An Introduction to Genetic Algorithms - Boente

To address the finite−population case, Nix and Vose (1991) modeled the simple GA as a Markov chain.
Markov chains are stochastic processes in which the probability that the process will be in state j at time t
depends only on the state i at time t 1. Many processes in nature can be described as Markov chains. (For an
introduction to the mathematics of Markov chains see Feller 1968.)
A "state" of a finite−population GA is simply a particular finite population. The set of all states is the set of
possible populations of size n. These can be enumerated in some canonical order and indexed by i. Nix and
Vose represent the ith such population as a vector of length 2 l . The yth element of is the number of
occurrences of string y in population Pi. It is clear that under the simple GA the current population Pj depends
(stochastically) only on the population at the previous generation. Thus, the GA can be modeled as a Markov
chain.
To construct such a model, we need to write down the probability of going from any given population to any
other under the simple GA. The set of all possible populations of size n can be represented by a giant matrix,
Z, in which the columns are all possible population vectors . How many possible populations of size n are
there? The answer is
(Deriving this is left as an exercise.) A given element Zy,i of Z is the number of occurrences of string y in
population i.
Here is a simple example of constructing the Z matrix: Let l = 2 and n = 2. The possible populations are
The array Z is
Chapter 4: Theoretical Foundations of Genetic Algorithms
A state for the Markov chain corresponds to a column of Z.
The next step is to set up a Markov transition matrix Q.Q is an N × N matrix, and each element Qi,j is the
probability that population Pj will be produced from population Pi under the simple GA. Once this matrix is
defined, it can be used to derive some properties of the GA's behavior.
Writing down the transition probabilities QI,j is a bit complicated but instructive. Let pi(y) be the probability
that string y will be generated from the selection and recombination process (i.e., steps 3 and 4) of the simple
GA acting on population Pi. The number of occurrences of string y in population Pj is Zy,j, so the probability
that the correct number comes from population Pi is simply the probability that Zy,j occurrences of y are
produced from population Pi. This is equal to the probability that y is produced on Zy,j different
selection−and−recombination steps times the number of ways in which these Zy,j different
selection−and−recombination steps can occur during the total n selection−and−reproduction steps. Following
109