<s>
Group	B-Algorithm
method	I-Algorithm
of	I-Algorithm
data	I-Algorithm
handling	I-Algorithm
(	O
GMDH	B-Algorithm
)	O
is	O
a	O
family	O
of	O
inductive	B-Algorithm
algorithms	O
for	O
computer-based	O
mathematical	O
modeling	O
of	O
multi-parametric	O
datasets	O
that	O
features	B-General_Concept
fully	O
automatic	O
structural	O
and	O
parametric	O
optimization	O
of	O
models	O
.	O
</s>
<s>
GMDH	B-Algorithm
is	O
used	O
in	O
such	O
fields	O
as	O
data	B-Application
mining	I-Application
,	O
knowledge	O
discovery	O
,	O
prediction	O
,	O
complex	O
systems	O
modeling	O
,	O
optimization	O
and	O
pattern	O
recognition	O
.	O
</s>
<s>
GMDH	B-Algorithm
algorithms	O
are	O
characterized	O
by	O
inductive	B-Algorithm
procedure	O
that	O
performs	O
sorting-out	O
of	O
gradually	O
complicated	O
polynomial	O
models	O
and	O
selecting	O
the	O
best	O
solution	O
by	O
means	O
of	O
the	O
external	O
criterion	O
.	O
</s>
<s>
A	O
GMDH	B-Algorithm
model	O
with	O
multiple	O
inputs	O
and	O
one	O
output	O
is	O
a	O
subset	O
of	O
components	O
of	O
the	O
base	O
function	O
(	O
1	O
)	O
:	O
</s>
<s>
where	O
fi	O
are	O
elementary	O
functions	O
dependent	O
on	O
different	O
sets	O
of	O
inputs	O
,	O
ai	B-Application
are	O
coefficients	O
and	O
m	O
is	O
the	O
number	O
of	O
the	O
base	O
function	O
components	O
.	O
</s>
<s>
In	O
order	O
to	O
find	O
the	O
best	O
solution	O
,	O
GMDH	B-Algorithm
algorithms	O
consider	O
various	O
component	O
subsets	O
of	O
the	O
base	O
function	O
(	O
1	O
)	O
called	O
partial	O
models	O
.	O
</s>
<s>
Coefficients	O
of	O
these	O
models	O
are	O
estimated	O
by	O
the	O
least	B-Algorithm
squares	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
GMDH	B-Algorithm
algorithms	O
gradually	O
increase	O
the	O
number	O
of	O
partial	O
model	O
components	O
and	O
find	O
a	O
model	O
structure	O
with	O
optimal	O
complexity	O
indicated	O
by	O
the	O
minimum	O
value	O
of	O
an	O
external	O
criterion	O
.	O
</s>
<s>
As	O
the	O
first	O
base	O
function	O
used	O
in	O
GMDH	B-Algorithm
,	O
was	O
the	O
gradually	O
complicated	O
Kolmogorov	B-Algorithm
–	I-Algorithm
Gabor	I-Algorithm
polynomial	I-Algorithm
(	O
2	O
)	O
:	O
</s>
<s>
The	O
inductive	B-Algorithm
algorithms	O
are	O
also	O
known	O
as	O
polynomial	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
Jürgen	O
Schmidhuber	O
cites	O
GMDH	B-Algorithm
as	O
one	O
of	O
the	O
first	O
deep	B-Algorithm
learning	I-Algorithm
methods	O
,	O
remarking	O
that	O
it	O
was	O
used	O
to	O
train	O
eight-layer	O
neural	B-Architecture
nets	I-Architecture
as	O
early	O
as	O
1971	O
.	O
</s>
<s>
This	O
inductive	B-Algorithm
approach	O
from	O
the	O
very	O
beginning	O
was	O
a	O
computer-based	O
method	O
so	O
,	O
a	O
set	O
of	O
computer	O
programs	O
and	O
algorithms	O
were	O
the	O
primary	O
practical	O
results	O
achieved	O
at	O
the	O
base	O
of	O
the	O
new	O
theoretical	O
principles	O
.	O
</s>
<s>
In	O
fact	O
,	O
this	O
approach	O
can	O
be	O
considered	O
as	O
one	O
of	O
the	O
implementations	O
of	O
the	B-Application
Artificial	I-Application
Intelligence	I-Application
thesis	O
,	O
which	O
states	O
that	O
a	O
computer	O
can	O
act	O
as	O
powerful	O
advisor	O
to	O
humans	O
.	O
</s>
<s>
genetic	O
selection	O
of	O
pairwise	O
features	B-General_Concept
,	O
Godel	O
's	O
incompleteness	O
theorems	O
and	O
the	O
Gabor	O
's	O
principle	O
of	O
"	O
freedom	O
of	O
decisions	O
choice	O
"	O
,	O
the	O
Adhémar	O
's	O
incorrectness	O
and	O
the	O
Beer	O
's	O
principle	O
of	O
external	O
additions	O
.	O
</s>
<s>
GMDH	B-Algorithm
is	O
the	O
original	O
method	O
for	O
solving	O
problems	O
for	O
structural-parametric	O
identification	O
of	O
models	O
for	O
experimental	B-General_Concept
data	I-General_Concept
under	O
uncertainty	O
.	O
</s>
<s>
GMDH	B-Algorithm
differs	O
from	O
other	O
methods	O
of	O
modelling	O
by	O
the	O
active	O
application	O
of	O
the	O
following	O
principles	O
:	O
automatic	O
models	O
generation	O
,	O
inconclusive	O
decisions	O
,	O
and	O
consistent	O
selection	O
by	O
external	O
criteria	O
for	O
finding	O
models	O
of	O
optimal	O
complexity	O
.	O
</s>
<s>
It	O
had	O
an	O
original	O
multilayered	O
procedure	O
for	O
automatic	O
models	O
structure	O
generation	O
,	O
which	O
imitates	O
the	O
process	O
of	O
biological	O
selection	O
with	O
consideration	O
of	O
pairwise	O
successive	O
features	B-General_Concept
.	O
</s>
<s>
Such	O
procedure	O
is	O
currently	O
used	O
in	O
Deep	B-Algorithm
learning	I-Algorithm
networks	O
.	O
</s>
<s>
This	O
initiated	O
the	O
development	O
of	O
the	O
GMDH	B-Algorithm
theory	O
as	O
an	O
inductive	B-Algorithm
method	O
of	O
automatic	O
adaptation	O
of	O
optimal	O
model	O
complexity	O
to	O
the	O
level	O
of	O
noise	O
variation	O
in	O
fuzzy	O
data	O
.	O
</s>
<s>
Therefore	O
,	O
GMDH	B-Algorithm
is	O
often	O
considered	O
to	O
be	O
the	O
original	O
information	O
technology	O
for	O
knowledge	O
extraction	O
from	O
experimental	B-General_Concept
data	I-General_Concept
.	O
</s>
<s>
The	O
convergence	O
of	O
multilayered	O
GMDH	B-Algorithm
algorithms	O
was	O
investigated	O
.	O
</s>
<s>
In	O
1977	O
a	O
solution	O
of	O
objective	O
systems	O
analysis	O
problems	O
by	O
multilayered	O
GMDH	B-Algorithm
algorithms	O
was	O
proposed	O
.	O
</s>
<s>
It	O
was	O
proved	O
,	O
that	O
non-physical	O
models	O
of	O
GMDH	B-Algorithm
are	O
more	O
accurate	O
for	O
approximation	O
and	O
forecast	O
than	O
physical	O
models	O
of	O
regression	O
analysis	O
.	O
</s>
<s>
Present	O
stage	O
of	O
GMDH	B-Algorithm
development	O
can	O
be	O
described	O
as	O
blossom	O
out	O
of	O
deep	B-Algorithm
learning	I-Algorithm
neuronets	O
and	O
parallel	O
inductive	B-Algorithm
algorithms	O
for	O
multiprocessor	O
computers	O
.	O
</s>
<s>
External	O
criterion	O
is	O
one	O
of	O
the	O
key	O
features	B-General_Concept
of	O
GMDH	B-Algorithm
.	O
</s>
<s>
Criterion	O
describes	O
requirements	O
to	O
the	O
model	O
,	O
for	O
example	O
minimization	O
of	O
Least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
Criterion	O
of	O
Regularity	O
(	O
CR	O
)	O
–	O
Least	B-Algorithm
squares	I-Algorithm
of	O
a	O
model	O
at	O
the	O
sample	O
B	O
.	O
</s>
<s>
Cross-validation	B-Application
criteria	O
.	O
</s>
<s>
For	O
modeling	O
using	O
GMDH	B-Algorithm
,	O
only	O
the	O
selection	O
criterion	O
and	O
maximum	O
model	O
complexity	O
are	O
pre-selected	O
.	O
</s>
<s>
The	O
very	O
first	O
consideration	O
order	O
used	O
in	O
GMDH	B-Algorithm
and	O
originally	O
called	O
multilayered	O
inductive	B-Algorithm
procedure	O
is	O
the	O
most	O
popular	O
one	O
.	O
</s>
<s>
Multilayered	O
procedure	O
is	O
equivalent	O
to	O
the	O
Artificial	B-Architecture
Neural	I-Architecture
Network	I-Architecture
with	O
polynomial	O
activation	O
function	O
of	O
neurons	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
algorithm	O
with	O
such	O
an	O
approach	O
usually	O
referred	O
as	O
GMDH-type	O
Neural	B-Architecture
Network	I-Architecture
or	O
Polynomial	B-Algorithm
Neural	I-Algorithm
Network	I-Algorithm
.	O
</s>
<s>
Li	O
showed	O
that	O
GMDH-type	O
neural	B-Architecture
network	I-Architecture
performed	O
better	O
than	O
the	O
classical	O
forecasting	O
algorithms	O
such	O
as	O
Single	O
Exponential	O
Smooth	O
,	O
Double	O
Exponential	O
Smooth	O
,	O
ARIMA	O
and	O
back-propagation	O
neural	B-Architecture
network	I-Architecture
.	O
</s>
<s>
This	O
approach	O
has	O
some	O
advantages	O
against	O
Polynomial	B-Algorithm
Neural	I-Algorithm
Networks	I-Algorithm
,	O
but	O
requires	O
considerable	O
computational	O
power	O
and	O
thus	O
is	O
not	O
effective	O
for	O
objects	O
with	O
a	O
large	O
number	O
of	O
inputs	O
.	O
</s>
<s>
An	O
important	O
achievement	O
of	O
Combinatorial	O
GMDH	B-Algorithm
is	O
that	O
it	O
fully	O
outperforms	O
linear	O
regression	O
approach	O
if	O
noise	O
level	O
in	O
the	O
input	O
data	O
is	O
greater	O
than	O
zero	O
.	O
</s>
<s>
In	O
contrast	O
to	O
GMDH-type	O
neural	B-Architecture
networks	I-Architecture
Combinatorial	O
algorithm	O
usually	O
does	O
not	O
stop	O
at	O
the	O
certain	O
level	O
of	O
complexity	O
because	O
a	O
point	O
of	O
increase	O
of	O
criterion	O
value	O
can	O
be	O
simply	O
a	O
local	O
minimum	O
,	O
see	O
Fig.1	O
.	O
</s>
<s>
—	O
GMDH-based	O
,	O
predictive	O
analytics	O
and	O
time	O
series	O
forecasting	O
software	O
.	O
</s>
<s>
—	O
Weka	B-Language
plugin	O
,	O
Open	O
source	O
.	O
</s>
