Long Memory Properties and Complex Systems
long-range dependency ( see Stock & Watson , 1996 ), initializing such discrete system far away from its stable fixed point may also generate a long memory behavior .
In order to derive a mathematical framework to establish the probabilities p * for each rule , the approach here used assumes that the previous state of the system is a Markov process , i . e . a random system that changes its states according to a transition rule that only depends on the current state . Thus , the transition rule can in fact describe the probability that the next cell will be 1 or 0 , according to a Binomial Probability , as described in Wolfram ( 1983 ).
So , for example , if rule 150 is considered :
Figure 12 . Encoding of Rule 150 .
and establish that x i , t represents the value of the cell at position i at time step t , it is possible to write the probability P [ x i , t + 1
= 1 ]. To accomplish that , it is necessary summing up the individual probabilities of the cases where x i , t + 1
=
1 , according to a Binomial Probability . For notation simplicity , writing
P [ x i , t + 1 = 1 ] = p t ,
P [ x i , t + 1
= 0 ] = ( 1 – p t ) and P [ x i , t + 1 = 1 ] = p t + 1
, one is able to develop the following equation :
which is a discrete nonlinear system that describes the underlying probability in the Markov Process . This Markovian assumption is made in order to reduce the complexity of the mathematical treatment of the system .
If one solves the equation by establishing p t + 1
= p t
= p *, a fixed point is found , and consequently , its respective properties may be studied , like stability , in order to check if there is an attractor present in this system .
In this example , the system is characterized by the fixed points m and , according to the Lyapunov stability criteria , only 0.5 is a stable fixed point . The other fixed points are unstable . This can be verified by calculating :
and checking if .
Therefore , from the assumption of a Markov Process , if the initial conditions of the system are set using p 0
≠ 0.5 , the system will converge towards p t = 0.5 .
31