Solution of practical problems and GMDH theory design lead to development
of broad spectrum of software algorithms. Each of them corresponds
to some definite conditions of it application [17]. Algorithms mainly
differ one from another by the modelscandidates set generator arrangement
for given basic function, by the way of models structure complexing
and, at last, by the external criteria accepted. Algorithm choice
depends on specifics of the problem, noise dispersion level, sufficiency
of data sample, and if data sample contains only continuous data.
Most often criteria
of accuracy, differential or informative type are used. The work
of GMDH algorithms has a straightforward analogy with the work of
gardener during selection of a new hybrid plant [11].

GMDH algorithms 
Variables 
Parametric 
Nonparametric

Continuous 
 Combinatorial (COMBI)
 Multilayered Iterative (MIA)
 GN
 Objective System Analysis (OSA)
 Harmonical
 Twolevel (ARIMAD)
 MultiplicativeAdditive (MAA) 
 Objective Computer Clusterization
(OCC);
 "Pointing Finger" (PF) clusterization algorithm;
 Analogues Complexing (AC) 
Discrete or binary

 Harmonical Rediscretization

 Algorithm on the base of Multilayered
Theory of Statistical Decisions (MTSD) 
The basic parametric GMDH algorithms listed in table, have been
developed for continuous variables. Among the parametric algorithms
[1,9] the most known are:
 the basic Combinatorial (COMBI)
algorithm. It is based on full or reduced sortingout of gradually
complicated models and evaluation of them by external criterion
on separate part of data sample;
 Multilayered Iteration (MIA)
algorithm use at each layer of sorting procedure the same partial
description (iteration rule). It should be used when it is needed
to handle a big number of variables;
 Objective System Analysis (OSA)
algorithm. The key feature of it, is that it examine not single
equations, but systems of algebraic or difference equations, obtained
by implicit templates (without goal function). An advantage of the
algorithm is that the number of regressors is increased consequently,
the information embedded in the data sample is utilized better;
 Twolevel (ARIMAD) algorithm for modeling of longterm
cyclic (such as stock or weather) processes. There are used systems
of polynomial or difference equations for identification of models
on two time scales and then choice of the best pair of models by
external criterion value. For this can be used any parametric algorithm
from described above [23].
Less known are parametric algorithms, which apply an exhaustive
search to difference, harmonic or harmonicexponential functions,
and the MultiplicativeAdditive algorithm, in which tested
polynomial models are obtained by taking the logarithm of the product
of input variables [18,19].
The parametric GMDH algorithms have proved to be highly efficient
in cases where one is to model objects with nonfuzzy characteristics,
such as engineering objects. In cases, where modeling involves objects
with fuzzy characteristics, it is more efficient to use the nonparametric
GMDH algorithms, in which polynomial models are replaced by a data
sample divided into intervals or clusters. Such type algorithms
completely solve the problem of coefficients estimates bias elimination.
Nonparametric algorithms are exemplified by:
 Objective Computer Clusterization (OCC)
algorithm that operates with pairs of closely spaced sample points
[5]. It finds physical clusterization that would as possible be
the same on two subsamples;
 "Pointing Finger" (PF)
algorithm for the search of physical clusterization. It is implemented
by construction of two hierarchical clustering trees and estimation
by the balance criterion [20];
 Analogues Complexing (AC)
algorithm, which use the set of analogues instead of models and
clusterizations [8]. It is recommended for the most fuzzy objects;
 Probabilistic algorithm,
based on the Multilayered Theory of Statistical Decisions
[6]. It is recommended for recognition and forecasting of binary
objects and for the veritability of input data control to avoid
the possible experts errors in it.
Recent developments of the GMDH have led to neuronets
with active neurons, which realise twicemultilayered structure:
neurons are multilayered and they are connected into multilayered
structure. This gives possibility to optimize the set of input variables
at each layer, while the accuracy increases. The accuracy of forecasting,
approximation or pattern recognition can be increased beyond the
limits which are reached by neuronet with single neurons [9,10,34].
In this approach, which corresponds to the actions of human nervous
system, the connections between several neurons are not fixed but
change depending on the neurons themselves. Such active neurons
are able during the learning selforganizing process to estimate
which inputs are necessary to minimise the given objective function
of neuron. This is possible on the condition that every neuron in
its turn is multilayered unit, such as modeling GMDH algorithm.
Neuronet with active neurons, is considered as a tool to increase
AI problems accuracy and leadtime with the help of regression area
extension for inaccurate, noisy data or small data samples.
The GMDH algorithms recently are applied in optimization to solve
the problems of normative forecasting (after "whatifthen"
scenario) and optimal control of multivariable illdefined objects.
Many illdefined objects in macroeconomy, ecology, manufacturing
etc. can be described accurately enough by static algebraic or by
difference equations, which can be transformed into problems of
linear programming by nomination of nonlinear members by additional
variables. GMDH algorithms are applied to evaluate deflections of
output variables from their reference optimal values [7,21]. Examples
of use of Simplified Linear Programming (SLP)
algorithm should be used for expert computer advisers construction,
normative forecasting and control optimization of averaged variables.
An important example [10] gives the prediction of effects of experiments.
The algorithm solves two problems: calculation of effects of a given
nuclear experiment and calculation of parameters which are necessary
to reach optimal effects. It means, that the realization of experiments
can often be replaced by computer experiments.
As already noted, the GMDH algorithms have been developed for
continuous variables. In practice, however the sample will often
include variables discretized into a small number of levels or even
binary values. To extend the GMDH algorithms to discretized or binary
variables, the Harmonic Rediscretization algorithm has been
developed [22].
The existence of a broad gamut of GMDH algorithms is traceable
to the fact, that it is impossible to define the characteristics
of the rest or controlled objects exactly in advance. Therefore,
it can be good practice to try several GMDH algorithms one after
another and to decide which one suits a given type of objects best.
All the questions, which arise during modeling process, are to
be solved by the comparison of the criterion values: that variant
is better, which leads to more deeper minimum
of basic external criteria. In this way, the type of algorithm is
chosen objectively, according to the value of the discriminating
criterion.
Information about dispersion of noise level is very useful to chose
the algorithm type. For small dispersion level we can use the learning
networks of GMDH type, based on the ordinary regression analysis
using internal criteria. For considerable noise level the GMDH algorithms
with external criteria are recommended. And for high level of noise
dispersion nonparametric algorithms of clusterization
or analogues complexing should
be applied [8].
