How far on average is the empirically determined invariant measure
measure of a cellular automaton in a given class from the value predicted
by the Markov equations which define the class? Above, we first
averaged the empirical invariant measures of rules in a class, and
then determined the distance of this average to the
fixed-point measure of the equations which define the class. We will
now first compute the distance from the empirical invariant measure of
each rule in the class to the fixed-point measure of the equations
which define the class, and then average over the class. This gives
a class average distance. For each class, the class average distance
must be less than or equal to the distance between the empirical and
theoretical class invariant measure.
Figure 5 shows
the average 
distance of the empirical invariant measure of rules
in a class from the empirical class invariant measure.
The contribution of each class to a distribution is weighted by
the number of rules in the class. The curves are then normalized so
that the total bin height is 1.
Error bars give 
1 standard deviation over the bin.
Orders of approximation 0-2 are shown.
Increasing dash length corresponds to increasing order of approximation.
Figure:
Distribution of
the average distance of the empirical invariant measure of rules
in a class from the empirical class invariant measure.
The contribution of each class to a distribution is weighted by
the number of rules in the class. The curves are then normalized so
that the total bin height is 1.
Error bars give 1 standard deviation over the bin.
Orders of approximation 0-2 are shown.
Increasing dash length corresponds to increasing order of approximation.
As the order of approximation is increased, the typical average distance decreases significantly. The distributions for 1st- and 2nd-order theories have long tails. It is clear, however, average predictions tend to improve with increase in order of approximation. This may be somewhat surprising in view of the results of previous subsections. Recall that in subsection xxx it was demonstrated that the greater the order, the lower the fraction of rules represented in these samples which are close to the standard measure. Then it was demonstrated in subsection xxx that the nearer the invariant measure of a class is to the standard measure the better, on average, the Markov estimates of that invariant measure. Hence, for this sampling distribution, lower orders of theory should yield smaller values for the distribution over rules of average distance in figure 3. In fact, increase in order results in smaller average distances. This implies that the nature of the sampling distribution does not mask the increase in predictive power with order of the Markov approximation.