Journal on Policy and Complex Systems
The same was done to obtain the results for p 0 = p * .
Table 3 . Acceptance Ratio of Statistic Tests for p 0 = p *
Rule
V / S Test Acceptance for Long Memory @ 95 %
R / S Test Acceptance for Long Memory @ 95 %
Rule 18 27 % 24 %
Rule 106 84 % 85 % Rule 110 30 % 22 % Rule 150 8 % 11 % Rule 182 35 % 24 %
As can be seen , in Tables 2 and 3 , these tests summarize the hypothesis of long memory behavior discussed in the other tests and on the evaluation of the autocorrelation function plots . Hence , basically , all rules exhibit the rise of long-range dependency , when initialized far away from their steady-state densities , except for rule 106 .
Furthermore , there is no consensus whether there is a rise of long memory behavior in rule 182 , given the fact that it converges very rapidly towards the equilibrium density . Thus , an interesting discussion can be made in terms of these two pathological cases .
First , if rule 106 is considered , it is worth mentioning that the transition rule for each cell can be defined by the following expression :
where |
denotes the “ not ” operation |
over the cell at the ith position , in the |
instant t . So , if |
, then |
, |
and vice versa . |
|
|
Additionally , if one considers the density of the system given by :
It is possible to realize that the system is stationary when :
So , by plugging the initial definition of the transition rule into the density expression one obtains :
Consequently , by factorizing :
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