[exe_ kalauzi et al 2012] modeling the relationship between higuchi’s fractal dimension

7/23/2019 [EXE_ Kalauzi Et Al 2012] Modeling the Relationship Between Higuchis Fractal Dimension

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O R I G I N A L A R T I C L E

Modeling the relationship between Higuchis fractal dimensionand Fourier spectra of physiological signals

Aleksandar Kalauzi Tijana Bojic

Aleksandra Vuckovic

Received: 27 December 2011 / Accepted: 26 April 2012 / Published online: 17 May 2012

International Federation for Medical and Biological Engineering 2012

Abstract The exact mathematical relationship between

FFT spectrum and fractal dimension (FD) of an experi-mentally recorded signal is not known. In this work, we

tried to calculate signal FD directly from its Fourier

amplitudes. First, dependence of Higuchis FD of mathe-

matical sinusoids on their individual frequencies was

modeled with a two-parameter exponential function. Next,

FD of a finite sum of sinusoids was found to be a weighted

average of their FDs, weighting factors being their Fourier

amplitudes raised to a fractal degree. Exponent dependence

on frequency was modeled with exponential, power and

logarithmic functions. A set of 280 EEG signals and

Weierstrass functions were analyzed. Cross-validation was

done within EEG signals and between them and Weierst-

rass functions. Exponential dependence of fractal expo-

nents on frequency was found to be the most accurate. In

this work, signal FD was for the first time expressed as a

fractal weighted average of FD values of its Fourier

components, also allowing researchers to perform direct

estimation of signal fractal dimension from its FFTspectrum.

Keywords Fractal dimension FFT spectra EEG signals Weierstrass functions Higuchis method

1 Introduction

Physiological, especially EEG signals have been analyzed

for decades through linear and nonlinear methods. Among

the first group of methods, Fast Fourier Transform is by far

the most frequently used approach, transferring the signal

from time to frequency domain and yielding amplitude and

phase spectra. More recently, nonlinear methods, origi-

nating from the chaos theory, provided a separate and

(partly) independent set of information about the signal,

consequently about the system generating it [1,2,14,22].

One of the most frequently used quantities was the fractal

dimension (FD), serving as a measure of signal complexity

[7, 12, 13, 17]. The term fractal dimension refers to a

non-integer or fractional dimension of a geometric object.

Applications of FD within this framework include those in

time domain and the ones in the phase space. In case of the

former, FD of the original waveform is directly estimated

in the time domain, where the original signal itself is

considered a geometric figure. Phase space approaches

estimate the FD of an attractor in statespace domain. The

phase space representation of a nonlinear, autonomous, and

dissipative system can contain one or more attractors with

generally fractional dimension. Geometric objects with

fractal dimension 1\FD\ 2, such as the famous Koch

curve, are characterized by the property of self-similarity.

Their points neither lie on a one-dimensional curve, nor

A. Kalauzi (&)

Department for Life Sciences, Institute for Multidisciplinary

Research, University of Belgrade, KnezaViseslava1,

11000 Belgrade, Serbiae-mail: [email protected]

T. Bojic

Laboratory for Radiobiology and Molecular Genetics,

Laboratory 080, Vinca Institute of Nuclear Sciences,

University of Belgrade, p.fah 522, 11001 Belgrade, Serbia

e-mail: [email protected]

A. Vuckovic

Biomedical Engineering Division, School of Engineering,

University of Glasgow, Glasgow, UK

e-mail: [email protected]

1 3

Med Biol Eng Comput (2012) 50:689699

DOI 10.1007/s11517-012-0913-9


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two-dimensional plane, but somewhere in between.

Strictly speaking, physiological signals do not meet this

requirement. However, signals can be observed as numer-

ical approximations of true fractal objects, more complex

ones having a greater FD. This forms the basis for the

construction of algorithms for FD estimation.

Many authors investigated a possible connection

between linear and nonlinear measures with various suc-cesses [6,8, 15,16,18,21,26]. Specifically, a number of

previous studies attempted to relate different partial spec-

tral signal properties with its FD value. Fox [6] and Hig-

uchi [8], among others, studied the relationship between

the power exponent and signal FD, but this result is valid

only for signals obeying the power law. On the other hand,

Navascues and Sebastian[16] derived a theoretical implicit

relationship between a signal FD and its discrete power

spectrum or its Fourier Transform [18], but related to sig-

nals that have previously been interpolated with a fractal

interpolation function. Recently, we tried to model a rela-

tionship between high frequency amplitudes ([8 Hz) andHiguchi FD of EEG signals in anesthetized rats [21].

However, a general and direct explicit mathematical rela-

tionship between amplitudes (or powers) of Fourier com-

ponents and fractal dimension of recorded experimental

signals is missing in the literature. In this article, we tried

to model one possible form of this link, directly relating

signal FD with Fourier amplitude spectra of recorded one-

dimensional signals. Optimal values of model parameters

were calculated on a set of 280 sixty-second EEG signals,

recorded from ten healthy adult individuals in two states of

vigilance (wake and drowsy) and on a set of Weierstrass

functions, having theoretically defined FD and covering

nearly the whole FD range.

2 Methods and results

2.1 Dependence of FD on the frequency of sinusoids

and signal sampling frequency

According to Higuchis algorithm [7], starting from N

signal samplesx(1), x(2), , x(N), a new self-similar time

seriesXmk , with progressively reduced sample frequency, is

calculated as:

Xmk :xm;xm k;xm 2k;. . .;x m N k

k

k

for m = 1, 2, , k, where m stands for the initial sample;

k = 2, , kmax, is the degree of sample frequency

reduction, kmax is an arbitrarily estimated parameter [19,

20] and square brackets stand for the integer part of a real

number. Signal length,Lm(k), is then computed for each of

the ktime series Xmk as

Lmk 1

k

XNmk i1

abs xm ik xm i 1k

0@

1A N 1

Nmk

k

1

where (N - 1)/([(N - m)/k]k) represents a normalization

factor. All signal lengths, Lm(k), are further averaged

forming the mean value of the signal length, L(k), for eachk = 2, , kmax:

Lk 1

k

Xkm1

Lmk:

Signal complexity, expressed as the fractal dimension,

FD, is finally estimated as the slope of least squares linear

best fit from the plot of log(L(k)) versus log(1/k).

If one applies Higuchis algorithm to surrogate sinu-

soidal waveforms, the resulting fractal dimension (FD) will

depend on two factors: frequency of the waveform itself

(fs), as well as the sampling frequency (fsamp). Dependenceof FD on the waveform frequency for two values of fsampis presented in Fig. 1. Fractal dimension calculated with

Higuchis algorithm does not depend on sinusoids

amplitude or initial phase.

It is possible to derive an exact analytical expression

for FD = u(fs, fsamp) strictly following the procedure

described in Higuchis work [7] (for details, please see

Appendix). However, due to its complexity, there is a

practical reason to model the process with a simpler

mathematical expression. From Fig.1 it can be seen that

0 5 10 15 20 25 30

1

1.2

1.4

1.6

1.8

2

fs(Hz)

FD

fb

Fig. 1 Dependence of Higuchis fractal dimension on frequency for

two series of computer-generated sinusoids with increasing frequen-

cies (fs). Solid line sampling frequency fsamp = 256 Hz, dashed line

128 Hz. White symbols FD values calculated according to the exact

formula [20, see the Appendix] for (open triangle)fsamp = 256 and

(open circle) 128 Hz

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Higuchis FD reaches its theoretical maximalvalue FD= 2 at

different breakpoint frequencies: fb & 30 Hz in case of

fsamp = 256 Hz and fb & 15 Hz forfsamp = 128 Hz. There-

fore, it is necessary to adopt FD = min{u(fs,fsamp), 2} if the

frequencies beyondfb are to be included (necessary in most

cases). Two simple models were tested:

Exponential model

FD ufs;fsamp min aefsamp expkefsamp fs 1; 2

2

Power model

FD ufs;fsamp min apfsamp fkpfsamp

s 1; 2n o

;

3

both presuming that FD ? 1 iffs ? 0. In order to compare

their validity, a series of 50 surrogate sinusoids was

generated, with equal amplitudes and random phases,

differing in fs: fi = 1, 2, , 50 Hz, i = 1,, 50. A

resampling procedure was then performed on each memberof this set, yielding a total of 500 signals (fsamp = 50, 100,

150, , 500 Hz). All their FD values were calculated using

Higuchis algorithm, FDh, with kmax = 8 [19] and each

curve, FDh(fi), differing in fsamp, was nonlinearly fitted

(Nelder-Mead algorithm, MATLAB 6.5) with models 2

and 3. The results are presented in Fig. 2. In the lowest two

panels, square fitting error was calculated according to the

expression

err2=point 1

Npofsamp

XNpofsampi1

FDhfi;fsamp

ub

fi;f

samp2

where ub(fi,fsamp) denotes nonlinear part of the model

u(fi,fsamp), where frequencies fi lie to the left of the

breaking point fb, i.e., where ub(fi, fsamp) B 2. Further,

Npo(fsamp) is the number of fitting points. It depends on

fsampand is equal to the number of generated sinusoids for

which FDh(fi,fsamp) B 2. For fsamp = 50, 100, 150, 200,

250, 300, 350, 400, 450, 500 Hz, these numbers were

Npo = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, respectively.

As can be seen on the lowest panels of Fig.2, exponential

model proved to be more accurate than the power model for

all sampling frequencies. More, parameter ae was very

weakly dependent on fsamp (ae = 0.02538 0.00037) andcould be treated as a constant. On the other hand, 1/ke was

directly proportional to fsamp: linear regression yielded

fsamp = 31.42144/ke ? 1.68349 Hz, with the corresponding

Pearsons coefficient r = 0.999997. Therefore, it can be

approximated withke = 31.42/(fsamp - 1.68) & 31.4/fsamp

0 200 4000

0.05

0.1

ae

0 200 4000

0.5

1

ke

0 200 4000

0.001

fsamp (Hz)

err

2/point

0 200 400 600

0

2

4

6

x 10-3

ap

0 200 400 6002.7

2.8

2.9

3

3.1

kp

0 200 4000

0.001

fsamp (Hz)

err

2/point

Fig. 2 Model parameter and

square error values, obtained by

fitting FD = u(fs, fsamp) with

exponential model (2) (left

panels) and power model (3)

(right panels), for different

sampling frequencies

Med Biol Eng Comput (2012) 50:689699 691

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s, since for most physiological signals fsamp 1.68 Hz. Incase of exponential model and for fsamp = 256 Hz, break-

point frequency could be calculated as:

aeexpkefsampfb 1 2 ) fb 1=ke256 log1=ae 29:74Hz:

Also, a relationship between fsamp and fb could beestablished:

fb log 1=ae =31:4 fsamp 0:117 fsamp:

2.2 Fractal dimension of a finite sum of sinusoids

Higuchis fractal dimension of the sum of two sinusoids is

dependent on their frequencies as well as their amplitudes.

In other words, FD of a signal given by:

yt y1t y2t A1sin2pf1t u1 A2sin2pf2t u2 4

depends on A1, A2, f1 and f2. As in this case, only relativeamplitudes contribute to total FD, it is convenient to introduce

reciprocal (or inverse) amplitude variability, where two

amplitudes vary in opposite directions, their sum remaining

constant. This can be formulated by introducing a relative

amplitude factor,ka = A1/(A1 ? A2),(0 B ka B 1), allowing

the signal to be written as:

yt y1t y2t A1 A2 kasin2pf1t u1

1 ka sin2pf2t u2: 5

As stated before, FD is generally a function ofA1,A2,f1

and f2. However, by introducing the concept of reciprocalamplitude variability, we actually reduced the number of

independent variables to three: ka, f1 and f2. For a chosen

pair of sinusoids, by fixing f1and f2, FD could be observed

as a function of only one variable: FD(y(t)) = f(ka). As

Higuchis FD is insensitive to signal amplitude, it is

convenient to assume that A1 ? A2 = 1. This dependence

is presented in Fig.3a, for two cases where f1 and f2 differ

for 5 Hz and for 17 Hz.

In both cases FD lies within the limits set by FD values

of the two componential sinusoids, i.e., FD(y1) B

FD(y1 ? y2) B FD(y2), where equalities apply whenka = 0

or ka = 1. Therefore, FD(y1 ? y2) might be expressed, in thefirst approximation, as a weighted average of FD(y1) and

FD(y2), where amplitudes act as weighting factors:

FDy1 y2 A1FDy1 A2FDy2

A1 A2: 6

By substituting (2) into (6), we obtain:

FDy1 y2 A1uf1;fsamp A2uf2;fsamp

A1 A2: 7

The same is valid in case of three componential sinusoids:

yt y1t y2t y3t A1 A2 A3

ka1 sin2pf1tu1 1 ka1

ka2 sin2pf2tu2

1 ka2 sin2pf3tu3

8

where ka1 = A1/(A1 ? A2 ? A3), ka2 = A2/(A2 ? A3). An

example of this type is presented in Fig. 3b, where the sum

contains sinusoids with frequencies f1 = 5 Hz, f2 = 10 Hz

and f3 = 15 Hz. Here FD(y1 ? y2 ? y3) lies between

FD(y1) = 1.0179 and FD(y3) = 1.1704 for all values of

ka1 and ka2. Therefore, for a finite sum of n sinusoids, by

generalizing (7) we obtain:

FDy FDXni1

yi

!

Pni1

Aiufi;fsamp

Pni1

Ai

: 9

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

ka

FD(

Higuchi)

FD(19Hz)=1.2871

FD(2Hz)=1.0029

FD(8Hz)=1.0462

FD(13Hz)=1.1256

a

0

0.5

1

0

0.5

1

1

1.05

1.1

1.15

1.2

ka2ka1

FD

b

Fig. 3 a Fractal dimension of the sum of two sinusoids with

reciprocally varying amplitudes. Solid line, two signals having

relatively close frequencies (f1 = 8 Hz; f2 = 13 Hz); dashed line, a

bigger difference in frequencies (f1 = 2 Hz; f2 = 19 Hz). b Fractal

dimension of the sum of three sinusoids (f1 = 5; f2 = 10;

f3 = 15 Hz), as function of the two corresponding relative amplitude

factorska1 and ka2, according to (8)


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In case of two sinusoids, if the weighted average was

exactly as in expressions (6) and (7), dependence of

FD(y1 ? y2) on kawould be linear, resulting in two straight

lines in Fig. 3a. As this is not the case, models for these

dependencies should be nonlinear. Even if an exact formula

for FD(y1 ? y2) = f(ka) was found, analogous to

FD = u(fi, fsamp) derived in the Appendix, it would

be extremely complicated. A most direct nonlineargeneralization of (7) is to introduce the power function of

amplitudes, Akex, and explore whether the introduced

exponent kex depends on fi. We fitted the solid line of

Fig.3a (FD of the sum of 8 Hz and 13 Hz sinusoids) with

two versions of the power function, differing in the number

of explanatory variables,

FDy1 y2 ka

kexFDy1 1 ka

kexFDy2

kakex 1 ka

kex 10

FDy1 y2 ka

kex1 FDy1 1 kakex2 FDy2

ka

kex1

1 ka

kex2:

11

A significant reduction in square error per point (more

than 73 times), and a rise in Radj2 from 0.880 to 0.998 was

obtained when expression (11) was applied (Fig. 4). The

fact that resulting values of kex1 = 4.07 and kex2 = 2.27

mutually differed (Fig.4b), clearly indicated that kex was

dependent on fi. For assessing the fitting quality, we used

the adjusted coefficient of determination [23]:

R2adj 1 1 R2

nd 1

nd par 1;

whereR2 stands for the ordinary determination coefficient;

nd is the number of data; par is the number of explanatory

variables (model parameters). As this coefficient is adjus-

ted to the number of explanatory variables, its use is rec-

ommended whenever qualities of regressions, differing in

par, are to be compared (as in Fig.4).

After substituting ka = A1/(A1 ? A2) and taking into

account that A1 ? A2 = 1, (11) reduces to

FDy1 y2

A1A1A2

kex1FDy1

A2A1A2

kex2FDy2

A1A1A2

kex1 A2

A1A2 kex2

A1

kex1 FDy1 A2kex2 FDy2

A1kex1 A2

kex2:

12

Considering Eqs. (2) and (3),

FDy1 y2 A1

kex1uf1;fsamp A2kex2uf2;fsamp

A1kex1 A2

kex2;

13

and in general case

FDy FDXni1

yi

!

Pni1

Aikexfiufi;fsamp

Pn

i1Ai

kexfi: 14

Finally, if the exponential model for u(fi,fsamp) is applied,

FDy FDXni1

yi

!

Pni1

Aikexfi min ae expkefsampfi 1; 2

Pni1

Aikexfi

:

15

0 0.2 0.4 0.6 0.8 11.04

1.06

1.08

1.1

1.12

1.14

ka

FD(

Hig

uchi)

err2/N = 1.20x10

-4

kex = 2.25

a

R2

adj= 0.88027

0 0.2 0.4 0.6 0.8 11.04

1.06

1.08

1.1

1.12

1.14

ka

FD(

Higuchi)

kex1= 4.07

kex2= 2.27

err2/N = 1.63x10

-6

b

R2

adj= 0.99837

Fig. 4 Modeling Higuchis fractal dimension of the sum of two sinusoids

(8 and13 Hz),with reciprocallyvarying amplitudes.Plus symbolsindicate

calculated FD, solid line indicates fitted with model. a One explaining

variable, kex, Eq.(10). b Twoexplainingvariables,kex1andkex2,Eq.(11)


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Equation (15) could be applied directly to an amplitude

FFT spectrum of a finite length one-dimensional signal,

whereAiand fi = if0 represent amplitude and frequency of

the ith Fourier component, respectively, f0 being the fre-

quency resolution. If calculated by (15), FD(y) can be

conveniently denoted as FD spectral. A mathematical

model for the function kex(fi) remained to be found. After

some elementary experimenting, we concluded that kex(fi)should be modeled with an ascending monotonous function

of frequency. Namely, by introducing descending functions

for kex(fi), we either obtained senseless results for FD

spectral (e.g., values were either beyond the natural range

1\FD\ 2), or FD spectral deviated from Higuchis FD

more than in case of ascending functions.

2.3 Application of the spectral method

on physiological signals

To test our method, we looked for a sufficiently rich set of

physiological signals, covering as much of the FD range aspossible. We chose the collection of 280 EEG signals

described in our previous papers [3, 24]. All recordings

were performed in accordance with the medical ethical

standards after the subjects signed the informed consent

form approved by the local ethical committee. Ten adult

healthy human individuals (seven males and three

females), age 2535 (mean SD: 28.3 6.5) years,

without mental disorders, were recorded after passing a

neurological screening. The subjects were lying in a dark

room with their eyes closed. A neurologist was monitoring

their state of alertness and preventing them from falling asleep

beyond N1 of NREM sleep. The individuals were not previ-

ously subjected to any sleep deprivation or deviation from

their circadian cycles and have not been taking any medicine.

Electrodes were positioned at 14 locations (F7, F8, T3, T4,T5,

T6, F3, F4, C3, C4, P3, P4, O1 and O2) according to the

International 1020 System with an average reference. Sig-

nals were sampled at a rate of 256 Hz, band pass filtered

between 0.5 and 70 Hz (with a software 50 Hz notch) and

artifacts were removed manually based on a visual inspection.

Signals from all individuals were classified into wake and

drowsy periods independently by two neurologists. Only

those signal sequences for which both experts agreed as being

either clearly awake or drowsy were used in thestudy (60 s for

each state and each subject).

However, this set of signals, consisting mainly of alpha

EEG oscillations superimposed on the Brownian-type back-

ground [3], covered only a part of the FD range (approx.

1\FD\1.5). This limitation was caused by the nature of

the Brownian fractal motion itself, known to be the first

integral of white noise. In order to stretch the signal FD values

to the remaining higher range (approx. 1.5\FD\ 2), we

included the set of their first derivatives, numerically obtained

by calculating first consecutive differences, using the diff

command in MATLAB 6.5. As expected, this operation

resultedin substituting the Brownianwith the white noise type

of background, leaving the oscillatory content essentially

intact (not shown). As expected, Higuchis FD of differenti-

ated signals stretched to occupy the upper FD range (initial

EEG signals 1.087\FD\1.506; differentiated 1.218\

FD\1.940). By uniting the two set of signals into onematrixof data (15,360 data 9 560 EEG channels) almost the entire

FD range was covered.

We tested three relatively simple three-parameter

mathematical expressions to model the ascending depen-

dence of exponent kex on frequency:

exponential kex(fi) = af 9 exp(-kf 9 fi) ? cf

power kex(fi) = af 9 (fi)kf

? cf

logarithmic kex(fi) = af 9 log(kf 9 fi) ? cf

and compared their performances, by minimizing square

error between Higuchis and spectral FD:

squerr=Nsig 1

Nsig

XNsigj1

FDhj FDsj

2; 16

whereNsigdenotes the number of signals, (FDh)jand (FDs)jstand for Higuchis and spectral FDs of the jth signal,

respectively. In order to calculate optimal numerical values

for parameters af, kfand cf, we developed originalprograms in

MATLAB 6.5 by which parameters were iteratively adjusted.

Namely, the system starts from an initial point in the param-

eter space, calculates the square error(16) for each ofthe eight

cube vertices in this space (cube size being determined by the

current accuracy) and moves to the minimal error vertex. Theprocedure is repeated until a local minimum is reached.

Resulting optimized parameters, square errors per signal, as

well as the corresponding Pearsons coefficients of linear

correlation, are presented in Table1.

As presented in Table1, all models proved to be rela-

tively successful, yielding Pearsons coefficient r[ 0.99

and squerr/Nsig\ 0.001. However, the exponential model

was most accurate according to both criteria (squerr/Nsigandr). Due to large signal ensembles, all correlations were

highly significant, p 0.001.

If we introduce the exponential model into (15), spectral

FD could be calculated as:

FDspectraly

Pni1

Aiafexpkfficf min ae expkefsamp fi 1; 2

Pni1

Aiafexpkffi cf

:

17

By using (17), the corresponding scatter plot for the 560

EEG signals, with parameter values contained in Table 1,


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and for fsamp = 256 Hz, is presented in Fig. 5a. Each dot

corresponds to one signal, with Higuchis FD on the abscissa

and spectral FD on the ordinate. All FFT spectra were

calculated using one 60 s epoch per signal, frequencies of

Fourier components beingfi = i/60 Hz,i = 1, 2, , 3,000,

in the frequency range 0.016750 Hz. However, according to

Eq. (17), for actual calculations only Fourier components

belowfb = 30 Hz were used.

2.4 Application of the spectral method on Weierstrass

functions

We also used Weierstrass functions (Wf), as another set of

data suitable for this analysis. They spread evenly across

the whole FD range and their FD values could be set as a

parameter during their generation [5, 9, 10, 12]. Analyti-

cally, they can be written as:

Wc

Ht

XiciH cos2pcit;

wherec[ 1 and 0\H\ 1,H= 2 - FD, i = 1, , 400.

For each value of parameter c = 1.1, 1.2, , 3.0, a set of

99 Wf was generated, having FD = 1.01, 1.02, , 1.99;

the total number of signals being 20 9 99 = 1,980. Fractal

dimensions of all functions were analyzed both by Higu-

chis algorithm, using kmax = 8 [19, 20], as well as using

expression (15), for each of the three kex(fi) models. These

results are also presented in Table1. Again, exponential

model showed the best results, according to both criteria.

The corresponding scatter plot is presented in Fig. 5b. Each

dot corresponds to one Wf, with Higuchis FD on the

abscissa, spectral FD on the ordinate. The latter were cal-

culated with exponential parameters contained in Table 1

and using fsamp = 256 Hz.

2.5 Testing the method by cross-validation

In order to test the reliability of the obtained results, we

formed a new ensemble of EEG signals by randomly

splitting the available set into two equally large halves,

reducing thus each subset to 280 signals. Parameters of all

Table 1 Optimized parameter values (af, kf and cf); minimized

square error per signal (squerr/Nsig); and Pearsons coefficient of

linear correlation (r) for the three models describing the dependence

of kex on frequency

Exponential

model

Power model Logarithmic

model

EEG signals

af -0.620 6.11 0.118

kf 0.0860 0.0180 17.65

cf 2.43 -4.39 1.38

squerr/Nsig 3.55 9 10-4 8.34 9 10-4 7.89 9 10-4

r 0.99613 0.99267 0.99307

Weierstrass functions

af -0.870 6.10 0.124

kf 0.119 0.0192 13.86

cf 2.23 -4.94 0.828

squerr/Nsig 5.20 9 10-4 8.24 9 10-4 8.10 9 10-4

r 0.99612 0.99398 0.99401

Results were obtained for 280 analyzed EEG signals, their 280derivatives and the set of Weierstrass functions, at sampling fre-

quency fsamp = 256 Hz, and calculated with at least three significant

digits

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 21

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

FD Higuchi

FDs

pe

ctral

0 50 1001.8

2

2.2

2.4

2.6

frequency (Hz)

kex

a

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 21

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

FD Higuchi

FDs

pectral

b

Fig. 5 Scatter plot depicting Higuchis (abscissa) and spectral

(ordinate) FD data. a For each of the 280 EEG signals as well as

their 280 derivatives (560 points); b for the analyzed Weierstrass

functions (1,980 points). Calculations were done using expression

(17), with parameter values contained in Tables1 and 2. The

corresponding dependence kex(fi) is presented as an inlet plotin the

upper left cornerof (a)


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three models were optimized on the first subset and used to

calculate spectral and Higuchis FD for each signal of the

second subset and vice versa, using fsamp = 256 Hz.

Square errors per signal, according to (16), and Pearsons

coefficients of linear correlation were calculated for eachtraining subset separately, as well as for the other, cross-

validated subset. In order to increase statistical weight of

the validation, this procedure was repeated ten times,

yielding 60 values for each of these quantities (20 per

model). The results are summarized in Table2.

It is important to note that all 120 coefficients of linear

correlation, 60 values for the training set and 60 for cross-

validated sets, were r[ 0.99 (not shown in Table2). As

can be seen in Table 2, in case of all three models, square

errors per signal were slightly greater for the cross-vali-

dated than those obtained for the training sets, while

Pearsons coefficients, as expected, were slightly reduced.

However, these differences were not significant. According

to MannWhitneyUtest (SPSS 13.0 for Windows), in case

of square errors Z = -1.298, p = 0.194 for the exponen-

tial; Z = -0.812, p = 0.417 for the power; Z= -0.920,

p = 0.358 for the logarithmic model. For Pearsons coef-

ficients, the values were Z= -0.514, p = 0.607; Z =

-0.054, p = 0.957; Z= -0.149, p = 0.882, respecting

the same order. One example of such a cross-validation for

the exponential model is presented in Fig. 6a.

Somewhat less favorable results (except for the expo-

nential model) were obtained when spectral FD for the set

of Weierstrass functions was analyzed with parameters

optimized for EEG signals, and vice versa. Specifically,

Pearsons coefficients were 0.99478 for the exponential,

0.98946 for the power and 0.98998 for the logarithmic

model when EEG was used as the training set; the values

were 0.99376 for the exponential, 0.97159 for the power

and 0.97225 for the logarithmic model when the system

was trained on Weierstrass functions. Square errors per

signal were 9.66 9 10-3 (exponential), 1.79 9 10-2

(power) and 1.71 9 10-2 (logarithmic) when EEG was the

Table 2 Mean standard deviation of 20 optimized parameter values (af,kfandcf); 20 minimized square errors per signal (squerr/Nsig) and 20

Pearsons coefficients of linear correlation (r), obtained for each model on each half from 10 random splits of the initial EEG signal set into two

Exponential model Power model Logarithmic model

af -0.620 0.009 6.22 0.13 0.118 0.003

kf 0.0860 0.0019 0.0178 0.0003 17.52 0.49

cf 2.43 0.01 -4.50 0.12 1.38 0.01

squerr/Nsig (3.53 0.30) 9 10-4

(8.28 0.78) 9 10-4

(7.84 0.75) 9 10-4

r 0.99615 0.00034 0.99273 0.00070 0.99312 0.00067

Cross-validated squerr/Nsig (3.63 0.30) 3 1024 (8.46 0.79) 3 1024 (8.03 0.76) 3 1024

Cross-validated r 0.99610 0.00036 0.99267 0.00092 0.99305 0.00089

BoldedMean standard deviation of the 20 square errors and 20 Pearsons coefficients after cross-validations performed between each of the

two corresponding halves of EEG signals

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 21

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

FD Higuchi

FDs

pectral

r = 0.99637

squerr/Nsig

= 0.000335

a

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 21

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

FD Higuchi

FDs

pectral

r = 0.99376

squerr/Nsig

= 0.00315

b

Fig. 6 Cross-validation results. a One case (out of 20) in which one

random half of the EEG set was tested by exponential model

parameters previously optimized on the other half. b Cross-validation

where the whole EEG set was tested by exponential model parameters

optimized on Weierstrass functions. Pearsons coefficient and square

error per signal are indicated in the upper left corners


1 3


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training set; 3.15 9 10-3 (exponential), 2.34 9 10-2

(power) and 2.29 9 10-2 (logarithmic) when Weierstrass

functions were used as the training set. Again, exponential

model proved to be most successful, as both corresponding

correlations exceeded 0.99. Square errors per signal were

smaller for one order of magnitude in case of exponential,

compared to the other two models, although greater than in

intra-EEG cross-validation. The result of EEG/Wf cross-validation is shown in Fig. 6b, where spectral FD values of

EEG signals were calculated with exponential model

parameters optimized on Weierstrass functions. A small

systemic FD spectral overestimation in the lower FD

Higuchi region appeared and is yet to be explored in future

work.

3 Discussion

A number of attempts were previously made to find a

relationship between signal spectral properties and itsfractal dimension. Fox [6] and Higuchi [8] established a

widely used connection between fractal FD and the expo-

nent a, FD = (5 - a)/2, but valid for signals with power-

law of the power spectral density, Pf / fa, and only inthe middle range of exponent a values (1\ a\ 3). A

positive complexityfrequency correlation was noticed in

some earlier works. Ziller et al. [26] obtained a high sta-

tistical correlation between an index of mean frequency of

EEG and FD, performing a comparative study between

nonlinear indices and Hjorth methods by means of factor

analysis.

It was already noticed that in complex spectral profiles,

amplitude variations in different frequency regions could

have opposite effects on signal FD, indicating that it might

be the result of some kind of averaging. Our experiments

with unilateral cerebral injury in rat [11] revealed that a

direct and significant correlation (r = 0.8115) existed

between FD and signal power in the EEG gamma fre-

quency range (32128 Hz), both quantities being increased

after the injury. As well, in our very recent work [21] we

found that an increase in signal FD could be obtained by

simultaneous increase in amplitudes in the higher fre-

quency range ([8 Hz) of surrogate-filtered white noise.

The process was modeled mathematically and well corre-

lated with the corresponding measurements of EEG in

anesthetized rats. However, in a somewhat lower frequency

range (alpha), an opposite phenomenon was observed in

humans. Using the same set of signals as in the present

study, we found an inverse relationship between signal

alpha amplitude and its FD and applied a four-parameter

logistic function to model the relationship [3]. Again, this

was a partial result, since the proposed model is valid only

for signals where one physiological oscillation is

superimposed on the existing background noise [3,4]. All

these findings treated the spectralfractal link partially,

lacking a full mathematical connection between the two

quantities, valid for any frequency range. Our present work

could therefore be regarded as a generalization of the

results described in these studies, the obtained relationship

being applicable to signals with any spectral profile.

Relationship between monofractal, multifractal andspectral measures of the human EEG, in different scalp

locations and sleep stages, was recently investigated [25].

The authors found that cross-correlations between these

measures had specific topographic and sleep stage prop-

erties. They also concluded that a multifractal measure

(range of fractal spectra, DD) was the best at sleep stage

classifications. Although we treated EEG signals in this

work as monofractal, as the first approximation, the results

reported in [25] suggest that it would be promising to

extend our approach to explore the relationship between

spectral and multifractal measures of EEG signals.

Dependence of such new modeling parameters on scalplocations, in different sleep stages, might also yield new

insights into properties of human EEG.

According to data presented in Tables1 and 2, the

question of choosing one among the three presented

models is still not resolved, as their performances turned to

be relatively comparable in case of physiological signals.

However, at this point, we would recommend using the

exponential model, as it performed best both when used on

separate sets as well as in cross-validation tests. This issue

will probably be resolved with the analysis of additional

signals, or with the appearance of a new and better model.

In an attempt to explain the nature of the obtained

relationship between a linear and a fractal quantity, let us

observe formula (17) and the inlet of Fig. 5. If kex was not

dependent on frequency but was rather fixed at values

kex = 1 or kex = 2, we would interpret signal FD as being

a weighted average of FDs of Fourier components with

either Fourier amplitudes or powers as weighting factors.

However, according to their fractal dimensionality,

weighting factors could be comprehended as new spectral

quantities, and we could even observeAi(fi)kex(fi) as a new

type of spectrum.

In conclusion, beside finding that signal fractal dimen-

sion can be estimated as a fractal weighted average of its

Fourier component FDs, the presented approach could

provide a practical help to researchers when analyzing

biological time-series characterized with complex dynam-

ics (EEG, ECG, respiration, etc). With our method, signal

FD could be calculated immediately after the FFT, rather

than running a separate FD algorithm. In future, we expect

our model parameters to be further refined, if tested on a

wider ensemble of signals. Finding a more suitable model

for the dependence of the exponent kex on frequency may


1 3


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also improve the technique, together with a sophistication

of the basic analytical expression itself (e.g., by introducing

a more realistic type of averaging).

Acknowledgments This work was financed by the Ministry of

Education and Science of the Republic of Serbia (Projects OI 173022

and III 41028). We thank Dr. Vlada Radivojevic and Mr. Predrag

Sukovicfor their help and cooperation in obtaining and analyzing the

data and Dr. Zarko Martinovicfor second opinions in visual analysisand scoring of EEG records.

Appendix

We shall derive an exact expression linking the frequency

of a mathematical sinusoid with its fractal dimension,

according to the procedure described in [7]. In order to

apply trigonometric summation rules, instead of the abso-

lute difference of signal samples, as in (1), we shall observe

the sum of squares

L2mk 1

k

XNmk i1

xm ik xm i 1k 2

0@

1A N 1

Nmk

k

:

18

Consequently, FD should be calculated according to a

slightly modified formula:

FD 1

2slopeln1=k; lnL2k 1 k 1;. . .; kmax;

where

L2k 1

k

Xkm1

L2mk: 19

For a given sinusoid xi a sin xiDt u, where

i = 1, ,N;a and u representing its amplitude and initial

phase and Dt= 1/fsamp, formula (18) becomes

L2mk 1

k

XNmk i1

a sin m ikxDt u

0@

a sin m i 1kxDt u 2

!N 1

Nmk k:

After transforming the difference of sinuses into

product, and application of summation rule of squared

cosines over the index i, we can write

L2mk A

B Ccos

2m

k

2xDt /

N m

k

kxDt

! cos2 m

k

2xDt /

!

where

A 4N 1a2

Nmk

k2

sin2 k

2xDt

; B

Nmk

1

2 ;

Csin Nm

k

1

kxDt

2sinkxDt

:

Asm appears within the square brackets as well, cosinesummation rule over index m cannot be performed

directly. Respecting that for most signals N m,

therefore Nmk

N

k

, after some elementary operations

equation (19) could be written as

L2k 1

k

Xkm1

AB ACcos m2xDt

N

k

1

kxDt u

A cos2 mxDtk

2xDt u

!:

Finally, according to the summation rules,

L2k AB

k

AC

kL1

A

kL2 20

where

L1cos N

k

kxDt u

sin k 1xDt

sin xDt

cos N

k

1

kxDt u

;

L2 k 1

2

sin k 1xDt cos2u

2sinxDt cos2

k

2xDt u

:

Expression (20) was verified by calculating FD ofsinusoids for f x2p

2; 4;. . .; 30Hz; for fsamp = 256 Hz,

and f x2p

= 1, 2,, 14 Hz, for fsamp = 128 Hz and the

results presented as symbols in Fig. 1.

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