Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

where the n-step transition matrix Nn is given by
Nn = I + Q + Q 2 + Q 3 + ……+ Q n - 1 (15.19)
As n approaches infinity,
Lt N n = ( I – Q ) - 1 . (15.20)
n→∝
Consequently, as n approaches infinity,
Lt P n =
n→∝
Goodman has shown [8] that the matrix (I – Q) -1 is guaranteed to exist.
Thus given an initial probability vector π 0, the chain will have a transition to
an absorbing state with probability 1. Further, there exists a non-zero
probability that absorbing state will be the globally optimal state [8].
We now explain: why the chain will finally terminate to an absorbing
state. Since the first ‘a’ columns for the matrix P n , for n →∝, are non-zero and
the remaining columns are zero, therefore, the chain must have transition to
one of the absorbing states. Further, note that the first ‘a’ columns of the row
vector π n for n →∝ denote the probability of absorption at different states,
and the rest of the columns denote that the probability of transition to nonabsorbing
states is zero. Thus probability of transition to absorbing states is
one. Formally,
a
∑ Lt (π n ) i
i=1 n→∝
I 0
(I – Q ) – 1 R 0