6. Relationship between fractal dimension and spectral exponent

We may continue this exercise of comparing our various statistical parameters by considering the spectral exponent β as a means of quantifying the nature of a fractal trace. In practice, it is impractical to utilize a spectral analysis to evaluate the fractal properties of a time-series structure, due to the imprecision (relative to the aforementioned fractal analysis techniques) of applying a power law best-fit curve to characterize a spectral decomposition of a trace. Nevertheless, we may investigate the relationship that exists between the spectral exponent β, the fractal dimension DF, and the Hurst exponent H, so long as we recognize the imprecisions of these comparisons. In particular, the spectral exponent β typically is said to relate to the Hurst exponent as β ¼ 2H þ 1, implying the relationship DF ¼ ð Þ 5 � β =2. This relationship may be derived by observing that the two-point autocorrelation function

$$G\_V(\tau) = \left< V(t)V(t+\tau) \right> - \left< V(t) \right>^2 \propto \tau^{\beta - 1} \tag{14}$$

for a trace V tð Þ is related to the quantity j j V tð Þ� <sup>τ</sup> V tð Þ <sup>2</sup> D E as

$$\left\langle \left| V(t\_{\tau}) - V(t) \right|^{2} \right\rangle = 2 \left[ \left\langle V^{2} \right\rangle - G\_{V}(\tau) \right];\tag{15}$$

comparing this result to the aforementioned relationship

$$
\left\langle \left| V(t+\tau) - V(t) \right|^2 \right\rangle \propto \tau^{2H} \tag{16}
$$

leads to the expression β � 1 ¼ 2H [14]. However, systematic study [15] demonstrates that such a relationship is generally not very robust. Indeed, it is straightforward to test this robustness: In analogy to the investigation performed in Ref. [15], we investigated the relationship between spectral exponent and fractal dimension by generating a set of 20 noise traces, each with a length of 16,384 points and with a β value between 0 and 2. Applying each of the previously discussed timeseries fractal analysis techniques to each of these traces produced a corresponding set of fractal dimensions (for the variational box-counting analysis and Dubuc's variation analysis) or Hurst exponents (for the variance analysis); these data are shown in Figure 15, with the Hurst exponents "converted" to fractal dimensions via

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 VHð Þt that extends over a total time span Δt ¼ 1 and a total vertical range ΔVH ¼ 1. Dividing the time span into n increments of width 1=n, we expect the vertical range of the portion of

on average, the portion of VHð Þt present in a given interval may be covered by <sup>Δ</sup>VH=Δ<sup>t</sup> <sup>¼</sup> <sup>1</sup>=n<sup>H</sup> <sup>=</sup>ð Þ¼ <sup>1</sup>=<sup>n</sup> <sup>n</sup>=n<sup>H</sup> square boxes of side length 1=n. Thus, the total number of square boxes of side length 1=n needed in order to cover the entire trace is expected to be n n=n<sup>H</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup>�<sup>H</sup>. If we recall that the spatial box-counting method

Deriving a relationship between the Hurst exponent and fractal dimension. A Brownian motion trace VHð Þt (H ¼ 0:5) is normalized in both dimensions to be circumscribed inside a unit square, and subsequently is divided into n intervals of width 1/n. The self-affinity of an fBm trace leads to an estimation of the number of square boxes needed to cover the trace at a given length scale, motivating a relationship between H and DF. See

<sup>H</sup> <sup>¼</sup> <sup>1</sup>=n<sup>H</sup> (see Figure 14). Accordingly,

the trace within each interval to scale as Δt

The cumulative sum of Gaussian white noise results in Brownian motion.

Figure 13.

Fractal Analysis

Figure 14.

text for details.

20

<sup>6</sup> Note that this relation only applies to time-series fractals, since the notion of a Hurst exponent is undefined for spatial fractals.

Fourier transform and subsequently computes the cumulative sum of the noise trace to yield a fractional Brownian motion trace with a specified well-defined

Fractal Analysis of Time-Series Data Sets: Methods and Challenges

DOI: http://dx.doi.org/10.5772/intechopen.81958

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 "Hin" and with a total length well in excess of its resolution limit is a suitable representative of a pure fractal structure characterized by Hin. Thus, we assume that such a trace may fairly be used as a control against which the fidelity of the above-mentioned analysis techniques may

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 Hin between 0.025 and 0.975. In this manner, we sought to evaluate not only the fidelity of each fractal analysis technique in returning the expected results for the longer 16,384-point traces, but also the effect of performing the same analyses on data sets of limited length. Next, we applied each analysis technique under consideration to each of these traces, returning either a measured Hurst exponent Hout or a measured fractal dimension Dout. In the case of the Dubuc variation analysis, which returns a measured fractal dimension, this value was "converted"<sup>7</sup> to a Hurst exponent via the relation Hout ¼ 2 � Dout. Having extracted these values of Hout for each sample fBm trace and for each analysis technique, we produced a plot of Hout vs Hin representing all fBm traces analyzed with each analysis technique; these results are displayed in Figures 16 and 17 for randomly-generated fBm traces with lengths of 16,384 points and 512 points, respectively. In each of Figures 16 and 17, each data point represents the average Hout value measured via the corresponding analysis method. Each corresponding logarithmic scaling plot was fit to a straight line between a fine-scale cutoff of five data points and a coarse-scale cutoff of 1/5 of the full length of the trace. Each error bar represents one standard deviation in the measured values averaged to yield the corresponding data point. The dashed

black line represents the ideal relationship Hout ¼ Hin; that is, data points representing traces whose measured Hout values exactly match their generating

<sup>7</sup> 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.

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 Hout vs. Hin is expected to be linear with unity slope. Based on the results of the analyses summarized in Figures 16 and 17, our results may be summarized as follows: the variational box-counting method tends to over-estimate H except in the case of high H values; the variance analysis tends to under-estimate H; the Dubuc

Hin values would fall on this line.

23

Hurst exponent.

be evaluated.

#### Figure 15.

Measured fractal dimensions of colored noise traces generated with well-defined power spectral densities β. Each data point represents the average value of DF measured with the respective fractal analysis method for the set of 20 traces at the corresponding value of β. Each error bar represents one standard deviation from the mean value of DF recorded for each set of 20 traces. Lines connecting the data points are provided as a guide to the eye. The dashed line corresponds to the relationship DF ¼ ð Þ 5 � β =2.

DF ¼ 2 � H. Plotting these measured parameters as a function of the well-defined spectral exponent used to generate each trace, we see that the relationship DF ¼ ð Þ 5 � β =2 breaks down for DF close to 1 or 2.

### 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 welldefined 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

#### Fractal Analysis of Time-Series Data Sets: Methods and Challenges DOI: http://dx.doi.org/10.5772/intechopen.81958

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 "Hin" and with a total length well in excess of its resolution limit is a suitable representative of a pure fractal structure characterized by Hin. Thus, we assume that such a trace may fairly be used as a control against which the fidelity of the above-mentioned analysis techniques may be evaluated.

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 Hin between 0.025 and 0.975. In this manner, we sought to evaluate not only the fidelity of each fractal analysis technique in returning the expected results for the longer 16,384-point traces, but also the effect of performing the same analyses on data sets of limited length. Next, we applied each analysis technique under consideration to each of these traces, returning either a measured Hurst exponent Hout or a measured fractal dimension Dout. In the case of the Dubuc variation analysis, which returns a measured fractal dimension, this value was "converted"<sup>7</sup> to a Hurst exponent via the relation Hout ¼ 2 � Dout. Having extracted these values of Hout for each sample fBm trace and for each analysis technique, we produced a plot of Hout vs Hin representing all fBm traces analyzed with each analysis technique; these results are displayed in Figures 16 and 17 for randomly-generated fBm traces with lengths of 16,384 points and 512 points, respectively. In each of Figures 16 and 17, each data point represents the average Hout value measured via the corresponding analysis method. Each corresponding logarithmic scaling plot was fit to a straight line between a fine-scale cutoff of five data points and a coarse-scale cutoff of 1/5 of the full length of the trace. Each error bar represents one standard deviation in the measured values averaged to yield the corresponding data point. The dashed black line represents the ideal relationship Hout ¼ Hin; that is, data points representing traces whose measured Hout values exactly match their generating Hin values would fall on this line.

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 Hout vs. Hin is expected to be linear with unity slope. Based on the results of the analyses summarized in Figures 16 and 17, our results may be summarized as follows: the variational box-counting method tends to over-estimate H except in the case of high H values; the variance analysis tends to under-estimate H; the Dubuc

DF ¼ 2 � H. Plotting these measured parameters as a function of the well-defined spectral exponent used to generate each trace, we see that the relationship

Measured fractal dimensions of colored noise traces generated with well-defined power spectral densities β. Each data point represents the average value of DF measured with the respective fractal analysis method for the set of 20 traces at the corresponding value of β. Each error bar represents one standard deviation from the mean value of DF recorded for each set of 20 traces. Lines connecting the data points are provided as a guide to the eye. The

7. Generating fractional Brownian motions and characterizing fractal

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 welldefined 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

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

DF ¼ ð Þ 5 � β =2 breaks down for DF close to 1 or 2.

dashed line corresponds to the relationship DF ¼ ð Þ 5 � β =2.

represent experimentally measured data sets.

analysis techniques

Figure 15.

Fractal Analysis

22

<sup>7</sup> 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.

variation analysis performs well only for H 0:5; and AFA provides an accurate estimate of H throughout the range of H values. In the case of the shorter, 512-point traces, the deviations from the ideal relationship Hout vs. Hin are more pronounced. Additionally, the precision of the estimated H values for these shorter traces suffers as well, as seen in the relatively large error bars on the data points corresponding to

Fractal Analysis of Time-Series Data Sets: Methods and Challenges

DOI: http://dx.doi.org/10.5772/intechopen.81958

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

Summarizing the fidelity of four fractal analysis methods in measuring the H value for randomly-generated 512-point fBm traces with a minimum feature size of 10 points. The scaling properties were observed over 1.01

Summarizing the fidelity of four fractal analysis methods in measuring the H value for randomly-generated 512-point fBm traces with a minimum feature size of 10 points. The scaling properties were observed over 0.71

the shorter traces.

Figure 19.

Figure 20.

25

orders of magnitude in length scale.

orders of magnitude in length scale.

Figure 16.

Plotting Hout vs. Hin for randomly-generated-16,384-point fBm traces as measured by the variational boxcounting method (yellow), adaptive fractal analysis (green), Dubuc's variation analysis (red), and the variance analysis (blue).

#### Figure 17.

Plotting Hout vs. Hin for randomly-generated 512-point fBm traces as measured by the variational boxcounting method (yellow), adaptive fractal analysis (green), Dubuc's variation analysis (red), and the variance analysis (blue).

#### Figure 18.

Comparison of a 512-point fBm trace with Hin ¼ 0:5 before (red) and after (blue) Fourier filtering to a minimum feature size of 10 points.

#### Fractal Analysis of Time-Series Data Sets: Methods and Challenges DOI: http://dx.doi.org/10.5772/intechopen.81958

variation analysis performs well only for H 0:5; and AFA provides an accurate estimate of H throughout the range of H values. In the case of the shorter, 512-point traces, the deviations from the ideal relationship Hout vs. Hin are more pronounced. Additionally, the precision of the estimated 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

#### Figure 19.

Figure 16.

Fractal Analysis

Figure 17.

Figure 18.

24

minimum feature size of 10 points.

variance analysis (blue).

variance analysis (blue).

Plotting Hout vs. Hin for randomly-generated-16,384-point fBm traces as measured by the variational boxcounting method (yellow), adaptive fractal analysis (green), Dubuc's variation analysis (red), and the

Plotting Hout vs. Hin for randomly-generated 512-point fBm traces as measured by the variational boxcounting method (yellow), adaptive fractal analysis (green), Dubuc's variation analysis (red), and the

Comparison of a 512-point fBm trace with Hin ¼ 0:5 before (red) and after (blue) Fourier filtering to a

Summarizing the fidelity of four fractal analysis methods in measuring the H value for randomly-generated 512-point fBm traces with a minimum feature size of 10 points. The scaling properties were observed over 1.01 orders of magnitude in length scale.

#### Figure 20.

Summarizing the fidelity of four fractal analysis methods in measuring the H value for randomly-generated 512-point fBm traces with a minimum feature size of 10 points. The scaling properties were observed over 0.71 orders of magnitude in length scale.

such spectral filtering is manifest in a fractal analysis even at length scales signifi-

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 Hout value measured via the corresponding analysis technique using the aforementioned cutoffs at the fine scale limit and 1/5 of the entire trace as the coarse scale cutoff limit. Each error bar represents one standard deviation in the measured values that were averaged to yield the corresponding data point. The dashed black line represents the

Examples of the logarithmic scaling plots that yielded the data summarized in

Comparison of scaling plots produced by the Dubuc variation method applied to a 512-point fBm trace with

Comparison of scaling plots produced by the adaptive fractal analysis method applied to a 512-point fBm trace with Hin ¼ 0:5 before (red) and after (blue) Fourier filtering to a minimum feature size of 10 points.

Hin ¼ 0:5 before (red) and after (blue) Fourier filtering to a minimum feature size of 10 points.

Figures 16–17 and 19–20 are provided in Figures 21–24. For purposes of

cantly greater than that of the minimum feature size.

DOI: http://dx.doi.org/10.5772/intechopen.81958

Fractal Analysis of Time-Series Data Sets: Methods and Challenges

ideal relation Hout ¼ Hin, as discussed above.

Figure 23.

Figure 24.

27

Figure 21. Comparison of scaling plots produced by the variational box-counting method applied to a 512-point fBm trace with Hin ¼ 0:5 before (red) and after (blue) Fourier filtering to a minimum feature size of 10 points.

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 Hin ¼ 0:5.

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 finescale 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

#### Figure 22.

Comparison of scaling plots produced by the variance method applied to a 512-point fBm trace with Hin ¼ 0:5 before (red) and after (blue) Fourier filtering to a minimum feature size of 10 points.

Fractal Analysis of Time-Series Data Sets: Methods and Challenges DOI: http://dx.doi.org/10.5772/intechopen.81958

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 Hout value measured via the corresponding analysis technique using the aforementioned cutoffs at the fine scale limit and 1/5 of the entire trace as the coarse scale cutoff limit. Each error bar represents one standard deviation in the measured values that were averaged to yield the corresponding data point. The dashed black line represents the ideal relation Hout ¼ Hin, as discussed above.

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

#### Figure 23.

spectrally filtered via Fourier transforms to exhibit a well-defined minimum feature size (i.e., a well-defined maximum frequency component). Specifically, each trace

Comparison of scaling plots produced by the variational box-counting method applied to a 512-point fBm trace with Hin ¼ 0:5 before (red) and after (blue) Fourier filtering to a minimum feature size of 10 points.

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

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 finescale 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

Comparison of scaling plots produced by the variance method applied to a 512-point fBm trace with Hin ¼ 0:5

before (red) and after (blue) Fourier filtering to a minimum feature size of 10 points.

was subjected to a Fourier filter that eliminates all frequency components

versions of an fBm trace with Hin ¼ 0:5.

Figure 22.

26

Figure 21.

Fractal Analysis

Comparison of scaling plots produced by the Dubuc variation method applied to a 512-point fBm trace with Hin ¼ 0:5 before (red) and after (blue) Fourier filtering to a minimum feature size of 10 points.

#### Figure 24.

Comparison of scaling plots produced by the adaptive fractal analysis method applied to a 512-point fBm trace with Hin ¼ 0:5 before (red) and after (blue) Fourier filtering to a minimum feature size of 10 points.

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.

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.

The authors wish to thank Drs. Adam Micolich, Rick Montgomery, Billy Scannell, and Matthew Fairbanks for fruitful discussions. Generous support for this

Acknowledgements

Author details

29

Ian Pilgrim\* and Richard P. Taylor

University of Oregon, Eugene, Oregon, USA

provided the original work is properly cited.

\*Address all correspondence to: pilgrim.ian@gmail.com

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

work was provided by the WM Keck Foundation.

Fractal Analysis of Time-Series Data Sets: Methods and Challenges

DOI: http://dx.doi.org/10.5772/intechopen.81958

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

#### 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

Fractal Analysis of Time-Series Data Sets: Methods and Challenges DOI: http://dx.doi.org/10.5772/intechopen.81958

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.
