Extended definition of the optimal
by the theory of discriminating criteria
It has been demonstrated theoretically and experimentally that
the exhaustive-search curves
are gradual and unimodal for the expected value of the criterion
. The number of candidate models tested in each exhaustive-search
layer cannot be infinitely large. Because of this, the curves take
on a slightly wavy shape, and a small error may creep into the optimal
model structure choice.
Therefore almost every GMDH algorithm consecutively uses two criteria.
At first, an exhaustive search is applied to all candidate models
for compliance with the main criterion, and a small number of models
whose structure is close to optimal is selected. Then only one optimal
model is selected that complies with a special discriminating criterion.
The forecast error variation criterion RR(s)
is proposed :
where: yi - is the variable value in the input
data sample; - is the variable value calculated according to the model and is the mean value of variable or the value of trend function.
The majority of criteria of structure quality
of model is separate case of general formulae :
CR(s,n,X,y) = h1(s,n)V(s,n,X,y) + h2(s,n)D2,
where V(.) - magnitude of model error;
h1(.), h2(.) - multiplicative and additive fine
functions for model complexity (limit growth of estimated parameters
D2 - estimation of unknown dispersion
Compromise between model accuracy and its complexity typical for
criteria of structural identification can be expressed explicitly
or implicitly. For external criteria implicit fine is reached by
division of data sample, for example into two parts: W=(ATBT)T,
Interesting examples of criteria in GMDH algorithms were proposed
- Hild and Bozdogan define the ICOMP criteria  based on information-theoretical
statistical theory, which take into account both parametric uncertainly
as well as complexity, thereby eliminating the need for arbitrarily
sub-dividing the data set for cross-validation;
- Lange in  gives a review of selection criteria for the
case of uncertainty;
- Wang introduces the "Innovation-Contribution" criterion 
in order to select independent variables automatically.