In the following the relationship between the statistical properties of
the equations which define an Markov class and the statistical properties
of the rules contained in an Markov class will be explored.
These investigations will concern cellular automata with two states
per cell and next-nearest neighbor (r=2) interaction rules.
The number of cellular automaton rules with radius 2 on two
states per cell is 
. Obviously, a complete survey
of r=2 rules not feasible. The large number of r=2 rules, however, presents the
opportunity to use a statistical analysis on the space of
all such automata. The main questions to be addressed below
are: 1) to what degree of accuracy do various
orders of approximation
predict the statistical properties of rules in a class? and
2) how homogeneous are the classes defined at various orders ?
These questions are distinct, but related. Consider using the classification to find a cellular automaton with some specified statistical property. Ideally, the equations which specify classes should accurately predict the expected properties of rules in the classes defined. Further, the rules in a class should have closely related properties. It could happen, for instance, that classes have wide variability over their members,yet the Markov approximation accurately predicts the average over a class of some statistical property. In this case, the variability renders the theoretical prediction useless for any practical question concerning individual cellular automaton rules. In the opposite extreme, it could happen that rules within a class have very similar statistical properties, but that these shared properties are different from those predicted by the equation which defines the class. It this case it would be difficult to find rules with some desired property by solving Markov equations. Once one such rule was found, however, many other rules with the same property could be constructed by inverting the equations corresponding to the given rule.
It is demonstrated below that both the accuracy with which the Markov approximation predicts the properties of rules in a class, and the homogeneity of classes increases with order of approximation. This is shown by a statistical analysis of samples of Markov classes of orders 0-2 of r=2 rules. A sample of rules from each class selected is examined empirically. Empirical estimates are made of invariant 1- and 2-block probabilities of these rules. The same estimates are made in Markov approximation. The primary tool used to assess accuracy and homogeniety is the computation of the distance in measure space between various Markov measures estimated in this way.
Apart from subsection xxx, all of the data to be discussed is presented in the form of distributions. It must be clearly understood that the distribution of measured properties depends on the way samples are chosen. An attempt was made to sample uniformly in the parameter space of the Markov approximation. It is anticipated that this should correspond to a sampling which is uniform as well over the set of automata themselves. Consider an ideal case in which all classes at a given order are sampled, all rules in each class have exactly the same properties (perfect homogeneity), and these properties are exactly those predicted by that order of approximation (perfect accuracy). In this case, the distribution of some property over all rules could be found by determining that property using the approximation for each class, and weighing the result by the number of rules in the class. We will see below that this ideal does not obtain for r=2 rules evaluated at low orders of approximation. Beyond 0th-order, not all classes can be sampled. At any order 0-2, not all rules in a class necessarily have the same properties. These properties are not necessarily exactly predicted by the Markov approximation. Below, we will obtain approximate distributions over the entire set of r=2 rules by theoretically and empirically evaluating the properties of a sample of classes, and weighting the results by the number of rules in the classes.