Recall a CA rule project a circular-boundary state space back to it self, that is
where , and more strictly , for simplicity we let , thus
In practical, we take n=20^2=400;
In practical, we sought a big enough representative subset, from , so that
Denote , we have
We then define a profile with Pearson correlation so that
, where , denotes standard deviation of X and Cov[X,Y] denotes covariance between X and Y.
We assert that classify the dynamics of the CA rule by capturing the periodic behavior in the underlying dynamics.
Importantly, we can find a good by taking a random and obtain
where T is a variable parameter.