There were published many papers and books devoted to the Group
Method of Data Handling theory and its applications. The GMDH can
be considered as further propagation of inductive selforganizing
methods to the solution of more complex practical problems. It solves
the problem of how to handle data samples of observations. The goal
is to get mathematical model of the object (the problem of identification
and pattern recognition) or to describe the processes, which will
take place at object in the future (the problem of process forecasting).
GMDH solves, by sortingout procedure, the multidimensional problem
of model optimization:
where:
G  set of considered models; CR is an external criterion
of model g quality from this set; P  number of variables
set; S  model complexity; z^{2}  noise dispersion;
T  number of data sample transformation; V  type
of reference function. For definite reference function, each set
of variables corresponds to definite model structure P = S.
Problem transforms to much simpler onedimensional
when
z^{2}= const, T = const, and V = const.
Method
is based on the sortingout procedure, i.e. consequent testing of
models, chosen from set of modelscandidates in accordance with
the given criterion. Most of GMDH algorithms, use the polynomial
reference functions. General connection between input and output
variables can be expressed by Volterra functional series, discrete
analogue of which is KolmogorovGabor polynomial:
where
X(x_{1},x_{2},...,x_{M},);  input
variables vector; A(a_{1},a_{2},...,a_{M},);
 vector of coefficients or weights;
Components
of the input vector X can be independent variables, functional
forms or finite difference terms. Other nonlinear reference functions,
such as difference, probabilistic, harmonic, logistic can also be
used. The method allows to find simultaneously the structure
of model and the dependence of modelled system output on the values
of most significant inputs of the system.
The
GMDH theory solve the problems of:
longterm forecasting [3,18];
shortterm forecasting of processes and events [2];
identification of physical regularities;
approximation of multivariate processes;
physical fields extrapolation
[4];
data samples clusterization
[5];
pattern
recognition in the case of continuousvalued or discrete variables;
diagnostics and recognition by probabilistic sortingout algorithms
[6];
decision
support after "whatif" scenario and vector process normative
forecasting [7];
modelless processes forecasting using analogues
complexing [8];
selforganization of twicemultilayered
neuronets with active neurons [9,10].
In [12] were obtained the theoretical grounds of GMDH effectiveness
as adequate method of robust forecasting models construction,
essence of which consists of automatically generation of models
in given class by sequential selection of the best of them by
criteria, which implicitly by sample dividing take into account
the level of indeterminacy.
Since 1968 a big number of GMDH technique implementations
for modeling of economic, ecological, environmental, medical,
physical and military objects were done in several countries.
Some outdated approaches are used in USA by Barron Associates
Co. in "ASPNII", AbTech Corp. in "ModelQuest"
(AIM), by Ward Systems Group, Inc. in "NeuroShell2",
and DeltaDesign Software in "SelfOrganize!" commercial
software tools.
Selforganizing modeling is based on statistical
learning networks, which are networks of mathematical functions
that capture complex, nonlinear relationships in a compact and
rapidly executable form. Such networks subdivide a problem into
manageable pieces or nodes and then automatically apply advanced
regression techniques to solve each of these simpler problems.
Special
GMDH peculiarities
The main peculiarity of GMDH algorithms is that, when it uses
continuous data with noise, it selects as optimal simplified nonphysical
model. Only for accurate and discrete data the algorithms
point out socalled physical model  the most simple optimal,
from all unbiased models. It is proved the convergence
of multilayered GMDH algorithms [25] and it is proved that shortened
nonphysical model is better
than full physical model on error criterion (for noisy and continuous
data for prediction and approximation solving, more simplified
Shennon's nonphysical models become more accurate [12]). It can
be noted, that this conclusion has place in model selection on
the basis of models entropy maximization (Akaike approach), in
average risk minimizing (Vapnik approach) and in another modern
approaches. The only way to get nonphysical models is to use
sortingout GMDH algorithms. Usage of sortingout procedure guarantees
selection of the best optimal model from all possible. Regularity
of optimal structure of forecasting models change in dependence
on general indexes of data indeterminacy (noise level, data sample
length, design of experiment, number of informational variables)
was shown in [24,25,27].
The
special peculiarities of GMDH are following :
 External
supplement:Following S.Beer work [13], only the external
criteria, calculated on new independent information,can
produce the minimum of sortingout characteristic. Because of
this data sampling is divided into parts for model construction
and evaluation.
 Freedom
of choice:Following D.Gabor work [14], in multilayered
GMDH algorithms are to be conveyed from one layer to the next
layer not all but F best results of choice to provide "freedom
of choice"
 The
rule of layers complication:Partial descriptions (forms
of a mathematical description for iteration) should be simple,
without quadratic members in them;
 Additional
model definition:In cases, when the choice of optimal
physical model is difficult, because of noise level or oscillations
of criterion minima characteristic, auxiliary discriminating
criterion is used [15]. The choice of the main criterion
and constrains of sortingout procedure is the main heuristic
of GMDH;
 All
algorithms have multilayered structure and parallel
computing can be implemented for their realization.
 All
questions that arise about type of algorithm, criterion, variables
set choice should be solved by minimal value of
external criterion.
The main criteria proposed
are: crossvalidation PRR(s), regularity AR(s) and
balance of variables criterion BL(s). Estimation of their
effectiveness (investigation of noise immunity, optimality and
adequateness) and their comparison with another criteria was done
in detail in [24,25,26,15]. The conditions, under which GMDH algorithm
produces the minimum of characteristics are following:

criterion of model choice is to be external, based on
additional fresh information, which was not used for model construction;

the data sample is not to be too long. Such data sample produce
the same model of usual regression analysis for exact data;

when difference type balance criterion BL(s) is used, small
noise is necessary or the variables in the data sample should
not be exactly measured [16].
Difference of the GMDH algorithms from another algorithms of structural
identification and best regression selection algorithms consists
of three main peculiarities:
usage of external criteria, which are based on data sample
dividing and are adequate to problem of forecasting models construction,
by decreasing of requirements to volume of initial information;
more diversity of structure generators: usage like in regression
algorithms of the ways of full or reduced sorting of structure variants
and of original multilayered (iteration) procedures;
better
level of automatization: there are needed to enter initial
data sample and type of external criterion only;
automatic
adaptation of optimal model complexity and external criteria to
level of noises or statistical violations  effect of noiseimmunity
cause robustness of the approach;
implementation of principle of inconclusive decisions in
process of gradual models complication.
