Measuring the Behavior of a cEA

We will introduce here some statistical measures that will be of especial interest for analyzing the mode of operation of cellular evolutionary algorithms. These statistical measures are both at genotypic (individuals) and phenotypic (population) levels. This study is taken from [CTTS98]. Refer to it for more complete information.

Basic Definitions and Notation

We can define the population as a vector of n genotypes (individuals) . The space of all possible populations is , where is the space of genotypes (so ). The fitness of an individual will be .

We can define an occupancy function such that , for all , is the number of individuals in x sharing the same genotype , i.e., the occupancy number of in x. The size of population x, , is defined as .

After that, we can now define a share function giving the fraction of individuals in x that have genotype ,i.e., .

Consider the probability space , where is the algebra of the parts of andis any probability measure on . Let us denote by the probability of generating a population by extracting n genotypes from according to measure. It can be shown that it is sufficient to know either of the two measures -(over the genotypes) or (over the populations)- in order to reconstruct the other.

The fitness function establishes a morphism from genotypes into real numbers. If genotypes are distributed over according to a given probability measure , then their fitness will be distributed over the real numbers according to a probability measure obtained fromby applying the same morphism:

 (1)

The probability of a given fitness value is defined as the probability that an individual extracted from according to measure has fitness (or, if we think of fitness values as a continuous space, the probability density of fitness ): for all , where .

An EA can be regarded as a time-discrete stochastic process

 , (2)

having the probability space as its base space, as its state space, and the natural numbers as the set of times, here called generations. must be thought of as the set of all the evolutionary trajectories, is a -algebra on , and is a probability measure over .

The transition function of the evolutionary process, in turn based on the definition of the genetic operators, defines a sequence of probability measures over the generations.

Let denote the probability measure on the state space at time t; for all populations ,

 . (3)

In the same way, let denote the probability measure on space at time t; for all ,

 . (4)

Similarly, we define the sequence of probability functions as follows: for all and ,

 . (5)

Genotypic statistics

This class of statistics is based in some diversity indices at individuals level -genotypic-. Following we will see come functions which are significant of the population evolution at a genotypic level.

• Occupancy and share functions. At any time , for allis a discrete random variable with binomial distribution
 (6)

thus, and . The share function is perhaps more interesting, because it is an estimator of the probability measure ; its mean and variance can be calculated from those of , yielding

 and (7)
• Structure. Statistics in this category measure properties of the population structure, that is, how individuals are spatially distributed.

The frequency of transitions of a population x of n individuals (cells) is defined as the number of borders between homogeneous blocks of cells having the same genotype, divided by the number of distinct couples of adjacent cells. Another way of putting it is that is the probability that two adjacent individuals (cells) have different genotypes, i.e., belong to two different blocks.

Formally, the frequency of transitions for a one-dimensional grid structure can be expressed as

 , (8)

where [] denotes the indicator function of proposition .

• Diversity. There are a number of conceivable ways to measure genotypic diversity, two of which we define below:
• Population entropy. The entropy of a population x of size n is defined as:  . (9)

Entropy takes on values in the interval and attains its maximum, , when x comprises n different genotypes.
• Diversity indices. The probability that two individuals randomly chosen from x have different genotypes is denoted by .
Let x be a population of n individuals with genotypes in . Then,  . (10)

Phenotypic statistics

Associated with a population x of individuals, there is a fitness distribution. We will denote by its (discrete) probability function.

• Performance. The performance of population x is defined as its average fitness, or the expected fitness of an individual randomly extracted from x, .
• Diversity. The most straightforward measure of phenotypic diversity of a population x is the variance of its fitness distribution, .
• Structure. Statistics in this category measure how fitness is spatially distributed across the individuals in a population.
Ruggeness measures the dependency of an individual's fitness on its neighbors' fitness. For a one-dimensional population, x, of size n, , ruggedness can be defined as follows:  . (11)

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