Content uploaded by Scott Bonnette

Author content

All content in this area was uploaded by Scott Bonnette on Jan 11, 2014

Content may be subject to copyright.

Available via license: CC BY 4.0

Content may be subject to copyright.

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

www.frontiersin.org September 2012 | Volume 3 | Article 371 |1

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

Frontiers in Physiology | Fractal Physiology September 2012 | Volume 3 | Article 371 |2

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=(w−1)/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 Mshouldbe1or2—i.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=1−l−1

nand w2=l−1

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

www.frontiersin.org September 2012 | Volume 3 | Article 371 |3

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

Frontiers in Physiology | Fractal Physiology September 2012 | Volume 3 | Article 371 |4

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

www.frontiersin.org September 2012 | Volume 3 | Article 371 |5

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

Frontiers in Physiology | Fractal Physiology September 2012 | Volume 3 | Article 371 |6

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).

REFERENCES

Beran, J. (1994). Statistics for Long-

Memory Processes. Boca Raton, FL:

CRC Press.

Brainard, D. H. (1997). The psy-

chophysics toolbox. Spat. Vis. 10,

433–436.

Cannon, M. J., Percival, D. B., Caccia,

D. C., Raymond, G. M., and

Bassingthwaighte, J. B. (1997).

Evaluating scaled windowed vari-

ance methods for estimating the

Hurst coefﬁcient of time series.

Physica A 241, 606–626.

Chen, Y., Repp, B., and Patel, A. (2002).

Spectral decomposition of variabil-

ity in synchronization and continu-

ation tapping: comparisons between

auditory and visual pacing and feed-

back coditions. Hum. Mov. Sci. 21,

515–532.

Delignieìres, D., Deschamps, T., Legros,

A, and Caillou, N. (2003). A

methodological note on non-linear

time series analysis: is Collins

and De Luca (1993)’s open- and

closed-loop model a statisti-

cal artifact? J. Mot. Behav. 35,

86–96.

Delignières, D., and Torre, K. (2011):

Event-based and emergent tim-

ing: dichotomy or continuum?

A reply to Repp and Steinman

(2010). J. Mot. Behav. 43,

311–318.

Delignieìres, D., Torre, K., and

Lemoine, L. (2005). Methodological

issues in the application of

monofractal analyses in psy-

chological and behavioral research.

Nonlinear Dynamics Psychol. Life

Sci. 9, 451–477.

Delignieìres, D., Torre, K., and

Lemoine, L. (2008). Fractal models

for event-based and dynamical

timer. Acta Psychol. 127, 382–397.

Di Matteo, T., Aste, T., and Dacorogna,

M. M. (2003). Scaling behaviors

in differently developed markets.

Physica A 324, 183–188.

Eke, H. A., Bassingthwaighte, P.

J.,Raymond,G.,Percival,D.,

Cannon, M., Balla, I., and Ikrényi,

C. (2000). Physiological time

series: distinguishing fractal noises

from motions. Pﬂügers Arch. 439,

403–415.

Farrell, S., Wagenmakers, E.-J.,

and Ratcliff, R. (2006). 1/f

noise in human cognition: is

it ubiquitous, and what does it

mean? Psychon. Bull. Rev. 13,

737–741.

Gao,J.B.,Cao,Y.H.,Tung,W.W.,and

Hu, J. (2007). Multiscale Analysis of

Complex Time Series: Integration of

Chaos and Random Fractal Theory,

and Beyond. Hoboken, NJ: Wile y

Interscience.

Gao,J.B.,Hu,J.,Mao,X.,andPerc,

M. (2012). Culturomics meets

random fractal theory: insights

into long-range correlations of

social and natural phenomena over

the past two centuries. J. R. Soc.

Interface 9, 1956–1964.

Gao,J.B.,Hu,J.,andTung,W.W.

(2011). Facilitating joint chaos and

fractal analysis of biosig nals through

nonlinear adaptive ﬁltering. PLoS

ONE 6:e24331. doi: 10.1371/jour-

nal.pone.0024331

Gao, J. B., Hu, J., Tung, W. W., Cao, Y.

H., Sarshar, N., and Roychowdhury,

V. P. (2006). Assessment of long

range correlation i n time series: how

to avoid pitfalls. Phys. Rev. E 73,

016117.

Gao,J.B.,Sultan,H.,Hu,J.,andTung,

W. W. (2010). Denoising nonlin-

ear time series by adaptive ﬁltering

and wavelet shrinkage: a compari-

son. IEEE Signal Process. Lett. 17,

237–240.

Gilden, D. L., Thornton, T., and

Mallon, M. (1995). 1/f noise in

www.frontiersin.org September 2012 | Volume 3 | Article 371 |9

Riley et al. Adaptive fractal analysis tutorial

human cognition. Science 267,

1837–1839.

Hu,J.,Gao,J.B.,andWang,X.

S. (2009). Multifractal analysis of

sunspot time series: the effects of

the 11-year cycle and Fourier trun-

cation. J. Stat. Mech. P02066.

Kuznetsov, N., Bonnette, S., Gao, J., and

Riley, M. A. (2012). Adaptive frac-

tal analysis reveals limits to fractal

scaling in center of pressure tra-

jectories. Ann. Biomed. Eng. doi:

10.1007/s10439-012-0646-9. [Epub

ahead of print].

Lebovitch, L. S., and Shehadeh, L. A.

(2005). “Introduction to fractals,”

in Tutorials in contemporary nonlin-

ear methods for the behavioral sci-

ences, edsM.A.RileyandG.C.Van

Orden 178–266. Retrieved June 28

2012, from http://www.nsf.gov/sbe/

bcs/pac/nmbs/nmbs.jsp

Lemoine, L., Torre, K., and Delignières,

D. (2006). Testing for the pres-

ence of 1/f noise in continuation

tapping data. Can. J. Exp. Psychol.

60, 247–257.

Lennon, J. L. (2000). Red-shifts and

red herrings in geographical ecol-

ogy. Ecography 23, 101–113.

Mandelbrot, B. B. (1997). Fractals and

Scaling in Finance. New York, NY:

Springer-Verlag.

Mandelbrot, B. B., and van Ness, J.

W. (1968). Fractional Brownian

motions, fractional noises and

applications. SIAM Rev. 10,

422–437.

Peng,C.K.,Buldyrev,S.V.,Havlin,S.,

Simons, S. M., Stanley, H. E., and

Goldberger, A. L. (1994). Mosaic

organization of DNA nucleotides.

Phys.Rev.E49, 1685–1689.

Torre, K., Balasubramaniam, R.,

Rheaume, N., Lemoine, L., and

Zelaznik, H. (2011). Long-range

correlation properties in motor

timing are individual and task

speciﬁc. Psychon. Bull. Rev. 18,

339–346.

Torre, K., and Delignières, D. (2008).

Unraveling the ﬁnding of 1/f noise

in self-paced and synchronized

tapping: a unifying mechanistic

model. Biol. Cybern. 99, 159–170.

Torre, K., Delignières, D., and Lemoine,

L. (2007). 1/fβﬂuctuations in

bimanual coordination: an addi-

tional challenge for modeling. Exp.

Brain Res. 183, 225–234.

Tung,W.W.,Gao,J.B.,Hu,J.,and

Yang, L. (2011). Recovering chaotic

signals in heavy noise environments.

Phys. Rev. E 83, 046210.

Wagenmakers, E. J., Farrel, S., and

Ratcliff, R. (2004). Estimation

and interpretation of 1/f-noise in

human cognition. Psychon. Bull.

Rev. 11, 579–615.

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,

distribution and reproduction in other

forums, provided the original authors

and source are credited and subjectt oany

copyright notices concerning any third-

party graphics etc.

Frontiers in Physiology | Fractal Physiology September 2012 | Volume 3 | Article 371 |10