Dynamical modelling of a genetic algorithm

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doi:10.1016/j.si

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Signal Processing 86 (2006) 2760–2770

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Dynamical modelling of a genetic algorithm

E.J. Solteiro Piresa,�, J.A. Tenreiro Machadob, P.B. de Moura Oliveiraa,c

aUniv. Tras-os-Montes e Alto Douro, Dep. de Engenharia Electrotecnica, 5000-911 Vila Real, PortugalbInstituto Superior de Engenharia do Porto, Dep. de Engenharia Electrotecnica, Rua Dr. Antonio Bernadino de Almeida,

4200-072 Porto, PortugalcCentro de Estudos Tecnologicos, do Ambiente e da Vida, 5000-911 Vila Real, Portugal

Received 12 April 2005; received in revised form 29 October 2005; accepted 6 December 2005

Available online 9 March 2006

Abstract

This work addresses the signal propagation and the fractional-order dynamics during the evolution of a genetic

algorithm (GA). In order to investigate the phenomena involved in the GA population evolution, the mutation is exposed

to excitation perturbations during some generations and the corresponding fitness variations are evaluated. Three distinct

fitness functions are used to study their influence in the GA dynamics. The input and output signals are studied revealing a

fractional-order dynamic evolution, characteristic of a long-term system memory.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Genetic algorithm; Signal processing; Dynamic; Fractional-order

1. Introduction

In the last twenty years genetic algorithms (GAs)have been applied in a plethora of fields such as inimage processing, pattern recognition, speech re-cognition, control, system identification, optimiza-tion, planning and scheduling [1].

Fractional calculus (FC) is a natural extension ofthe classical mathematics. In fact, since the begin-ning of theory of differential and integral calculus,several mathematicians investigated the calculationof non-integer order derivatives and integrals.Nevertheless, the application of FC has been scarce

e front matter r 2006 Elsevier B.V. All rights reserved

gpro.2006.02.019

ing author.

esses: [email protected] (E.J. Solteiro Pires),

t (J.A. Tenreiro Machado), [email protected]

Oliveira).

until recently, but the advances in the theory ofchaos motivated a renewed interest in this field.

The fundamental aspects of the FC theoryare addressed in [2–5]. In what concerns the FCapplication we can mention research on viscoelas-ticity/damping, chaos/fractals, biology, electronics,signal processing, diffusion and wave propagation,percolation, modelling, control and irreversibility[6–19,23].

Bearing these ideas in mind, this paper analyzesthe system signal evolution and the fractional-orderdynamics in the population of a GA-based optimi-zation. The article is organized as follows. Section 2introduces the problem and the GA method for itsresolution. Based on this formulation, Section 3presents the results for several simulations involvingdifferent working conditions and studies the resul-tant dynamic phenomena. Section 4 present thesame simulation with others fitness functions and

.

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ARTICLE IN PRESSE.J. Solteiro Pires et al. / Signal Processing 86 (2006) 2760–2770 2761

compare the results. Finally, Section 5 outlines themain conclusions.

2. The optimization GA

This section presents the optimization GA used inthe study of the dynamic system. To investigate theinfluence of the fitness function in the GA, are usedthree simple functions ff A; f B; f Cg, each at a time,

(a) (b)

Fig. 1. System dynamics. (a) Initial system, (b) perturbed system.

(a)

(b)

Fig. 2. Input signal dpm perturbation over T exc ¼ 2 generations

with seed i ¼ 1 (Dp ¼ 0:04, fitness f A). (a) Generation domain,

(b) polar diagram.

given by the expressions

f AðbÞ ¼ 1þ jb� 41j, ð1aÞ

f BðbÞ ¼ 1þ jb� 41j2, ð1bÞ

f CðbÞ ¼ 2� sincðb� 41Þ, ð1cÞ

b ¼ fb1; b2; b3; . . . ; blg. (2)

The objective consists on minimizing each functionthat depends on the parameter b 2 ½�50000; 50000�.In the experiments, involving the three test func-tions, it is adopted a binary Gray code, with thestring length of l ¼ 24 bits to represent the solutionsof the population (2) while having an identicalinitial population. The same applies for the geneticparameters, namely (i) a 50-population GA, (ii) anexecution during 200 generations, (iii) a rankselection having simple crossover and mutationwith probabilities pc ¼ 0:8 and pm ¼ 0:05, respec-tively. Moreover, the best solution of each genera-tion is always passed to the next generation. The

(a)

(b)

Fig. 3. Output variation df ðTÞ for the input excitation over

T exc ¼ 2 generations with seed i ¼ 1 (Dp ¼ 0:04, fitness f A).

(a) Generation domain, (b) polar diagram.

ARTICLE IN PRESSE.J. Solteiro Pires et al. / Signal Processing 86 (2006) 2760–27702762

influence of several factors can be analyzed in orderto study the dynamics of the GA, particularly theprobabilities. This influence can be distinct accord-ing to the type of selection, elitism, fitness functionand string length used in the GA. In this work,

Fig. 4. Transfer function H1ðjwÞ using seed i ¼ 1 (T exc ¼ 2,

Dp ¼ 0:04, fitness f A).

(a)

(c)

Fig. 5. Polar diagrams of HAðjwÞ for Dp ¼ f0:03; 0:04; 0:05g, n ¼ 17

Dp ¼ 0:04, (c) polar diagram with Dp ¼ 0:05.

across each test function experiment, it is changedonly the initial seed of the mutation probabilitynoise that is added to pm.

3. Evolution, signal propagation and fractional-order

dynamics

This section studies the dynamical phenomenainvolved in the signal propagation within the GApopulation. In this perspective, small amplitudeperturbations are superimposed over biasing signalsof the GA system and its influence upon thepopulation fitness is evaluated. The experimentsreveal a fractional-order dynamics capable of beingdescribed by systems theory tools.

3.1. Introduction to fractional calculus

Since the foundation of the differential calculusthe generalization of the concept of derivative andintegral to a non-integer order a has been thesubject of distinct approaches. Due to this reasonthere are several alternative definitions of fractional

(b)

01. (a) Polar diagram with Dp ¼ 0:03, (b) polar diagram with


derivatives. For example, the Laplace definition of aderivative of fractional order a 2 C of the signalxðtÞ, Da½xðtÞ�, is a ‘direct’ generalization of theclassic integer-order scheme yielding to Eq. (3). Thismeans that frequency-based analysis methods havea straightforward adaptation (for zero initial con-ditions):

LfDa½xðtÞ�g ¼ saX ðsÞ. (3)

(a)

(b)

(d)(e

Fig. 6. Estimated gain, poles and fractional-orders vs. T exc for fitness f A

vs. T exc, (d) pole b vs. Texc, (e) pole fractional-order b vs. Texc.

An alternative approach, based on the concept offractional differential, is the Grunwald–Letnikovdefinition given by Eq. (4)

Da½xðtÞ� ¼ limh!0

1

ha

Xþ1k¼0

ð�1Þk

"

�Gðaþ 1Þ

Gðk þ 1Þða� k þ 1Þxðt� khÞ

#. ð4Þ

(c)

)

. (a) Gain k vs. T exc, (b) pole a vs. Texc, (c) pole fractional-order a


An important property revealed by this equation isthat while an integer-order derivative implies just afinite series, the fractional-order derivative requiresan infinite number of terms. This means that integerderivatives are ‘local’ operators in opposition withfractional derivatives that have, implicitly, a ‘mem-ory’ of all past events.

The characteristics revealed by fractional-ordermodels make this mathematical tool well suited todescribe phenomena such as irreversibility andchaos because of its inherent memory property. In

Table 1

Parameters of gi, i ¼ f1; 2g for fk; a; a; b;bg approximation of f A optim

Dp k a a

g1ð105Þ g2 g1 g2 g1

0.03 10.0 �0.40 2.90 �1.40 10.8

0.04 10.0 �0.44 2.12 �1.26 7.4

0.05 9.7 �0.39 2.50 �1.31 7.2

(a)

(c)

Fig. 7. Polar diagrams of HBðjwÞ for Dp ¼ f0:03; 0:04; 0:05g, n ¼ 17

Dp ¼ 0:04, (c) polar diagram with Dp ¼ 0:05.

this line of thought, the propagation of perturba-tions and the appearance of long-term dynamicphenomena in a population of individuals subjectedto an evolutionary process seem to be a case whereFC tools fit adequately.

3.2. The GA dynamics

In this section the GA system is stimulated byperturbing the mutation probability, pm, through awhite noise signal, dpm, and the corresponding

ization function

b b

g2 g1 g2 g1 g2

6 0.99 0.93 �0.67 0.83 0.04

2 �1.11 1.14 �0.76 0.75 0.09

3 �1.10 1.16 �0.79 0.75 0.10

(b)

01. (a) Polar diagram with Dp ¼ 0:03, (b) polar diagram with

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(c)

(d) (e)

(a)

(b)

Fig. 8. Estimated gain, poles and fractional-orders vs. Texc for fitness f B. (a) Gain k vs. T exc, (b) pole a vs. T exc, (c) pole fractional-order avs. T exc, (d) pole b vs. Texc, (e) pole fractional-order b vs. Texc.

Table 2

Parameters of gi, i ¼ f1; 2g for fk; a; a; b;bg approximation of f B optimization function

Dp k a a b b

g1ð108Þ g2 g1 g2 g1 g2 g1 g2 g1 g2

0.03 6.0 �0.40 6.39 �2.26 6.87 �1.44 2.02 �1.05 0.70 0.11

0.04 5.0 �0.38 4.66 �2.03 5.65 �1.25 1.93 �1.09 0.66 0.15

0.05 4.0 �0.30 6.83 �2.27 7.39 �1.40 1.42 �0.95 0.70 0.12

E.J. Solteiro Pires et al. / Signal Processing 86 (2006) 2760–2770 2765

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Fig. 9. Output variation df ðTÞ for fitness f C and for the input

excitation over Texc ¼ 2 generations with seed i ¼ 1 (Dp ¼ 0:04).

Fig. 10. Transfer function H1ðjwÞ for fitness f C and using seed

i ¼ 1 (T exc ¼ 2, Dp ¼ 0:04).

E.J. Solteiro Pires et al. / Signal Processing 86 (2006) 2760–27702766

population fitness modification df is evaluated, seeFig. 1. The crossover and other probability signalsused, pc and po, remain unchanged. Therefore, thevariation of the mutation probability and theresulting fitness modification on the GA population,during the evolution, can be viewed as the systeminputs and output signals varying during thesuccessive generations. This analysis is evaluatedusing several experiments with different seeds for asmall amplitude white noise perturbation signal. Allthe other seeds remain unchanged.

In this perspective, a white noise signal dpmðTÞ isadded to the mutation probability pm of thestrings during a time period T exc and the newmutation probability pn is calculated by Eq. (5)where Z is the white noise signal with maximumamplitude �Dp

pn ¼

0 if pm þ ZðDpÞo0;

1 if pm þ ZðDpÞ41;

pm þ ZðDpÞ otherwise:

8><>: (5)

Consequently, the input signal, at generation T, isthe difference between the two cases, that isdpmðTÞ ¼ pnðTÞ � pmðTÞ. On the other hand, theoutput signal is the difference in the populationfitness with and without noise, that is df ðTÞ ¼

f nðTÞ � f ðTÞ.Fig. 2 shows the input signal dpmðTÞ, with seed

i ¼ 1, in the generation domain and the correspond-ing polar diagram, for f A, Dp ¼ 0:04 and T exc ¼ 2,where F½dpmðTÞ� represents the Fourier transformof the signal perturbation. Fig. 3 shows thecorresponding output signal variation df ðTÞ.

Once having de Fourier description of the inputand output signals it is possible to calculate thecorresponding transfer function (6) for seed i

HiðjwÞ ¼Ffdf iðTÞg

FfdpmiðTÞg; i ¼ f1; . . . ; ng. (6)

The transfer function HiðjwÞ, with seed i ¼ 1,between the input and output signals, is depictedin Fig. 4.

After repeating for all seeds a ‘representative’transfer function is obtained by using the median ofthe statistical sample [20,21] of n experiments.

3.3. Transfer function identification

In this section the median of the numerical systemtransfer function, Fig. 5, is approximated byanalytical expressions with gain k 2 Rþ and two

poles ða; bÞ 2 Rþ of fractional orders ða;bÞ 2 Rþ,respectively, given by Eq. (7) where l ¼ A forfunction f A

GlðjwÞ ¼k

½ðjw=aÞ þ 1�a½ð�jw=bÞb þ 1�. (7)

Note that expression (7) is a transfer functionadopting left and right half-plane poles a and b,respectively.

In order to estimate the transfer function para-meters, an identification GA adopting a real stringis executed with the representation fk; a; b; a;bg. Theidentification GA is executed during T ide ¼ 600generations with a 100 strings population. It is used


the simulated binary crossover [22] and, when onemutation occurs, the corresponding valuefx1; . . . ;x5g � fk; a; b; a;bg is changed accordingwith the equations:

xiþ1 ¼ 10ui xi, (8a)

ui�U½�ei;þei�, (8b)

where ui is a random number generated through theuniform probability distribution U and ei is fixedaccording with the range of estimation.

The fitness function f ide measures the distancebetween the median HðjwkÞ and GðjwkÞ:

f ide ¼Xnf

k¼1

kHðjwkÞ � GðjwkÞk, (9)

where H represents the median of the n transferfunctions resulting for each different seed, nf isthe total number of sampling points and wk,k ¼ f1; . . . ; nf g is the corresponding vector offrequencies.

(a)

(c)

Fig. 11. Polar diagrams of HðjwÞ for fitness f CðbÞ and Dp ¼ f0:03; 0:diagram with Dp ¼ 0:04, (c) polar diagram with Dp ¼ 0:05.

Since the optimization GA has a stochasticdynamics, each execution of the GA, with adifferent noise seed, leads to a different transferfunction. Consequently, in order to obtain anumerical convergence [20] are performed n ¼

1701 perturbation experiments, for each optimiza-tion function, with different seeds for the whitenoise signal, dpmðTÞ (all the other seeds for pm, pc

and selection remains unchanged). So, the transferfunction of the optimization GA dynamics isevaluated by computing the Fourier transform(FT) for each pair of input and output signals.After that, the medians of the transfer functionscalculated previously (i.e., for each real andimaginary part and for each frequency) are takenas the final part of the numerical transfer functionHðjwÞ (see Fig. 5).

For evaluating the influence of the excitationperiod T exc several simulations are performedranging from Texc ¼ 2 to T exc ¼ 12 generations.The relation between the transfer function para-meters and Texc are shown in Fig. 6.

(b)

04; 0:05g, n ¼ 1701. (a) Polar diagram with Dp ¼ 0:03, (b) polar


The charts of fk; a; b; a;bg can be approxi-mated through Eq. (10) leading to the parametersof Table 1.

fk; a; a; b;bg ’ g1ðT excÞg2 . (10)

These results reveal that the transfer functionparameters fk; a; a; b;bg vary with a power law vs.the excitation time Texc. Moreover, all the para-meters have a strong dependence with Texc on thetransfer function. Furthermore, k varies with Dp

and pm while fa; a; b; bg have a low dependence withDp and pm.

(a)

(c)

(e)

Fig. 12. Estimated parameters vs. T exc for fitness f C . (a) Gain k vs. Texc

vs. Texc, (e) pole b vs. T exc, (f) pole fractional-order b vs. T exc.

By enabling the zero/pole order to vary freely, weget non-integer values for a and b. Alternatively, theadoption of an integer-order transfer functionwould lead to a larger number of zero/poles to getthe same quality in the analytical fitting to thenumerical values. The ‘requirement’ of fractional-order models, in opposition with the classical caseof integer models, is a well-known discussion andeven nowadays final conclusions are not clear sinceit is always possible to approximate a fractionalfrequency response through an integer one as longas we make use of a larger number of zeros and

(b)

(d)

(f)

, (b) delay T vs. T exc, (c) pole a vs. T exc, (d) pole fractional-order a


poles. Nevertheless, the present experiments con-stitute a complementary point of view towards theadoption of FC.

4. Experiments with the fitness functions f B and f C

In this section is presented the polar diagram andthe correspondent parameters approximation whenis used the f B and f C optimization functions. Theexperiments are carried out in the same conditionsas the previous section.

The polar diagrams obtained for f B can be seen inFig. 7. Adopting again the transfer function (7),with l ¼ B, it yields the estimated parametersfk; a; a; b; bg depicted in Fig. 8. These parameterscan be approximated also through the powerexpressions (10) leading to the results shown inTable 2. Function f B reveals a behavior withsimilarities to the previous case, with exception ofthe gain k that has a small variation with Dp.

The study of the fitness function f C follows anidentical strategy. Nevertheless, it results a differenttype of output and transfer function as can be seenin Figs. 9 and 10. The output is less sensitive andresponds only after a delay of several generations.Consequently, this optimization GA leads to adifferent polar diagrams for HðjwÞ (see Fig. 11). Inthis case the median HðjwÞ can be approximated by

GCðjwÞ ¼ke�jwT

½ðjw=aÞ þ 1�a½ðjw=bÞ þ 1�b(11)

leading to a minimum phase system in contrast tothe previous cases.

Table 3

Parameters of gi, i ¼ f1; 2g for fk;T ; a; a; b;bg approximation of f C opt

Dp k

g1 g2

(a) k;T parameters vs. Dp

0.03 28328 �0.51

0.04 21467 �0.51

0.05 18418 �0.53

(b) a; a; b;b parameters vs. Dp

Dp a a

g1 g2 g1 g2

0.03 0.58 �0.38 11.71 �0.32

0.04 0.16 �0.35 1.50 �0.35

0.05 0.25 �0.18 34.51 �0.21

In this simulation only k (Fig. 12) follows clearlyEq. (10), resulting the estimated values depicted inTable 3. The other parameters reveal a kind of noisybehavior and no clear trendlines emerge. Never-theless, in order to easy comparisons it was decidedto maintain the parameters estimation through (10)leading to exponents g2 close to zero.

5. Conclusions

This paper analyzed the signal propagation andthe dynamic phenomena involved in the timeevolution of a population of individuals. The studywas established on the basis of a simple GAoptimization using three different functions. Whilethe study of GA schemes has been extensivelystudied, the influence of perturbation signals overthe operating conditions is not well known.

From the results can be concluded that theoptimization functions and the time duration ofperturbation have a strong influence upon thebehavior of the GA while the value adopted forthe mutation probability seems to be less significant.

Bearing these ideas in mind, a FC calculusperspective was introduced in order to developapproximating transfer functions analytically simpleyet meaningful. It was shown that fractional-ordermodels capture phenomena and properties thatclassical integer-order sometimes overlook due tothe approximation requirement for the introductionof a large number of parameters that makes thingsunclear.

imization function

T

g1 g2

0.34 1.17

57.22 0.02

87.80 �0.36

b b

g1 g2 g1 g2

0.45 �0.04 9.65 0.23

0.61 �0.23 19.72 �0.16

0.88 �0.27 3.09 �0.08


Acknowledgment

This paper is partially supported by the grantProdep III (2/5.3/2001) from FSE.

References

[1] T. Back, U. Hammel, H.-P. Schwefel, Evolutionary compu-

tation: comments on the history and current state, IEEE

Trans. Evolutionary Comput. 1 (1) (1997) 3–17.

[2] B. Ross, Fractional Calculus and its Applications, Lecture

Notes in Mathematics, vol. 457, Springer, Berlin, 1974.

[3] K.S. Miller, B. Ross, An Introduction to the Fractional

Calculus and Fractional Differential Equations, Wiley,

New York, 1993.

[4] K.B. Oldham, J. Spanier, The Fractional Calculus: Theory

and Application of Differentiation and Integration to

Arbitrary Order, Academic Press, New York, 1974.

[5] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional

Integrals and Derivatives: Theory and Applications, Gordon

and Breach Science Publishers, New York, 1993.

[6] A. Gement, On fractional differentials, Proc. Philos. Mag.

25 (1938) 540–549.

[7] A. Oustaloup, La Commande CRONE: Commande Ro-

buste d’Ordre Non Intier, Hermes, 1991.

[8] A. Oustaloup, La Derivation Non Entier: Theorie, Synthese

et Applications, Editions Hermes, 1995.

[9] C.G. Koh, J.M. Kelly, Application of fractional derivatives

to seismic analysis of base-isolated models, Earthquake Eng.

Struct. Dyn. 19 (1990) 229–241.

[10] J.A. Tenreiro Machado, Analysis and design of fractional-

order digital control systems, J. System Anal.—Modelling-

Simulation 27 (1997) 107–122.

[11] J.A. Tenreiro Machado, System modeling and control

through fractional-order algorithms, FCAA—J. Fractional

Calculus Appl. Anal. 4 (2001) 47–66.

[12] I. Podlubny, Fractional Differential Equations, Academic

Press, San Diego, 1999.

[13] B.M. Vinagre, I. Petras, I. Podlubny, Y.Q. Chen, Using

fractional order adjustment rules and fractional order

reference models in model-reference adaptive control, Non-

linear Dyn. 1–4 (29) (2002) 269–279.

[14] P.J. Torvik, R.L. Bagley, On the appearance of the

fractional derivative in the behaviour of real materials,

ASME J. Appl. Mech. 51 (1984) 294–298.

[15] O.P. Agrawal, Solution for a fractional diffusion-wave

equation in a bounded domain, Nonlinear Dyn. 29 (1–4)

(2002) 145–155.

[16] S. Westerlund, Dead Matter Has Memory! Causal Consult-

ing, Kalmar, Sweden, 2002.

[17] T.J. Anastasio, The fractional-order dynamics of brainstem

vestibulo-oculomotor neurons, Biol. Cybern. 72 (1994) 69–74.

[18] Y. Chen, K.L. Moore, Discretization schemes for fractional-

order differentiators and integrators, IEEE Trans. Circuits

Systems 49 (3) (2002) 363–367.

[19] A.L. Mehaute, Fractal Geometries: Theory and Applica-

tions, Penton Press, Golden, Columbia, 1991.

[20] J.A. Tenreiro Machado, A.M.S.F. Galhano, A statistical

perspective to the Fourier analysis of mechanical manip-

ulators, J. Systems Anal.—Modelling-Simulation 33 (1998)

373–384.

[21] L. Figueiredo, J.A. Tenreiro Machado, J.R. Ferreira,

Dynamical analysis of freeway traffic, IEEE Trans. Intell.

Transp. Systems 5 (4) (2004) 259–266.

[22] K. Deb, Multi-Objective Optimization Using Evolutionary

Algorithms, Wiley, New York, 2001.

[23] E.J. Solteiro Pires, J.A. Tenreiro Machado, P.B. de Moura

Oliveira, Fractional order dynamics in a GA planner, Signal

Process. 83 (11) (2003) 2377–2386.

Dynamical modelling of a genetic algorithm

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