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The authors present a tutorial description of adaptive fractal analysis (AFA). AFA utilizes an adaptive detrending algorithm to extract globally smooth trend signals from the data and then analyzes the scaling of the residuals to the fit as a function of the time scale at which the fit is computed. The authors present applications to synthetic mathematical signals to verify the accuracy of AFA and demonstrate the basic steps of the analysis. The authors then present results from applying AFA to time series from a cognitive psychology experiment on repeated estimation of durations of time to illustrate some of the complexities of real-world data. AFA shows promise in dealing with many types of signals, but like any fractal analysis method there are special challenges and considerations to take into account, such as determining the presence of linear scaling regions.
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METHODS ARTICLE
published: 28 September 2012
doi: 10.3389/fphys.2012.00371
A tutorial introduction to adaptive fractal analysis
Michael A. Riley1*, Scott Bonnette1,Nikita Kuznetsov 1,Sebastian Wallot2and Jianbo Gao 3,4
1Department of Psychology, Center for Cognition, Action, and Perception, University of Cincinnati, Cincinnati, OH, USA
2MINDLab, Aarhus University, Aarhus, Denmark
3PMB Intelligence, LLC, West Lafayette, IN, USA
4BME, School of Life Sciences and Technology, Xi An Jiao Tong University, Xian, PR China
Edited by:
John G. Holden, University of
Cincinnati, USA
Reviewed by:
Andras Eke, Semmelweis
University, Hungary
Dranreb E. Juanico, Ateneo de
Manila University, Philippines
*Correspondence:
Michael A. Riley, Department of
Psychology, Center for Cognition,
Action, and Perception, University of
Cincinnati, Cincinnati, OH
45221-0376, USA.
e-mail: michael.riley@uc.edu
The authors present a tutorial description of adaptive fractal analysis (AFA). AFA utilizes
an adaptive detrending algorithm to extract globally smooth trend signals from the data
and then analyzes the scaling of the residuals to the fit as a function of the time scale
at which the fit is computed. The authors present applications to synthetic mathematical
signals to verify the accuracy of AFA and demonstrate the basic steps of the analysis.
The authors then present results from applying AFA to time series from a cognitive
psychology experiment on repeated estimation of durations of time to illustrate some
of the complexities of real-world data. AFA shows promise in dealing with many types of
signals, but like any fractal analysis method there are special challenges and considerations
to take into account, such as determining the presence of linear scaling regions.
Keywords: adaptive fractal analysis, time series analysis, fractal physiology, biosignal processing, non-linear
analysis
INTRODUCTION
Adaptive fractal analysis (AFA; Hu et al., 2009; Gao et al., 2010,
2011) is a relatively new fractal analysis method that may hold
promise in dealing with many types of real-world data. In this
paper we present a step-by-step tutorial approach to using AFA.
We begin by reviewing some basic principles of fractal processes
that will be helpful for our presentation of AFA. We then discuss
AFA and provide a guide for implementing it. We conclude with
an analysis of some synthetic signals and of some real data from
an experiment in human cognition.
FRACTAL PROCESSES
Many physiological and behavioral processes exhibit fractal
dynamics. This means the measured patterns of change over
time—the behavioral time series—exhibit certain properties,
including self-similarity and scaling (Lebovitch and Shehadeh,
2005). Self-similarity means that the patterns of fluctuations at
faster time scales mimics the patterns of fluctuations at slower
time scales. Scaling means that measures of the patterns (such as
the amount of variability present) depend on the resolution or
the time scale at which the measurements have been taken. Many
fractal analyses, including AFA, focus explicitly on how a measure
of variability scales with the size of a time window over which the
measure is calculated. Gao et al. (2007)providedasuccinctand
comprehensive treatment of various fractal analysis methods.
When conducting fractal analysis of a time series it is impor-
tant to understand the concepts of fractional Gaussian noise
(fGn) and fractional Brownian motion (fBm), and the differ-
ences between the two. fGn is a stationary, long-memory pro-
cess, whereas fBm is a non-stationary, long-memory process
(Mandelbrot and van Ness, 1968; Beran, 1994; Mandelbrot,
1997). Roughly speaking, stationary processes fluctuate by a rela-
tively constant degree around amean value that remains relatively
constant over time, whereas for a non-stationary process the
statistical moments of the process (e.g., mean and variance)
are time-dependent. “Long-memory” means that the processes
exhibit statistical dependencies (correlations) over very long time
scales, as opposed to a process for which only adjacent or nearly
adjacent data points are correlated with each other. Figure 1
depicts sample time series of fBm and fGn processes.
fGn and fBm are, nominally, dichotomous types of signals.
While this is true in an important sense, fGn and fBm are
nonetheless related. The increments of a fBm process (created
by differencing the signal, i.e., subtracting each value in the time
series from the prior value) form a fGn signal [see Eke et al.
(2000), for a detailed description of the fGn-fBm dichotomy].
Stated differently, successively summing the data points in a fGn
time series will produce a fBm time series. As described below,
fGn and fBm require different treatment when using fractal meth-
ods to analyze their temporal structure, and the results of a fractal
analysis on these two different types of signals will necessarily
have different interpretations.
A parameter called the Hurst exponent, H,providesawayto
quantify the “memory” or serial correlation in a time series. The
exact meaning of Hdepends on whether a signal is fGn or fBm.
Hvalues indicate the correlation structure of a fGn signal, but
for a fBm signal the Hvalues refer to the correlation structure of
the increments obtained by differencing the time series (Cannon
et al., 1997). It is therefore necessary to carefully classify a signal
as fGn or fBm (or some other kind of signal) before proceeding
with fractal analysis of the signal.
With that caveat noted, different Hvalues indicate different
types of long-memory. Actually, H=0.5 indicates the absence
of long-memory (i.e., the process is random—it possesses no
memory meaning that data points are uncorrelated with each
other) or possesses only short-memory (correlations across very
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Riley et al. Adaptive fractal analysis tutorial
FIGURE 1 | Top: A time series of white noise, a fGn process. Bottom:a
time series of brown noise, a fBm process. A brown noise process can be
obtained by successively summing data points in the white noise process.
smallscalesonly).Thiscanbeconsideredanullhypothesisof
sorts when conducting a fractal analysis; one is often interested
in determining whether the data possess some sort of tempo-
ral structure rather than being just a truly random, uncorrelated
process.
Afindingof0<H<0.5 indicates an anti-correlated or anti-
persistent process for cases of fGn and fBm, respectively. This
means that increases in the signal (for fGn) or in the incre-
ments of the signal (for fBm) are likely to be followed by
decreases (and decreases are likely to be followed by increases)—a
negative long-range correlation. In contrast, 0.5<H<1indi-
cates a correlated process for fGn or what is termed a persistent
process for fBm. In this case, increases in the signal (for fGn) or
in the increments of the signal (for fBm) are likely to be followed
by further increases, and decreases are likely to be followed by
decreases (i.e., a positive long-range correlation). Anti-persistent
and persistent processes contain structure that distinguishes them
from truly random sequences of data.
To reiterate the point made earlier, and as Eke et al. (2000)
carefully explained, an important first step in any type of fractal
analysis is to determine the basic type of signal one has measured,
i.e., whether the signal is fGn or fBm (see also Cannon et al.,
1997). Simply plotting the time series can sometimes help the user
make a first-pass determination about whether a pre-processing
stage of integrating the data is required. Integration is required
only if the data are a stationary, noisy increment process (such as
fGn; Figure 1). Integration is not advised if the data are a non-
stationary random-walk process (such as fBm; Figure 1). The
consequences of this choice are important; Hestimates can be
artificially inflated by integration of a signal which should not
be integrated, for example, whereas a lack of integration when
it should be performed could suggest the appearance of multi-
ple scaling regions separated by a cross-over point when only one
scaling region actually exists (see Delignieìres et al., 2003).
Ofcourse,itisoftenthecasethataplotofthetimeseries
cannot be easily classified as an increment or random-walk pro-
cess based on its appearance alone. Eke et al. (2000)presenteda
strategy for determining the signal type, termed the signal sum-
mation conversion (SSC) method, in the context of a broader
approach to analyzing physiological signals that might exhibit
fractal dynamics. The method essentially involves comparison of
results obtained when the signal is integrated versus not inte-
grated. If Hvalues for the non-integrated data approach or exceed
a value of 1, then integration of the signal is generally not recom-
mended. Hvalues for non-integrated and integrated time series
generated by an ideal fBm process should differ by a value of
1; if the difference is considerably greater or less than 1 further
scrutiny of the data is required, because in that case the data
may not fit within the fBm-fGn framework (Gao et al., 2006;
Kuznetsov et al., 2012).
ADAPTIVE FRACTAL ANALYSIS
AFA is similar in some regards to detrended fluctuation analysis
(DFA; Peng et al., 1994), and many aspects of AFA will be familiar
for readers who already understand DFA. We point out some of
these similarities in our presentation of AFA to help those readers,
although familiarity with DFA is not required. Because of these
similarities, AFA shares many of the same advantages as DFA over
other fractal methods, such as the fact that Hestimated by DFA
and AFA do not saturate at 1 as is the case for other methods (Gao
et al., 2006).
But despite the similarities between the methods, there are
important differences which provide AFA with some advantages
over DFA. For example, AFA can deal with arbitrary, strong non-
linear trends while DFA cannot (Hu et al., 2009; Gao et al., 2011),
AFA has better resolution of fractal scaling behavior for short time
series (Gao et al., 2012), AFA has a direct interpretation in terms
of spectral energ y while DFA does not (Gao et al., 2011), and there
is a simple proof of why AFA yields the correct Hwhile such a
proof is not available for DFA [see Equations 6 and 7 in Gao et al.
(2011)].
It is important to note that like many other analyses used
to quantify fractal scaling AFA cannot be used independently
to assert that a process is or is not a fractal process. Because
there are non-fractal processes that can falsely give the appear-
ance of fractal scaling and long-range correlations, it is desirable
to use other methods for this purpose (e.g., Wa g e n ma kers et a l.,
2004; Delignieìres et al., 2005; Farrell et al., 2006; Torre et al.,
2007).
The first step in AFA is to identify a globally smooth trend sig-
nal that is created by patching together local polynomial fits to
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Riley et al. Adaptive fractal analysis tutorial
the time series. This is one of the primary differences between
DFA and AFA; DFA does not involve the creation of this glob-
ally smooth trend, and instead relies on discontinuous, piece-wise
linear fits. Basically, creating a globally smooth trend signal means
that one tries to recreate local features of the data using sim-
ple polynomial functions. An example is shown in Figure 2.
Small segments of the time series can be approximated reason-
ably well by adjusting the parameters of a polynomial regression
model.
We can now express these ideas in more precise terms [see
also Tung et a l . (2011), who provided a thorough description of
the detrending scheme that forms the basis of AFA]. The goal
of this step of the analysis is to create a global trend—a syn-
thetic time series v(i), i=1,2,...,N,whereNis the length of
the original time series. We denote the original time series as
u(i). Determination of the global trend is achieved by partition-
ing the original data u(i)into windows of length w=2n+1,
with the windows overlapping by n+1 points. Since setting
w(a process we describe below) determines the value of n
[i.e., n=(w1)/2], nisnotafreeparameterthatmustbe
chosen.
Within each window the best fitting polynomial of order
Mis identified. This is done through standard least-squares
regression—the coefficients of the polynomial model are adjusted
until the polynomial fits the data with the least amount of residual
error. Increasing the order Mcan usually enhance the qual-
ity of the fit, but one must be cautious about over-fitting the
FIGURE 2 | An illustration of the process of identifying a globally
smooth trend signal. Linear (Top ;M=1) or polynomial (Bottom;M=2)
trends are fit to pieces of the signal of length w(257 in this case). These
fits are shown as black lines superimposed on the original data series (gray
curves). The local fits are then stitched together (see Equation 1) to create a
smooth global trend signal, depicted in red. Notice that when the end of
the series is encountered only half of the data points in that window are
used for the trend without smoothing.
data. Typically Mshouldbe1or2i.e.,alinearorquadratic
function. The goal is not to fit every squiggle or variation in
u(i)with the polynomial model, but simply to capture any rel-
atively global trends in the data while leaving enough residual
variability to analyze further. Presently, there are no validated,
objective criteria for selecting M, so careful exploration of dif-
ferent Mvalues may be required when analyzing a given time
series.
The local fits then have to be “stitched” together in such a way
that they provide a smooth global fit to the time series. Without
this stitching, the local polynomial fits would be disconnected
with each other, as is the case for DFA. The stitching and the
resulting smooth trend signal thus represents a major distinction
between DFA and AFA. The fit to overlapping regions is created by
taking a weighted combination of the fits of two adjacent regions
to ensure that the concatenation of the local fits is smooth [math-
ematically, this means that v(i) is continuous and differentiable],
according to
y(c)(l)=w1y(i)(l+n)+w2y(i+1)(l), l=1,2,...,n+1(1)
where w1=1l1
nand w2=l1
n. According to this scheme,
the weights decrease linearly with the distance between the point
and the center of the segment. This ensures symmetry and effec-
tively eliminates any jumps or discontinuities around the bound-
aries of neighboring regions. In fact, the scheme ensures that the
fitting is continuous everywhere, is smooth at the non-boundary
points, and has the right- and left-derivatives at the boundary. By
choosing the parameters of each local fit to maximize the good-
ness of fit in each case, and then applying Equation 1 to stitch the
local fits together, the global fit will be the best (smoothest) fit to
the overall time series. Furthermore, this fitting scheme will work
with any arbitrary signal without any a priori knowledge of the
trends in the data.
Thenextstepistodetrendthedatabyremovingtheglobal
trend signal that was just created. We remove the trend because
are interested in how the variance of the residuals of the fit—the
more fine-grained fluctuations in the original time series u(i)
scale with w, as described below. This type of detrending is very
different than simply removing a linear (or higher-order) fit to
the original time series prior to data analysis (cf. Di Matteo et al.,
2003); the detrending method in AFA (and DFA) is done locally
over windows of varying length wbutnottotheentiretime
series as a whole. The residuals of the fit of the data to the trend
signal are identified by subtracting the global trend from the orig-
inal time series—we compute u(i)v(i).(Thisissimilartothe
detrending step performed in DFA, except that as noted for DFA
the local linear fits are not smoothly stitched together to create a
globally smooth trend signal, but rather are discontinuous with
respect to one another.)
These steps that have been described are then repeated for a
range of wvalues (i.e., for a range of time scales). Thus, one
must choose a minimum and maximum w,aswellasthesize
of the time steps (i.e., increases in w) used for the analysis. It
is perhaps best to begin with the smallest and largest possible w
values, i.e., w=3 samples and w=N/2samples(orN/2+1if
the time series has an even number of samples) where Nis the
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Riley et al. Adaptive fractal analysis tutorial
length of the time series. However, as discussed by Cannon et al.
(1997), exclusion of some of the smaller and larger window sizes
can increase the reliability of Hestimates. This may be a help-
ful step when analyzing signals that show a single scaling region
over some intermediate range of time scales, and where issues
such as measurement noise or insufficient time series length could
cause an apparent breakdown of scaling at smaller and larger
time scales, respectively. However, one should first ensure that the
regions under consideration for exclusion do not themselves con-
tain distinct types of fractal scaling (i.e., that the signal contains
multiple scaling regions) to avoid loss of information about the
signal. In light of such considerations, we used a wrange of 3
to (29+1=)513 samples for the analyses reported here. Any
further adjustments to the wrange can be determined after the
next step in the analysis, when one plots log2F(w)as a function
of log2w,aswedescribebelowinouranalysesofsampledata
(and see Kuznetsov et al., 2012). Typically it is sufficient to use
a step size of 1, although there may be occasions when a smaller
step size is desired to obtain better resolution for identifying lin-
ear scaling relations in the plot. In our experience, a step size
of less than 0.5 typically does not provide useful new informa-
tion, but this is an issue that should be explored for each unique
data set.
The next step is to examine the relation between the variance
of the magnitude of the residuals, F(w), and the window size, w.
For a fractal process, the variance of the residuals scales with w
(i.e., is proportional to wraised to the power H) according to
F(w)=1
N
N
i=1
(u(i)v(i))21/2
wH.(2)
Fractal scaling can be quantified through the slope (obtained
using simple linear regression) of a linear relation in a plot of
log2F(w)as a function of log2w(Figure 3). This slope provides
an estimate of the Hurst exponent, H.
It should be noted that two qualitatively different signals (one
fGn, the other fBm) could have the same Hvalue. For example, a
white noise signal (fGn, so it is integrated prior to analysis) and
a brown noise signal (fBm, so it would not be integrated prior to
analysis) would both yield H=0.5. Because of this one should
use caution performing statistical comparisons of Hfor signals
that may differ in regard to being fGn or fBm, and it is partly for
this reason that Eke et al. (2000) emphasize the need to report
signal classification along with Hvalues. For clarity, here we dis-
tinguish between Hfor these two processes using the labels HfGn
and HfBm.
The above steps constitute the basic process of applying AFA.
Often one would perform AFA on each time series in an experi-
mental data set to obtain an Hvalue(s) for each, and then submit
the set of Hvalues to standard statistical analyses (e.g., t-test or
analysis of variance) to determine if Hchangesacrossexperimen-
tal conditions or between groups of subjects. That is, Hbecomes
a dependent variable that is analyzed to determine if it changes
across levels of some factor.
In the next sections, we apply AFA to known, mathemati-
cal fractal processes and then to real-world data obtained from
an experiment on human cognition (repeated estimation of the
duration of a time interval). The application to known fractal
signals demonstrates how AFA is capable of classifying signals
in terms of H. The application to real-world data reveals the
complexities and challenges of using fractal analysis methods to
signals that are not idealized fractal processes, like most real sig-
nals in the biological, behavioral, and physical sciences. One of
these challenges is the matter of deciding how to identify linear
scaling regions for AFA (and this challenge applies to other fractal
methods, including DFA).
APPLICATIONS OF AFA
APPLICATION TO KNOWN FRACTAL PROCESSES
Here we present applications of AFA to artificially created time
series including some well-studied fractal processes. The advan-
tage of doing so is that we can compare the results of AFA to
what should be the “right” answers based on a priori, mathe-
matical knowledge of the artificial time series. Consistent with
the goal of this paper to serve as a tutorial for using AFA, we
do not mean for this to represent a fully comprehensive test of
the method, but rather a straightforward, minimal demonstration
that the method correctly identifies these simple “toy” signals. We
present results of AFA applied to time series of random, white
noise, and two idealized fractal processes known as pink noise and
brown noise.
Synthetic time series properties
The artificial time series were generated using MATLAB (The
MathWorks, Inc.; Natick, MA). Ten time series of length
N=10,000 were generated for each of three categories of signals
using an inverse Fourier transform (Lennon, 2000): White, pink,
and brown noise (see Figure 4). Initially, DFA was used to verify
that the synthetic time series we created indeed had the desired
mathematical characteristics. The integrated white, integrated
pink, and non-integrated brown series were found to have mean
(±1SD)Hvalues of HfGn =0.49 ±0.01, HfGn =0.97 ±0.01,
and HfBm =0.51 ±0.01, respectively. The close correspondence
between those results and the theoretical values of HfGn =0.5,
HfGn =1.0, and HfBm =0.5, respectively, indicates that the sim-
ulations produced accurate simulations of fractal processes. Based
on our a priori knowledge of the signals, confirmed by visual
inspection of stationarity of the time series and these preliminary
checks using DFA, only the white and pink noise time series were
integrated prior to AFA. The brown noise time series were not
integrated.
Data reduction and analysis
The AFA steps described above were implemented on the set
of 30 synthetic time series. Parameters of window size w=0.5
and polynomial orders of M=1andM=2werechosenforthe
analyses (AFA was performed once with each polynomial order).
Sample AFA plots are shown in Figure 5.
Results
For the white noise time series, using polynomial orders of M=1
and M=2, AFA returned mean Hvalues of HfGn =0.49 ±0.01
and HfGn =0.50 ±0.01, respectively. The pink noise time series
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Riley et al. Adaptive fractal analysis tutorial
FIGURE 3 | On the left is depicted a demonstrations of how the fits to
different window sizes wrelate to the AFA plot, shown on the right. The
AFA plot is a plot of log2F(w)(i.e., variance of the residual to the globally
smooth trend signal) as a function of log2w(i.e., time scale or window size).
A linear relation in this plot captures fractal scaling, and the slope of the line
of best fit provides an estimate of the Hurst exponent H. For visual simplicity
we only depicted non-overlapping window edges with the dotted gray line,
while the analysis uses overlapping windows.
were also effectively categorized by AFA in the original time series.
AmeanHvalue of HfGn =0.98 ±0.01 was obtained using a
polynomial order M=1andameanvalueofHfGn =0.99 ±
0.02 was found using a polynomial order M=2. Lastly, AFA
successfully characterized the non-integrated synthetic brown
noise time series. Using polynomial orders of M=1andM=2,
AFA returned mean HfBm values of 0.51 ±0.02 and 0.52 ±0.01,
respectively.
Discussion
The application of AFA to the synthetic time series indicated that
AFA is able to characterize the types of noise with a similar accu-
racy as DFA. The obtained Hvalues corresponded very closely
to the theoretically expected values and to the values obtained by
DFA (presented earlier). The estimates also exhibited high reli-
ability (low SD values). Changing the polynomial order Mhad
very small consequences for these synthetic data; M=2resulted
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Riley et al. Adaptive fractal analysis tutorial
FIGURE 4 | Sample time series of white (top), pink (middle), and brown (bottom) noise.
in slightly better estimates for white and pink noise (and for this
polynomial order AFA produced slightly more accurate estimates
than did DFA), but slightly worse estimates for brown noise.
APPLICATION TO REAL-WORLD DATA FROM A COGNITIVE
PSYCHOLOGY EXPERIMENT
We analyzed time series produced by a single participant who
repeatedly performed a cognitive task (estimating the duration of
a temporal interval) over the course of multiple experimental ses-
sions. The task of repeated temporal estimation is frequently used
to study the variability of human time estimation (Delignières
and Torre, 2011) and was one of the first reported cases of 1/f
noise in human cognitive behavior (Gilden et al., 1995).
Experimental methods
A single female undergraduate student who gave informed con-
sent participated voluntarily in the study which was approved by
the Institutional Review Board at the University of Cincinnati.
She was paid $10 per session. The task required the partic-
ipant to provide repeated estimates of a 1-s time interval.
Time estimates were recorded from the presses of the spacebar
of a millisecond-accurate keyboard (Apple A1048, Empirisoft).
Response times were recorded using the Psychophysics Toolbox for
Matlab (Brainard, 1997), which recorded the time of each key
press during the experiment. We defined one time interval esti-
mate as the time from the beginning of one space bar press to the
next one.
At the beginning of each experimental session the partici-
pant listened to 20 metronome beats of the 1-s interval to be
estimated. The metronome was then turned off, and the partic-
ipant then immediately began performing the time estimation
task. A total of 1050 estimates were produced consecutively in
each experimental session, and each session lasted approximately
20 min. There were two experimental conditions that varied with
regard to the presence or absence of feedback about the accu-
racy of the estimates. In the no-feedback condition the participant
did not receive any explicit feedback about timing performance.
This condition was similar to tasks used previously in contin-
uation tapping experiments (Gilden et al., 1995; Chen et al.,
2002; Wagenmakers et al., 2004; Torre and Delignières, 2008).
In the feedback condition a computer monitor was used to
present feedback specifying the error (in ms) of the most recent
estimate on every trial. For example, if the participant hit the
space bar 250ms after 1s had passed since the previous press,
the feedback on the screen would read “250 ms late.” The par-
ticipant first completed 10 no-feedback trials, one per day on
consecutive days, and then completed 10 feedback trials (again
one per day on consecutive days). For present purposes we focus
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Riley et al. Adaptive fractal analysis tutorial
FIGURE 5 | Example log2F(w)vs. log2wplots returned by AFA for the
time series depicted in Figure 4.The plots on the left side (panels A, C,
and E) are from AFA using a polynomial order of M=1 while those on the
right side (panels B, D, and F) are from AFA using a polynomial order of
M=2. Plots Aand Bare for white noise, plots Cand Dare for pink noise,
and plots Eand Fare for brown noise. The respective HfGn (A, B, C, and D)
and HfBm (Eand F) values are shown for each signal.
on just the first and the last trial in each of the two feedback
conditions.
Data processing and results
We followed the standard procedure in the literature on tempo-
ral estimation to remove all observations less than 300 ms and
any observations falling beyond 3 SD from the mean. Such val-
ues are likely to originate from accidents such as double-tapping
the space bar or not initially pressing the bar hard enough, and
a significant number of these kinds of outlying values can have
detrimental results. From looking at plots of the data processed
in this way (Figure 6), it was clear that the time series of tem-
poral estimates were more similar to fGn than fBm (compare to
Figure 1)1. Therefore, we integrated our data prior to performing
1Following suggestions by Cannon et al. (1997)andEke et al. (2000), we per-
formed spectral analyses on the data to provide a more objective classification
of our time series as fGn or fBm. The spectral exponents ranged from 0.48 to
0.75, indicating the signals were consistent with fGn.
Session 1 Session 1
Session 10Session 10
No Feedback Feedback
FIGURE 6 | Trial series of continuous time estimates with and without
accuracy feedback after removing observations faster than 300 ms and
beyond 3 SD from the mean. The participant performed the task 10 times
in each feedback condition.
AFA. Then, the same basic steps for AFA described previously
were again implemented, but with the following additional con-
siderations taken into account. We used M=1(giventhatusing
M=2 did not show consistently better results in our analysis of
thesampletimeseries)andlog
2wstep sizes of 0.5 (because we
wanted to enhance the resolution of the AFA plots to facilitate the
identification of linear scaling regions).
When dealing with real-world data, if fractal scaling is present
it may be limited to a range of time scales (i.e., wvalues). If this is
not taken into account, it may lead to inaccuracies in the estima-
tion of H.BeforeestimatingH, then, it was important to visually
inspect the plots of log2F(w)as a function of log2wto identify
regions where linear scaling might be present. If fractal scaling
appears limited, it may be necessary to restrict the range of the lin-
ear fit to the plot to exclude regions where linear scaling does not
occur. Inclusion of regions where fractal scaling is actually absent
can lead to inaccuracies and reduce the reliability of Hestimates
(Cannon et al., 1997), and may present an unrealistic picture
of the degree to which fractal scaling really is a major feature
of the signal being analyzed. In practice, it is desirable to make
this process as objective and automated as possible to avoid bias.
Elsewhere (Kuznetsov et al., 2012) we have described this issue in
more detail, and presented a quantitative procedure designed for
this process. For the sake of this tutorial, however, we chose the
linear regions visually after inspecting the AFA plots for each trial
without the linear fits imposed to examine the possibility of linear
scaling.
As often occurs with empirical data (as opposed to pure math-
ematical fractals), some of our time series yielded slightly curved
log2F(w)functions (cf. Di Matteo et al., 2003)andhadcut-
off edge effects especially at larger time scales (w>8 or 256
estimates). Visual inspection of the AFA plots (see Figure 7)
suggested two distinct regions of linear scaling, one for low w
www.frontiersin.org September 2012 | Volume 3 | Article 371 |7
Riley et al. Adaptive fractal analysis tutorial
FIGURE 7 | AFA plots for the time series of time estimates presented in
Figure 6.The HfGn values are indicated for each scaling region.
(i.e., fast time scales) and a longer region for higher w(i.e., slower
time scales), for both feedback conditions and for both the first
and last experimental sessions. Such a finding was expected based
on previous studies that revealed HfGn <0.5overthefaster
scales and HfGn >0.5 at the slower scales (Lemoine et al., 2006;
Delignieìres et al., 2008).
In the first experimental session the fast scaling region for the
no-feedback condition spanned windows log2wfrom 1.58 to 3.17
(in terms of actual number of time estimates this corresponded
to a range of 3–9). The HfGn value associated with this region
was 0.50, indicating the presence of uncorrelated white noise. The
slower scaling region for the no-feedback condition had an HfGn
value of 0.91 (indicating a positive correlation at this scale) and
spanned windows log2wfrom 3.17 to 9 (13–513 estimates). On
the last trial, after a period of practice, the fast scaling region
showed a tendency to become slightly anti-correlated but was still
very close to white noise (HfGn =0.48) and its length decreased
compared to the first session (it now spanned 1.58–2.81 log2w,or
3–7 estimates). The slow scaling region increased in length (it now
spanned from log2w=3.17 to 9; 9 to 513 estimates) and became
more uncorrelated because its HfGn value decreased to 0.77.
A similar pattern of results was found for performance in the
feedback condition (see Figure 6,rightpanel).Thefastscaling
region during the first session spanned windows from 1.58 to
3.17 log2w(in terms of actual number of time estimates this
corresponded to a range of 3–9) and had a HfGn = 0.48, indi-
cating uncorrelated white noise dynamics on this scale. One
major difference compared to the no-feedback condition was the
shorter length of the slow scaling region in the first session, which
now spanned values of log2wfrom 3.17 to 8 (9–257 estimates).
Similar to the no-feedback condition, the dynamics at this scale
exhibited positive correlation as indexed by HfGn = 0.87. The
breakdown at larger log2wis likely due to an initial transient evi-
dent in the time series plot for this session—for about the first
100 estimates the participant consistently underestimated the 1-s
interval, but then began to estimate it more accurately. Because
this only happened during one part of the trial, this affected the
slowest scaling region of the AFA plot. At trial number 10, simi-
larly to the no-feedback condition, the fast scaling region showed
a tendency to become slightly more anti-correlated but was still
veryclosetowhitenoise(HfGn = 0.44) and its length decreased
compared to the first session (it now spanned 1.58–2.81 log2w,
or 3–7 estimates). The slow scaling region increased in length (it
now spanned log2w=3.17–9; 9–513 estimates) and became less
correlated because its HfGn value decreased to 0.79.
Discussion
Finite, real-world time series are typically more complex than
the ideal simulated noises of mathematics. For example, as was
apparent in these time series, experimental data can contain mul-
tiple scaling regions. Partly, this may be because experimental data
contain both the intrinsic dynamics of the process that generated
the signal plus the measurement noise inherent in any recording
device. Apart from that, the intrinsic dynamics of real-world sig-
nals may have singular events and non-stationarities that if severe
enough often can complicate many analyses (including AFA).
Because of this it is very important to carefully examine the raw
data and the corresponding scaling plots before conducing any
quantitative analyses.
With regard to the dynamics of cognitive performance in this
temporal estimation task, these results provide preliminary evi-
dence of the presence of practice effects in the continuous time
estimation task. Practice led to a decrease in the Hexponent of the
slow scaling region, suggesting that the responses became some-
what more uncorrelated at this scale with practice. Of course our
preliminary results have to be interpreted with caution because
they are based on single participant and there are individual dif-
ferences in the slow scaling region Hvalues in this task (To r r e
et al., 2011). The differences between feedback conditions at the
fast time scales were not expected because previous literature
reported anti-correlated dynamics at this scale (Lemoine et al.,
2006; Delignieìres et al., 2008). Feedback clearly resulted in an
increased tendency for anti-correlated, corrective dynamics at
faster time scales because participants were displayed their per-
formance with regard to the benchmark 1 s time. They appeared
to use that information to correct performance on a trial-by trial-
basis. In the no-feedback condition, this information was not
readily available, which led to essentially random performance at
thefasttimescales.
GENERAL DISCUSSION
We applied AFA to known fractal signals and to real-world data
from an experiment in human cognitive psychology that involved
the repeated reproduction of a time interval. AFA recovered the
Hvalues of the known mathematical signals with high accuracy.
This was generally true for both M=1andM=2. The choice of
polynomial order did not have a very large effect, although M=2
yielded slightly better results for the white and pink noise signals
but slightly worse results for the brown noise signal. Linear scaling
was well defined over a single region for these signals.
Application of AFA to the experimental data revealed some
of the complexities in applying fractal analyses to real data, par-
ticularly the issue of identification of linear scaling regions. We
determined the scaling regions visually and then fit lines to them
to obtain estimates of H. Often this is sufficient, but it is not an
objective process and it could be subject to bias in an experiment
that involves testing a particular hypothesis or an initial effort to
classify a previously unanalyzed type of signal. If visual selection
Frontiers in Physiology | Fractal Physiology September 2012 | Volume 3 | Article 371 |8
Riley et al. Adaptive fractal analysis tutorial
of the scaling region is used, it should be done by multiple
observers (so that inter-rater reliability can be computed) who
are blind to the experimental conditions and study hypotheses (to
avoid bias). In Ku znet sov et a l. (2012)wepresentanobjective,
quantitative technique based on model-selection methods that
could be used to identify scaling regions, but more work remains
to be done on this issue.
For the experimental time series we analyzed two linear scaling
regions were apparent rather than one. Consistent with previous
results using other analysis methods including spectral analysis
(Lemoine et al., 2006; Delignieìres et al., 2008), these regions
showed distinct slopes. The faster time scale yielded lower HfGn
and were basically random white noise processes (especially for
the no-feedback condition) with a slight tendency toward exhibit-
ing anti-correlated fluctuations. The longer time scale yielded
higher HfGn values consistent with a correlated process that was
close to idealized pink-noise. The presence of feedback had some
influence on the structure of the fluctuations of the repeated tem-
poral estimates, as did the practice afforded by performance on
consecutive experimental sessions. One of these effects was that
linear scaling for the slower time scale broke down at larger wfor
the first session in the no-feedback condition, but spanned the
entire upper range of wfor the last session. These results show
that AFA may be sensitive to experimental manipulations that
affect the temporal structure of data series both with regard to the
estimated Hvalues and the range of wover which fractal scaling
occurs.
Besides the issue of identifying linear scaling region, AFA
requires several other choices such as the step size for the win-
dow size w. Typically 0.5 or 1 log2ware used, with smaller values
providing greater resolution in the AFA plot. In principle this
choice should have little impact on Hestimates, and would not
seriously impact computation time except perhaps for extremely
long time series. It could, however, have a strong impact on the
ability to identify linear scaling regions, especially with regard
to resolving the existence of linear scaling regions at faster time
scales. The choice of polynomial order Mfor the local fits is also
important, especially for signals that may have oscillatory or non-
linear trends as higher-order polynomials may be more effective
at extracting those trends. Typical choices of 1 or 2 seemed to pro-
vide about the same accuracy in estimates of Hfor the known
signals we analyzed.
Other factors that impact the ability to identify linear scaling
include the sampling rate and the trial length, which, respec-
tively, will affect the ability to resolve faster and slower time scales.
These are important choices. A very high sampling rate might
indicate the appearance of scaling at very fast time scales, but if
those time scales are not physically realistic, one should be cau-
tious about interpreting them. Increasing trial length may help
reveal or resolve scaling over very long time scales, which may
be very important when dealing with apparently non-stationary
time series.
Ideally, AFA should be used in conjunction with other meth-
ods, and converging results should be sought. But because AFA
but has several advantages over similar methods such as DFA
(Gao et al., 2011) the results may not always agree, so care
should be taken in interpreting the results. Like all fractal analysis
methods, AFA requires careful consideration of signal properties,
parameter settings, and interpretation of results, and should not
be applied blindly to unfamiliar signals. It is particularly impor-
tant to plot and carefully inspect the time series and the AFA
plots to ensure that the apparent signal properties match with the
obtained results. In addition, as we noted previously the appear-
ance of linear scaling regions in an AFA plot is not a definitive test
for fractal scaling. When used carefully AFA may provide another
useful tool for analyzing signals that may exhibit fractal dynamics.
ACKNOWLEDGMENTS
Supported by NSF grants CMMI 1031958 (Jianbo Gao) and BCS
0926662 (Michael A. Riley).
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Conflict of Interest Statement: The
authors declare that the research
was conducted in the absence of any
commercial or financial relationships
that could be construed as a potential
conflict of interest.
Received: 29 June 2012; paper pend-
ing published: 03 August 2012; accepted:
29 August 2012; published online: 28
September 2012.
Citation: Riley MA, Bonnette S,
Kuznetsov N, Wallot S and Gao J (2012)
A tutorial introduction to adaptive
fractal analysis. Front. Physio. 3:371.
doi: 10.3389/fphys.2012.00371
This article was submitted to Frontiers
in Fractal Physiology, a specialty of
Frontiers in Physiology.
Copyright © 2012 Riley, Bonnette,
Kuznetsov, Wallot and Gao. This is
an open-access article distributed under
the terms of the Creative Commons
Attribution License,whichpermitsuse,
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