Journal on Policy and Complex Systems
That said , an additional noise is added to the system in order to have a true stochastic system and then , study its long memory properties . To accomplish that , it is introduced a probability based rule , where each cell has a probability p N of having its state randomly modified , with equal probabilities of assuming 0 ( white ) or 1 ( black ), and ( 1 – p N
) of having its state unmodified .
Having proposed the methodology , the rules 150 , 110 , 106 , 182 , and 18 are investigated .
These rules , according to the methodology discussed before , have the following fixed points between 0 and 1 ( where valid probabilities can reside ), noticing that “ S ” denotes a stable fixed point and “ U ” denotes an unstable fixed point .
Table 1 . Rule comparisons in relation to Fixed Point 1 , Fixed Point 2 , and Fixed Point 3
Hence , all studied rules presented here have , in theory , one valid attractor . In order to confirm that in practice , it was simulated 100 simulations of each rule , with an initial configuration of p 0 = 0.01 and p N
= 0.01 . In other words , the systems were initialized away from their respective equilibrium points , with a low amount of noise . After that , a mean evolution of each one of these processes were calculated and plotted in Figure 13 .
First , it is worth noticing that the aggregated series seem to behave like having an implicit regime switch , when analyzed globally . In the first part of the evolution of the system , there is a first-order behavior in the system , followed by saturation within a range where the steady state resides . This is what was exactly expected in terms of possibility to generate long-range dependency , as can be seen in Diebold and Inoue ( 2001 ).
Nonetheless , according to Figure 13 , it is possible to see that not all rules converge properly to the stable fixed points inferred before . Wolfram ( 1983 ) states that the main cause of such behavior is the violation of the assumptions made ( i . e . Markov Processes ) and the presence of serial correlation / non-
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