

is the sample mean and

is the sample standard deviation. The quantity

is then proportional to *H*.

The Hurst exponent also may be described as a measure of long-range correlations within a data set, such that measuring these correlations as a function of interval width may provide another measurement of the Hurst exponent. As an example of such an analysis, the “variance method”^{3} calculates the scaling properties of the trace’s autocorrelation as a function of time interval^{4} via calculation of the quantity

for a range of values of *H*. In practice, however, the variance method is found to produce a poor estimate of Hurst exponent.

As another means of quantifying the fractal properties of time-series traces, we now turn our attention to a method proposed by Benoit Dubuc in a 1989 paper [9] on the fractal dimension of profiles. Dubuc’s proposed “variation method”^{5} is conceptually similar to the variational box-counting method described above, but improves upon this method by resolving the fundamental arbitrariness of drawings boxes on a time-series trace. In short, Dubuc’s variation method probes the “space-filling” characteristics of a time-series trace through measurement of the scaling behavior of the amplitude of the trace within an

In practical terms, Dubuc’s variation method may be implemented is as follows: Consider a time-series data set

where

That is, for a given value of

Having constructed the traces

Conceptually, *x* and *y* dimensions. In continued analogy with spatial box-counting analyses, the fractal dimension of the trace is then determined via the relationship *D*.

As a final means of quantifying the fractal properties of time-series traces, we consider a technique known as “adaptive fractal analysis” (AFA) [10]. Similar to Dubuc’s variation method, AFA may be broadly described as investigating the geometrical properties of a time-series trace (in contrast to the aforementioned analyses that are best understood as probing numerical correlations). For example, and as discussed above, Dubuc’s variation method may be described as quantifying the generalized “area” needed to cover a time-series trace as analyzed at different characteristic time scales; in the case of AFA, approximations to the time-series trace are generated at varying resolutions, and the fidelity of such approximations is recorded as the resolution is varied. The AFA algorithm may be executed as follows: Again, consider a time-series data set *w* such that each pair of adjacent subsets overlap by *n* data points at either end of the trace). Next, these best-fit lines are “stitched” together to form a single, smoothly continuous curve in the following manner: Label the windows that span the trace with consecutive integers, and label the windows’ corresponding best-fit lines as *j*, construct the curve

*w* (see Figure 12).

As *w* is decreased, *w* is varied is used to determine the Hurst exponent. Specifically,

such that a plot of *H*.

## 5. Evaluating fractal analysis techniques

Each of the fractal analysis techniques discussed above is best understood as providing an *estimate* of the fractal dimension or Hurst exponent that characterizes a given time-series data set. The sections that follow present a method for evaluating the fidelity of these estimates that was developed and applied by the authors to the fractal analysis techniques under consideration. To objectively and quantifiably evaluate the fidelity of each of these techniques, it is desirable to investigate the accuracy of each technique when applied to traces with known Hurst exponents/fractal dimensions. To introduce a method for producing such “control” traces, we begin with a general discussion of noise traces.

A noise trace, as an example of a time-series structure, may be described as a single-valued function of a single independent variable. A variety of methods exist for quantifying the statistical properties of noise traces. For example, in addition to the aforementioned measurements of space-filling characteristics and long-range correlations, a spectral analysis of a noise trace may offer a natural quantification of the trace’s statistical properties.

Power-law noise represents a significant and broad class of noise traces. Specifically, a power-law noise trace has a power spectral density given by

A “brown noise” trace characterized by

Relaxing the restriction that the Gaussian noise trace consists of statistically independent increments permits consecutive increments to be positively or negatively correlated, such that the plot formed by the cumulative sum of the noise trace may be characterized by a Hurst exponent that deviates from

A fractional Brownian motion trace is thus self-affine in the sense that

where

denotes that the two random functions

Quantifying self-affinity using the formalism of the Hurst exponent motivates drawing a parallel between the Hurst exponent and the fractal dimension, as follows. Following the argument of Ref. [4], consider an fBm trace *n* increments of width ^{6}

The relationship *global* correlations, while the fractal dimension may be understood as describing a trace’s *local fine-scale structure* [13].

## 6. Relationship between fractal dimension and spectral exponent

We may continue this exercise of comparing our various statistical parameters by considering the spectral exponent

for a trace

comparing this result to the aforementioned relationship

leads to the expression

## 7. Generating fractional Brownian motions and characterizing fractal analysis techniques

The framework of the investigation summarized in Figure 15 may be applied to a more thorough investigation of the fidelity of each fractal analysis technique discussed above. That is, if we generate a fBm trace with a well-defined Hurst exponent and subject such a trace to the analysis techniques under consideration, we may evaluate the robustness of each analysis technique. In so doing, we may evaluate not only the fidelity of each analysis method, but also may explore how the analysis methods (individually and/or collectively) respond to less-idealized data sets. That is, by generating fBm traces with well-defined Hurst exponents and modifying the traces to better resemble real-world data sets, we may gain insight into how best to interpret our analytical results of experimentally derived data. Specifically, in addition to testing these analysis techniques on “full-size” 16,384-point fBm traces (with 16,384 arbitrarily chosen as a “sufficiently large” number), we additionally tested these analyses on traces of reduced length and/or reduced spectral content, which may better represent experimentally measured data sets.

A variety of methods exist for generating a fractional Brownian motion trace that exhibits a well-defined predetermined Hurst exponent. Examples of such methods include random midpoint displacement, Fourier filtering of white noise traces, and the summation of independent jumps [14]. This chapter considers randomly generated fBm traces that were created using a MATLAB program that generates a fractional Gaussian noise trace with the desired Hurst exponent via a Fourier transform and subsequently computes the cumulative sum of the noise trace to yield a fractional Brownian motion trace with a specified well-defined Hurst exponent.

While such computer-generated fBm traces are accurately described as exhibiting a well-defined Hurst exponent, the inherently finite nature of these traces precludes the traces from being fully “fractal.” That is, as with any natural structure with finite extent, the generated fBm traces necessarily exhibit a fine-scale resolution limit (owing to the point-wise granularity of the traces) as well as a coarse-scale size limit (owing to the finite total length of the traces). With this in mind, we must be content to forge ahead with the simplifying assumption that the effects of these particular limitations on our estimates of the underlying fractal scaling properties are negligible when considering a computer-generated fBm trace whose total length exceeds its step increment by several orders of magnitude. Accordingly, for the purposes of this analysis, we assume that an fBm trace generated with a predetermined Hurst exponent “

The procedure for evaluating each of these analysis techniques is thus as follows: We first generated a set of 50 16,384-point fBm traces as well as 50 512-point fBm traces at each of 39 input Hurst exponents ^{7} to a Hurst exponent via the relation

In the ideal case of a perfectly fractal fBm trace subjected to an analysis technique that produces a precise and accurate value of the Hurst exponent, a plot of *H*; the Dubuc variation analysis performs well only for *H* throughout the range of *H* values. In the case of the shorter, 512-point traces, the deviations from the ideal relationship *H* values for these shorter traces suffers as well, as seen in the relatively large error bars on the data points corresponding to the shorter traces.

We also investigated the effect on the measured *H* values resulting from another common deviation from ideal fractal behavior. Specifically, in experimentally measured time-series data sets, the smallest-scale measured features often are significantly larger than the resolution limit of the trace. Such is very often the case for experimentally measured data sets that are asserted to represent fractal behavior, in which the finest-scale features may exhibit a characteristic scale that is well over an order of magnitude larger than the point-wise resolution of the trace. To probe the effect of this limitation on a fractal analysis of such a trace, we repeated the above technique on a set of randomly-generated 512-point fBm traces that had been spectrally filtered via Fourier transforms to exhibit a well-defined minimum feature size (i.e., a well-defined maximum frequency component). Specifically, each trace was subjected to a Fourier filter that eliminates all frequency components corresponding to periods shorter than 10 data points, such that the resultant traces have a minimum feature size of 10 points. Figure 22 illustrates a characteristic result of this filtering procedure by comparing the original and Fourier filtered versions of an fBm trace with

Performing a fractal analysis of time-series traces with limited spectral content requires a reassessment of the length scales over which one expects to observe the fractal scaling properties. Whereas our analysis of fBm traces whose spectral content extended to the resolution limit of the traces examined scaling properties to a minimum length scale of five data points, we now cannot expect to see such scaling properties at length scales smaller than our minimum feature size of 10 data points. Given this well-defined minimum feature size, it may be tempting to set our fine-scale analysis cutoff at 10 data points and expect to observe the desired scaling properties at all length scales greater than this. In practice, however, the effect of such spectral filtering is manifest in a fractal analysis even at length scales significantly greater than that of the minimum feature size.

The results of passing the 512-point Fourier filtered fBm traces through the fractal analysis techniques under consideration are displayed in Figures 19 and 20, which illustrate the results obtained when applying fine-scale cutoffs of 10 data points (i.e., the traces’ minimum feature size) and 20 data points, respectively. In each of Figures 19 and 20, each data point represents the average

Examples of the logarithmic scaling plots that yielded the data summarized in Figures 16–17 and 19**–**20 are provided in Figures 21–24. For purposes of 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 these logarithmic scaling plots.

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

## 8. Conclusions

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 0.71 orders of magnitude.

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 limitation.

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 examine the behavior of fractal analysis applied to known fractal structures such as fBm traces that have been artificially subjected to such constraints. For example, one may argue that an fBm trace that is Fourier filtered to exhibit a coarser minimum feature size is analogous to a natural structure or phenomenon that has been subjected to exterior influences such as weathering effects or measurement limits: both may be considered examples of structures that are legitimately generated via processes associated with fractal behavior, but whose true fractal nature has been obfuscated by secondary considerations. In the eyes of the authors, such effects do not necessarily render the resulting structures “less fractal” than their idealized counterparts. Nevertheless, such effects demand careful consideration when choosing an analysis method and an acknowledgment of the inherent limitations thereof.

## Acknowledgments

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.

## Notes

- Though Mandelbrot discusses the concept of fractional dimension in this 1967 paper, he did not introduce the term “fractal” until 1975 [3].
- While the box-counting method is typically applied to structures embedded in two dimensions, it is straightforward to generalize the technique to higher- or lower-dimensional systems.
- Not to be confused with the variational box-counting method.
- In all discussions of time-series traces, we refer to the independent variable as “time” as a matter of convention unless otherwise specified. Additionally, as a matter of convention, we refer to an interval of the independent variable as a “length” unless otherwise specified.
- Not to be confused with the variational box-counting method or the variance method.
- Note that this relation only applies to time-series fractals, since the notion of a Hurst exponent is undefined for spatial fractals.
- As discussed above, such a conversion is at best an approximation. Nonetheless, utilizing this conversion serves as a self-consistent means of evaluating the response of this analysis technique when applied to fBm traces of a known Hurst exponent, as well as deviations from this behavior.