J. Shapiro, A. Prugel-Bennett, and M. Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In Terence C. Fogarty, editor, Evolutionary Computing, AISB Workshop, volume 993 of Lecture Notes in Computer Science. Springer, 1994. 12  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... in Harik, Cantu Paz, Miller, and Goldberg [59] Modeling binary GA as a dynamical system on a macroscopic level, i.e. by expected value dynamics (similar to the approach used in real valued ES theory) has been proposed by Prugel Bennett and Shapiro [60] Shapiro, Prugel Bennett, and Rattray [61]. One of the basic ideas is to describe the population s fitness distribution by expansions of a Gaussian (also used in ES theory, see Beyer [62, 63] The peculiarity of this approach is, however, that the underlying microscopic description level is bypassed using inference methods gleaned from ....

J. L. Shapiro, A. Prugel-Bennett, and L.M. Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In Lecture Notes in Computer Science, Vol. 865, pages 17--27. Springer, Heidelberg, 1994.

....the concept of selection intensity. The validity of this model is limited to the case of in nite population. This selection intensity based model provides a description of the dynamics of the tness mean. To extend the model to nite populations, separate works by Blickle and Thiele (1995) and Shapiro, Pr ugel Bennett, and Rattray (1994) addressed the problem of modeling higher order moments. Blickle and Thiele (1995) introduced a model of the evolution of the tness variance under the e ect of the selection strategy. Shapiro, Pr ugel Bennett, and Rattray (1994) used a statistical mechanics approach to model the evolution of ....

....to nite populations, separate works by Blickle and Thiele (1995) and Shapiro, Pr ugel Bennett, and Rattray (1994) addressed the problem of modeling higher order moments. Blickle and Thiele (1995) introduced a model of the evolution of the tness variance under the e ect of the selection strategy. Shapiro, Pr ugel Bennett, and Rattray (1994) used a statistical mechanics approach to model the evolution of cumulants of the tness distribution. In contrast to studies that use selection intensity, the last approach does not yield closed form solutions. Also, it does not reduce to convenient forms. On the other side, the gambler s ruin ....

Shapiro, J., Prugel-Bennett, & Rattray, M. (1994). A statistical mechanical formulation of the dynamics of genetic algorithms. Evolutionary Computing: AISB Workshop 1994 , 17-27.

....distribution is skewed to the right. The kurtosis is negative if the distribution is atter than a normal, and positive if the distribution is more peaked than a normal. One possibility to calculate the expected value of the cumulants is to integrate over all possible populations after selection [18]. Our approach is di erent: we calculate the expected tness of each individual, and then use equation 1 to obtain the cumulants. The critical observation is that we may interpret the tness values f i as samples of random variables F i with a common distribution. We may arrange the random ....

J. Shapiro, Prugel-Bennett, and M. Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In Evolutionary Computing: AISB Workshop 1994, pages 17-27. Springer-Verlag, 1994.

....[11] Whereas ad hoc statistics of fitness distributions (online performance, offline performance, etc. have historically been used as indicators of GA performance, classical statistics (mean, variance, skewness, excess) have been used for the purpose of modeling evolutionary trajectories [16]. Therefore, the point here is not to introduce the field of genetic algorithms to the concept of equivalence classes as noted above that has been done before, 31 the most notable examples being schema, and fitness distributions. The point is rather to give a coherent general account of ....

J. Shapiro, A. Prugel-Bennett, and M. Rattray, A Statistical Mechanical Formulation of the Dynamics of Genetic Algorithms, in: Lecture Notes in Computer Science, 865 (Springer-Verlag, Berlin, 1994) 17--27. 46

....EAs to address coevolutionary algorithm dynamics. We then employ concepts from evolutionary game theory to examine design aspects of conventional coevolutionary algorithms that are poorly understood. 1 Introduction While formal models of evolutionary algorithm (EA) dynamics burgeon in general [24, 26, 21], co evolutionary algorithms, in particular, still have few formal tools for their analysis investigations of coevolutionary algorithm dynamics are typically empirical in nature [7, 3, 1, 14, 16] The reason for this divide stems from the need to formally account for the de ning characteristic of ....

J. Shapiro, A. Prugel-Bennet, and M. Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In T. C. Fogarty, editor, Evolutionary Computing (AISB Workshop), pages 17-27. Springer-Verlag, 1994.

....of a population P . The characterization of the population by its fitness distribution has also been used by other researches, but in a more informal way. In [ Muhlenbein and Schlierkamp Voosen, 1993 ] the fitness distribution is used to calculate some properties of truncation selection. In [ Shapiro et al. 1994 ] a statistical mechanics approach is taken to describe the dynamics of a Genetic Algorithm that makes use of fitness distributions, too. 6 Selection (whole population) Randomly created Initial Population End Yes No Problem solved Recombination p 1 p c c Figure 2.1: Flowchart of the Genetic ....

Jonathan Shapiro, Adam Prugel-Bennett, and Magnus Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In Terence C. Fogarty, editor, Evolutionary Computing AISB Workshop. Springer , LNCS 865, 1994.

....inapplicable or inappropriate. This underpins the formal approach of this thesis, in which we seek to link explicitly the performance of the optimisation algorithm to measurable characteristics of the problem domain. Recent efforts in a similar vein include the statistical mechanical treatment of Shapiro et al. 1994), Prugel Bennett Shapiro (1995) and Rattray (1995) They have developed a macroscopic approach for modelling genetic algorithms based on the evolution of population fitness and correlation cumulants. While this does not have the precision of Vose s detailed models, it has the advantage of ....

....no scope for distinguishing any of these biased sampling methods from enumeration random or fixed as we demonstrate below. There are exceptions to this, for example the convergence results for simulated annealing which exploit knowledge of the problem domain (e.g. van Laarhoven Aarts, 1989, Shapiro et al. 1994) but much effort continues to be devoted to the general case, Vose, 1992; Goldberg, 1989c, 1989a, 1989b) While such studies lead to elegant mathematics, and provide occasional insights into stochastic search, their practical utility in guiding or even making contact with practitioners ....

[Article contains additional citation context not shown here]

J. Shapiro, A. Prugel-Bennett, and M. Rattray, 1994. A statistical mechanical formulation of the dynamics of genetic algorithms. In T. C. Fogarty, editor, Evolutionary Computing: AISB Workshop, pages 17--27. Springer-Verlag, Lecture Notes in Computer Science 865.

....the literature. Prugel Bennett and Shapiro [9] apply statistical mechanics to show analytically the evolution of the distribution of the population fitness for the one dimensional spin glass problem. The underlying GA employs classical operators. This approach is applied to other problems in [11] [12] and [10] The work presented below di#ers from these papers in two aspects. First, di#erent GA operators are analyzed. Second, we develop chromosome level dynamics that produce population dynamics as emergent behavior. The papers above do a population level analysis of summary statistics. Baum, ....

J. Shapiro, A. Prugel-Bennett and L. Rattray. A Statistical Mechanical Formulation of the Dynamics of Genetic Algorithms. Lecture Notes in Computer Science, 865, pages 17-27, 1994.

....rates fl = g=N , where g is the average number of genes that are mutated per sequence. The change in the fitness due to mutation will be given by F 0 = F Gamma 2 g(1 k 1 ) 2 2 g(1 Gammak 1 ) 2 : 4.2) 14 0 10 20 30 40 50 t 0. 0 100.0 200.0 300.0 400.0 500.0 k 2 k 2 [3] k 2 [5] k 2 [7] k 2 [9] k 2 [8] k 2 [9] 2 k 2 [4] k 2 [6] k 2 [8] Figure 3: The curves show the predictions for 2 versus the number of generations t using sets of ensemble variables with n c = 3 to 9. The predictions are for Muller s ratchet (k 1 = 1) with P = 100, fi = 0:005 and fl = 4=N . ....

Shapiro J. L., A. Prugel-Bennett, and M. Rattray. A Statistical Mechanical Formulation of the Dynamics of Genetic Algorithms. In T. C. Fogarty, editor, Lecture notes in Computer Science 865, pages 17--27, Berlin, 1994. Springer-Verlag.

....it is still difficult to get quantitative results concerning the effectiveness of GA search for different parameter settings, for different types of problems, or for different types of representations. This is an important goal of a theoretical description of genetic algorithms. In previous papers [7, 8], we proposed a statistical mechanics approach to the study of genetic algorithm dynamics. This approach was fairly crude and approximate, although we found good agreement with simulations in the cases we tested. We have since developed a more rigorous formulation [9] and better understanding of ....

....2 log(P ) s 2 2 2 5 2 20 Fig. 3. The change in width of fitness distribution after selection starting from a Gaussian distribution for P = 2 5 , 2 10 and 2 20 . The solid lines are calculated by numerical integration. 4. 1 Selection Selection has been described in detail elsewhere [7, 8], so will be considered briefly here. We have studied Boltzmann selection in which each member of the population is selected with probability p ff = e fiF ff Z ; Z = P X ff=1 e fiF ff : 1) where fi is determines the strength of selection and F ff is the fitness of string ff. The effect ....

J. L. Shapiro, A. Prugel-Bennett, and M. Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In Terence C. Fogarty, editor, Evolutionary Computing: AISB Workshop, Leeds, U.K., April 1994, Selected Papers, volume 864 of Lecture Notes in Computer Science, pages 17--27. Springer-Verlag, 1994.

....properly account for finite population effects. The theory is applied to two problems for which the full dynamics can be solved, extending a formalism developed by Prugel Bennett, Shapiro, and Rattray for modelling the dynamics of the GA using methods from statistical mechanics (Prugel Bennett Shapiro, 1994; Prugel Bennett Shapiro, 1995; Rattray, 1995; Rattray Shapiro, 1996; Shapiro et al., 1994) This formalism does not require that the population be sufficiently large to ensure convergence to the global optimum and properly accounts for correlations accumulated under selection. Under this ....

.... for which the full dynamics can be solved, extending a formalism developed by Prugel Bennett, Shapiro, and Rattray for modelling the dynamics of the GA using methods from statistical mechanics (Prugel Bennett Shapiro, 1994; Prugel Bennett Shapiro, 1995; Rattray, 1995; Rattray Shapiro, 1996; Shapiro et al., 1994). This formalism does not require that the population be sufficiently large to ensure convergence to the global optimum and properly accounts for correlations accumulated under selection. Under this formalism, the population is de scribed by a small number of macroscopic statistics and a ....

[Article contains additional citation context not shown here]

J. L. Shapiro, A. Prugel-Bennett, L. M. Rattray (1994) "A Statistical Mechanical Formulation of the Dynamics of Genetic Algorithms," Lecture Notes in Computer Science 865, 17--27.

....solutions. These two properties give GAs their interest, they also make them hard to analyse. In this paper we use statistical mechanics techniques to derive a set of deterministic dynamical equations describing the average behaviour of simple GAs. Preliminary results have been reported elsewhere [2, 3]. We show how this technique can be used to predict the evolution of Genetic Algorithms on a simple problem. In the simplest realization of a GA [4] each solution is represented by a genetic string or chromosome. A population of strings is generated at random to form the first generation. The ....

.... throughout the entire run of the GA (it is not exactly constant as selection also depends on the higher cumulants which change during a run) This rescaling plays the same role as linear fitness rescaling [4] a discussion of the connections between these two types of scaling is given in [3]. In the present paper we have used scaled selection with a selection parameter defined by fi s = fi q 2 =2 log(P ) The cumulants after selection are given by s n = n t n log P X ff=1 n ff e tF ff fi fi fi fi fi t=0 (2.2) where n ff is the number of times member ff has ....

Shapiro J. L., A. Prugel-Bennett, and M. Rattray. A Statistical Mechanical Formulation of the Dynamics of Genetic Algorithms. In T. C. Fogarty, editor, Lecture notes in Computer Science 865, pages 17--27, Berlin, 1994. Springer-Verlag.

....to create better solutions. Although there is a large body of theoretical work on GAs, the established theory does not yet provide a complete picture. Prugel Bennett and Shapiro (PBS) have introduced a formalism for modeling the dynamics of the GA, using methods from statistical mechanics [4, 5, 6]. They considered two closely related toy problems; the random field paramagnet and the Ising spin chain, for which the dynamics can be solved exactly [5] In this paper their formalism is generalized to a harder combinatorial optimization problem, subset sum. Although strictly np complete, this ....

J. L. Shapiro, A. Prugel-Bennett, L. M. Rattray, "A Statistical Mechanical Formulation of the Dynamics of Genetic Algorithms," Lecture Notes in Computer Science, 865, special edition on Evolutionary Computing edited by T. C. Fogarty, (1994).

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J. Shapiro, A. Prugel-Bennett, and M. Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In Terence C. Fogarty, editor, Evolutionary Computing, AISB Workshop, volume 993 of Lecture Notes in Computer Science. Springer, 1994. 12

No context found.

J. Shapiro, A. Prugel-Bennett, and M. Rattray. A Statistical Mechanical Formulation of the Dynamics of Genetic Algorithms. In Terence C. Fogarty, editor, Evolutionary Computing, AISB Workshop, volume 993 of Lecture Notes in Computer Science. Springer, 1994.

No context found.

Jonathan Shapiro, Adam Prugel-Bennett, and Magnus Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In Terence C. Fogarty, editor, Evolutionary Computing AISB Workshop. Springer , LNCS 865, 1994.

No context found.

Jonathan Shapiro, Adam Prugel-Bennett, and Magnus Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In Terence C. Fogarty, editor, Evolutionary Computing AISB Workshop. Springer , LNCS 865, 1994.

No context found.

J. L. Shapiro, A. Prugel-Bennett, and M. Rattray. A statistical mechanical formulation of the dynamics of genetic algorithms. In T. C. Fogarty, editor, Lecture notes in Computer Science 865, pages 17--27, Berlin, 1994. Springer-Verlag.

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