An Introduction to Genetic Algorithms - Boente

with E ± the energy of individual ±,P the population size, and ² a variable controlling the amount of selection.
This method is similar to "Boltzmann selection" with ² playing the role of temperature. This selection method
has some desirable properties for GAs (to be described in the next chapter), and also has useful features for
Prügel−Bennett and Shapiro's analysis.
This is a rather easy problem, even with no mutation, but it serves well to illustrate Prügel−Bennett and
Shapiro's approach. The goal was to predict changes in distribution of energies (the negative of fitnesses) in
the population over time. Figure 4.4 plots the observed distributions at generations 0, 10, 20, 30, and 40
(going from right to left), averaged over 1000 runs, with P = 50,N + 1 = 64, and ² = 0.05. Prügel−Bennett and
Shapiro devised a mathematical model to predict these changes. Given Át(E), the energy distribution at time t,
they determine first how selection changes Át(E) into Á s t(E) (the distribution after selection), and then how
crossover changes Á s t(E) into Á sc t (E) (the distribution after selection and crossover). Schematically, the idea
is to iterate
(4.11)
starting from the initial distribution Á0(E).
Chapter 4: Theoretical Foundations of Genetic Algorithms
Figure 4.4: Observed energy distributions for the GA population at generations 0, 10, 20, 30, and 40. Energy
E is plotted on the x axis; the proportion of individuals in the population at a given energy Á(E) is plotted on
the y axis. The data were averaged over 1000 runs, with P = 50, N + 1 = 64, and ² = 0.05. The minimum
energy for the given spin glass is marked. (Reprinted from Prügel−Bennett and Shapiro 1994 by permission of
the publisher. © 1994 American Physical Society.)
Prügel−Bennett and Shapiro began by noting that distributions such as those shown in figure 4.4 can be
uniquely represented in terms of "cumulants," a statistical measure of distributions related to moments. The
first cumulant,k1, is the mean of the distribution, the second cumulant,k2, is the variance, and higher cumulants
describe other characteristics (e.g., "skew").
Prügel−Bennett and Shapiro used some tricks from statistical mechanics to describe the effects of selection
and crossover on the cumulants. The mathematical details are quite technical. Briefly, let kn be the nth
cumulant of the current distribution of fitnesses in the population, k s n be the nth cumulant of the new
distribution produced by selection alone, and k c n be the nth cumulant of the new distribution produced by
crossover alone. Prügel−Bennett and Shapiro constructed equations for k s n using the definition of cumulant
and a recent development in statistical mechanics called the Random Energy Model (Derrida 1981). For
example, they show that k s 1 Hk1 ²k 2 and k s 2 H (1 1/P) k 2 ²k 3. Intuitively, selection causes the mean and
the standard deviation of the distribution to be lowered (i.e., selection creates a population that has lower
mean energy and is more converged), and their equation predicts precisely how much this will occur as a
function of P and ². Likewise, they constructed equations for the k c n:
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