Journal on Policy and Complex Systems
linear behavior , which can be verified by the convergence towards to the limiting probabilities — the systems do not converge instantaneously .
Still , if a similar inference procedure is carried out , but assuming that only k-steps before the current realization the system is initialized by a random process , more consistent results can be obtained . For this , the recurrence relation is rebuilt in order to analyze the probabilities of P [ x i , t + 1
= 1 ] in terms of φ ( x ( i – 2k , t – k ) , ... , x ( i , t – k )
, ... , x ( i + 2k , t – k )
), assuming an uniform probability for every initial configuration , i . e . for each case . Hence , Figure 14 is obtained .
Consequently , it is possible to verify that , despite the fact that there are some deviations when a more robust calculation of the limit is made , the initial framework is relatively robust enough to point out where the limiting probabilities do not reside .
So , given such confirmation , a direct comparison can be carried out between the processes initialized with
p 0
= 0.01 and p N = 0.01 ; and p 0
= p * and
p N
= 0.01 in order to verify the possible rise of long-range dependency , in terms of statistical tests , autocorrelation functions plot and distributions of the d fractional parameter .
In Figure 15 , the rise of longrange dependency is shown , according to the GPH ( 1983 ) test , when the systems are initialized away from the limiting density , except for the rule 182 , which seems to be insensitive to the fact that the initial state is initialized away from the limiting density ; and rule 106 , which seems to display long-range dependency independently from the initial state .
Such affirmation can be verified by inspecting the histograms and checking out that their mean are different from zero , which suggest the presence of long memory behavior .
The same procedure was also carried out by calculating the Künsch ( 1987 ) procedure , according to Figure 16 .
According to these results , it seems that the Künsch ( 1987 ) procedure is more sensitive than the GPH ( 1983 ), in terms of detecting long-range dependency . This procedure was able to detect a possible long-range dependency in all cases , as consequence of initializing the cellular automata away from their limiting densities , except for rule 106 . Therefore , basically , according to the Künsch ( 1987 ) procedure there is long-range dependency in rule 182 ( when initializing it far away from its limiting probabilities ), while it was not found according to the GPH ( 1983 ) procedure .
Hence , in order to complement these tests , it was generated the mean of the autocorrelation functions of all these systems , according to Figure 17 .
According to Figure 17 , it is possible to see that whenever the system is initialized away from its steady state , it seems to generate some sort of slow decay in the autocorrelation function , except for the rule 106 , which seems to always exhibit a slow decay independently from the initial state ; and rule 182 , which displays a behavior
34