Нечіткий МГУА та його застосування для прогнозування фінансових процесів
This paper is devoted to the investigation and application of the fuzzy inductive modeling method known as Group Method of Data Handling (GMDH) in problems of Data Mining, in particularly its application to solving the forecasting tasks in financial sphere. The advantage of the inductive modeling me...
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| author | Zaychenko, Yuriy Zaychenko, Helen |
| author_facet | Zaychenko, Yuriy Zaychenko, Helen |
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| description | This paper is devoted to the investigation and application of the fuzzy inductive modeling method known as Group Method of Data Handling (GMDH) in problems of Data Mining, in particularly its application to solving the forecasting tasks in financial sphere. The advantage of the inductive modeling method GMDH is a possibility of constructing an adequate model directly in the process of algorithm run. The generalization of GMDH in case of uncertainty — a new method fuzzy GMDH is described which enables to construct fuzzy models almost automatically. The algorithm of fuzzy GMDH is considered. Fuzzy GMDH with Gaussian and bell-wise membership functions MF are considered and their similarity with triangular MF is shown. Fuzzy GMDH with different partial descriptions orthogonal polynomials of Chebyshev and Fourier are considered. The problem of adaptation of fuzzy models obtained by FGMDH is considered and the corresponding adaptation algorithm is described. The extension and generalization of fuzzy GMDH in case of fuzzy inputs is considered and its properties are analyzed. The experimental investigations of the suggested FGMDH were carried out. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2019.1.07 |
| first_indexed | 2025-07-17T10:25:08Z |
| format | Article |
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© Yuriy Zaychenko, Helen Zaychenko, 2019
Системні дослідження та інформаційні технології, 2019, № 1 91
UDC 681.513.
DOI: 10.20535/SRIT.2308-8893.2019.1.07
FUZZY GMDH AND ITS APPLICATION TO FORECASTING
FINANCIAL PROCESSES
YURIY ZAYCHENKO, HELEN ZAYCHENKO
Abstract. This paper is devoted to the investigation and application of the fuzzy in-
ductive modeling method known as Group Method of Data Handling (GMDH) in
problems of Data Mining, in particularly its application to solving the forecasting
tasks in financial sphere. The advantage of the inductive modeling method GMDH
is a possibility of constructing an adequate model directly in the process of algo-
rithm run. The generalization of GMDH in case of uncertainty — a new method
fuzzy GMDH is described which enables to construct fuzzy models almost auto-
matically. The algorithm of fuzzy GMDH is considered. Fuzzy GMDH with Gaus-
sian and bell-wise membership functions MF are considered and their similarity
with triangular MF is shown. Fuzzy GMDH with different partial descriptions or-
thogonal polynomials of Chebyshev and Fourier are considered. The problem of ad-
aptation of fuzzy models obtained by FGMDH is considered and the corresponding
adaptation algorithm is described. The extension and generalization of fuzzy GMDH
in case of fuzzy inputs is considered and its properties are analyzed. The experimen-
tal investigations of the suggested FGMDH were carried out.
Keywords: fuzzy GMDH, membership functions, models adaptation, forecasting.
INTRODUCTION
One of the most important problems in the sphere of economy and finance is the
problem of forecasting economical and financial processes. The distinguishing
features of the problem are the following:
• the form of functional dependence is unknown and only model class is
determined, for example, polynomial of any degree or Fourier time series;
• short data samples;
• time series ( )ix t in general case is non-stationary.
In this case, the application of conventional methods of statistical analysis
(e.g. regression analysis) is impossible and it is necessary to utilize methods based
on computational intelligence (CI). The Group Method of Data Handling
(GMDH) developed by acad. A.G. Ivakhnenko [1] and extended by his col-
leges [2] belongs to this class. GMDH is a self-organizing method allowing to
discover internal hidden laws in the appropriate object area. The advantages of
GMDH algorithms are the possibility of constructing optimal models from sam-
ples with a small number of observations and unknown relationships among vari-
ables. This method does not demand to know the model structure a priori; it is
constructed by algorithm itself in the process of its run.
In case if input data are measured with errors which distribution is not nor-
mal and is unknown to a decision-maker then it is reasonable to construct fuzzy
model using Fuzzy GMDH method. FGMDH was suggested in [3] where main
ideas of the method and an algorithm of fuzzy GMDH was presented and investi-
Yuriy Zaychenko, Helen Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2019, № 1 92
gated in case when fuzzy coefficients of models have triangular membership
functions. For finding optimal parameters of fuzzy model, the LP problem was
introduced and solved. Later this method was extended to Gaussian and bell-wise
MF [4]. It was shown that the structure of LP model for this problem is the same
as for triangular MF. Further FGMDH method was extended to orthogonal poly-
nomials as partial descriptions [5, 6].
Problem of adaptation of fuzzy models in FGMDH was considered in [5]
and several adaptation methods were suggested and investigated. Numerous ex-
perimental investigations of fuzzy GMDH with different partial descriptions and
MF were carried out and comparison with classical GMDH was performed [7].
The extension of FGMDH and its generalization in the case when input data
is fuzzy were considered in [8]. The math model for determination of fuzzy coef-
ficients was constructed and general FGMDH algorithm for fuzzy inputs was de-
veloped and investigated.
The goal of this paper is to present a review of the main results in the field of
developing the fuzzy GMDH and experimental results of its applications to the
forecasting in financial sphere.
OTHER FUZZY-BASED APPROACHES IN GMDH NETWORKS
The application of GMDH for structure optimization of fuzzy polynomial neural
networks (FPNN) were developed and investigated in numerous works of Witold
Pedrycz and his colleagues since 2002 [9-16]. They implemented a like approach
as in fuzzy GMDH. In [9, 10], W. Pedrycz et al suggested hybrid neural network
called fuzzy polynomial neural networks (FPNNs), a hybrid modeling architec-
ture combining polynomial neural networks (PNNs) and fuzzy neural networks
(FNNs).
The development of the FPNNs is based on the technologies of computa-
tional intelligence (CI). The structure of the FPNN results from a synergistic us-
age of FNN and PNN. FNNs contribute to the formation of the premise part of the
rule-based structure of the FPNN while the consequence part of the FPNN is de-
signed using PNNs. Each PN of the network realizes a polynomial type of partial
description (PD) of the mapping between input and out variables: linear, quadratic
or modified quadratic. The structure of the PNN is not fixed in advance but it is
generated by GMDH to produce dynamic topology of the network.
The authors continued their investigations in the sphere of FPNN and for op-
timization of it topology suggested genetic algorithms (GAs) [11, 12]. Opposite to
the conventional HFPNN which use the GMDH method for structure synthesis,
they suggested to apply GA for structure optimization of both FNN and PNN. As
a result, genetically optimized HFPNN (gHFPNN) was constructed. The aug-
mented gHFPNN results in a structurally optimized structure and comes with a
higher level of flexibility in comparison to the conventional HFPNN. In the se-
quel, two general optimization mechanisms are explored. First, the structural op-
timization is realized via GAs whereas the ensuing detailed parametric optimiza-
tion is carried out in the setting of a standard least square method-based learning.
In [13, 14, 15], genetically optimized fuzzy relation-based polynomial neural
networks were introduced and investigated using information granulation (IG
gFRPNN). With the aid of the information granules based on C-Means clustering,
the initial location of membership functions were determined and initial values of
Fuzzy GMDH and its application to forecasting financial processes
Системні дослідження та інформаційні технології, 2019, № 1 93
polynomial function being used in the premised and consequence part of the
fuzzy rules respectively. The GA-based design procedure being applied at each
layer of the IG_gFRPNN leads to the selection of preferred nodes with specific
local characteristics (such as the number of input variables, the order of the poly-
nomial, a collection of the specific subset of input variables, and the number of
membership functions) available within the network.
In the sequel, the structural optimization is realized via GAs, whereas the en-
suing detailed parametric optimization is carried by the standard least square
method-based learning. The development of gFRPNN was continued in [16]
where the problem of constructing the FRPNN under conditions of high dimen-
sions was considered. Parallel fuzzy polynomial neural networks (PFPNNs) with
the aid of heterogeneous partition of the input space were suggested. In the design
of the premise part of the rule, a weighted fuzzy clustering method is used not
only to realize a non-uniform partition of the input space but to overcome a possi-
ble curse of dimensionality. While in the design of consequent part, fuzzy poly-
nomial neural networks are utilized to construct optimal local models (high order
polynomials) that describe the relationship between input variables and the output
variable within some local region of the input space. Particle swarm optimization
(PSO) was employed to adjust the design parameters of parallel fuzzy polynomial
neural networks.
The development and investigations of hybrid GMDH- fuzzy neural net-
works were performed by joint group of scientists in NTUU “KPI” and KTURE
(Kharkiv University of Radio-Electronics). In [17, 18], GMDH-wavelet neuro-
fuzzy system was suggested and investigated using advantages of neuro-fuzzy
networks and GMDH. In [19], GMDH-cascade neo-fuzzy networks were sug-
gested and investigated in the problem of forecasting. In this work the structure of
cascade neo-fuzzy network was constructed using GMDH which enabled to find
the structure of network and weights of neurons. In [20], GMDH-neural network
with spiking neurons was suggested.
This approach of developing hybrid GMDH–FNN systems was continued in
[21] where the authors suggested evolving a hybrid GMDH-neuro fuzzy system.
The hybrid system is grounded on both GMDH and the concept of evolving sys-
tems that makes it possible to define both optimal parameter values and the best
structure in every specific case. The important property of the suggested system is
that it does not require any high data volumes to get trained. Adjusting parameters
in a parallel fashion gives an option of increasing a processing speed of data
handling.
FUZZY GMDH: PRINCIPAL IDEAS AND MATHEMATICAL MODEL
CONSTRUCTION
As it is well-known, the drawbacks of classical GMDH are the followings [3, 4]:
– GMDH utilizes least squared method (LSM) for finding the model coeffi-
cients but matrix of linear equations may be close to degenerate and the corre-
sponding solution may appear non-stable and very volatile. Therefore, the special
methods for regularization should be used;
– after application of GMDH point-wise estimations are obtained but in
many cases it is needed find interval value for coefficient estimates;
Yuriy Zaychenko, Helen Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2019, № 1 94
– GMDH does not work in case of incomplete or fuzzy input data.
Therefore, in the last 10 years the new variant of GMDH – fuzzy GMDH
was developed and refined which may work with fuzzy input data and is free of
classical GMDH drawbacks [3]. As it is well known, GMDH is based on the fol-
lowing principles [1–3]:
1) the principle of multiplicity of models;
2) the principle of external complement which means that the whole sample
should be divided into two parts – training subsample and test subsample;
3) the principle of self-organization;
4) the principle of freedom of choice.
Fuzzy GMDH is also based on these principles but construct fuzzy models.
Let us consider its main ideas.
In [3–5], the linear interval model regression was considered:
nnZAZAZAY +++= ...1100 , (1)
where iA is a fuzzy number of triangular form described by pair of parameters
( )iii cA ,α= , where iα is interval center, ic is its width, 0≥ic .
Then Y is a fuzzy number with parameters determined as follows:
– the interval center:
ziziy
Tα=∑α=α ;
– the interval width:
∑ == zczicc T
iy .
For example, for the partial description of the kind
2
5
2
43210),( jijijiji xAxAxxAxAxAAxxf +++++=
it is necessary to assign in the general model (1),
10 =z , ixz =1 , jxz =2 , ji xxz =3 , 2
4 ixz = , 2
5 jxz = .
Let the training sample be },...,,{ 21 Mzzz , },...,,{ 21 Myyy . Then for the
model (1) to be adequate it is necessary to find such parameters ( ) nicii ,1, =α
satisfying the following inequalities:
Mk
yzcz
yzcz
kk
T
k
T
kk
T
k
T
,1
,
,
=
⎪⎩
⎪
⎨
⎧
≥+α
≤−α
.
Let us formulate the basic requirements for the linear interval model of par-
tial description of a kind (1).
It is necessary to find such values of the parameters ( )ii c,α of fuzzy coeffi-
cients for which:
1) real values of observed outputs ky would drop in the estimated interval
for kY ;
2) the total width of the estimated interval for all sample points would be
minimal.
Fuzzy GMDH and its application to forecasting financial processes
Системні дослідження та інформаційні технології, 2019, № 1 95
These requirements lead to the following linear programming problem [3, 4]:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+++++ ∑∑∑∑∑
=====
M
k
kj
M
k
ki
M
k
kjki
M
k
kj
M
k
ki xCxCxxCxCxCMС
1
2
5
1
2
4
1
3
1
2
1
10min ; (2)
under constraints:
+++−+++++ kjkikjkikjkikjki xCxCСxaxaxxaxaxaa 210
2
5
2
43210 (
kkjkikjki yxСxСxxС ≤+++ )2
5
2
43 , (3)
+++++++++ kjkikjkikjkikjki xCxCСxaxaxxaxaxaa 210
2
5
2
43210 (
kkjkikjki yxСxСxxС ≥+++ )2
5
2
43 , (4)
where 5,0,0 =≥ pC p , Mk ,1= and k is an index of a point.
As we can easily see the task (2) – (4) is linear programming (LP) problem.
However, the inconvenience of the model (2) – (4) for the application of standard
LP methods is that there are no constraints of non-negativity for variables ia .
Therefore for its solution it is reasonable to pass to the dual LP problem by intro-
ducing dual variables }{ kδ and }{ Mk+δ , Mk ,1= . Using simplex-method for the
dual problem and after finding the optimal values for the dual variables }{ kδ , the
optimal solutions ),( ii ca of the initial direct problem will be also found.
THE DESCRIPTION OF FUZZY GMDH ALGORITHM
Let us present the brief description of the algorithm FGMDH [3, 4].
1. Choose the general model type by which the sought dependence will be
described.
2. Choose the external criterion of optimality (criterion of regularity or un-
biasedness).
3. Choose the type of partial descriptions (for example, linear or quadratic
one).
4. Divide the sample into training trainN and test testN subsamples.
5. Put zero values to the counter of model number k and to the counter of
layers r (iterations number).
6. Generate a new partial model kf (1) using the training sample. Solve the
LP problem (2) – (4) and find the values of parameters iα , iс .
7. Calculate using test sample the value of external criterion ( )(r
ubkN or
)()2( rkδ ).
8. 1+= kk . If 2
NCk > for r=1 or 2
FCk > for r>1, then 1=k , 1+= rr and
go to step 9, otherwise go to step 6.
9. Calculate the best value of the criterion for models of r-th iteration. If
1=r , then go to step 6 otherwise, go to step 10.
Yuriy Zaychenko, Helen Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2019, № 1 96
10. If ε≤−− )1()( rNrN ubub or )()2( rkδ ≥ )()2(
1 rk−δ , then go to 11, otherwise
select F best models and assigning 1+= rr , 1=k , go to step 6 and execute
(r+1)-th iteration.
11. Select the best model out of models of the previous layer (iteration) using
external criterion.
ANALYSIS OF DIFFERENT MEMBERSHIP FUNCTIONS
In the first paper devoted to fuzzy GMDH [3], the triangular membership func-
tions (MF) were considered. But as fuzzy numbers may also have the other kinds
of MF it is important to consider the other classes of MF in the problems of mod-
eling using FGMDH. In [4], fuzzy models with Gaussian and bell-shaped MF
were investigated.
Consider a fuzzy set with MF of the form:
2
2)1(
2
1
)( c
x
B ex
−
−
=μ .
Let the linear interval model for partial description of FGMDH take the form
(1). Then the problem is to find such fuzzy numbers iB with parameters ),( ii ca
that:
• the observation ky would belong to a given estimate interval for the set
Y(k) with degree not less than α , 10 <α< ;
• the width of the estimated interval of the degree α would be minimal;
In [4, 6] it was shown that the problem of finding optimal fuzzy model will
be finally transformed to the following LP problem:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+++++ ∑∑∑∑∑
=====
M
k
kj
M
k
ki
M
k
kjki
M
k
kj
M
k
ki xCxCxxCxCxCMС
1
2
5
1
2
4
1
3
1
2
1
10min ; (5)
under constraints:
⎪⎭
⎪
⎬
⎫
≤α−+++−+++
≥α−+++++++
,ln2)...(...
,ln2)...(...
2
510
2
510
2
510
2
510
kkjkikjki
kkjkikjki
yxСxCСxaxaa
yxСxCСxaxaa
Mk ,1= . (6)
To solve this problem like in the case of triangular MF it is reasonable to
pass to the dual LP problem of the form:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
δ−δ ∑∑
==
+
M
k
kk
M
k
kk yy
11
Mmax , (7)
with constraints of equalities and inequalities:
0
11
M =δ−δ ∑∑
==
+
M
k
k
M
k
k ;
0
1 1
=δ−δ∑ ∑
= =
+
M
k
M
k
kkiMkki i
XX ; (8)
Fuzzy GMDH and its application to forecasting financial processes
Системні дослідження та інформаційні технології, 2019, № 1 97
0
1 1
22 =δ−δ∑ ∑
= =
+
M
k
M
k
kkMkkj j
XX ;
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
⎬
⎫
α−
≤δ+δ
α−
≤δ+δ
α−
≤δ+δ
∑
∑ ∑
∑
∑∑
∑∑
=
= =
+
=
==
+
=
+
=
,
ln2
............................................................
,
ln2
,
ln2
1
2
1 1
22
1
11
11
M
k
kjM
k
M
k
kkjMkkj
M
k
kiM
k
kki
M
k
Mkki
M
k
Mk
M
k
k
X
XX
X
XX
M
(9)
0≥δk , Mk 2,1= . (10)
Analyzing the dual LP program (5)–(10), it is easy to notice that this prob-
lem is always solvable as there is trivial solution 1=δk , Mk 2,1= . Therefore the
initial problem is also always solvable with any data.
Thus, fuzzy GMDH allows constructing fuzzy models and has the following
advantages:
1. The problem of optimal model determination is transferred to the problem
of linear programming which is always solvable.
2. There is so called interval regression model built as the result of the meth-
od performance.
FUZZY GMDH WITH DIFFERENT PARTIAL DESCRIPTIONS:
ORTHOGONAL POLYNOMIALS
As it is well known from the general GMDH theory, models-pretenders are gen-
erated in this method on the base of so called partial descriptions being elemen-
tary models of two variables. Usually as partial descriptions linear or quadratic
polynomials are used. The alternative to this class of those elementary models is
application of orthogonal polynomials. The choice of such polynomials as partial
descriptions is determined by the following advantages:
• Owing to orthogonal property, the calculation of polynomial coefficients
which approximate the simulated process goes faster than for non-orthogonal pol-
ynomials.
• The coefficients of polynomial approximating equation do not depend on
the degree of initial polynomial model so if a priori the real polynomial degree is
not known we may check the polynomials of various degrees and by this property
the coefficients obtained for polynomials of lower degrees remain the same after
transfer to higher polynomial degrees. This property is the most important during
investigation of real degree of approximating polynomial when solving applied
problems.
One of the properties of orthogonal polynomials widely used in this work is
the property of almost equal errors. Owing to this the very large errors do not
Yuriy Zaychenko, Helen Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2019, № 1 98
happen, on the contrary, in most cases the error values are small. Therefore the
damping of approximation errors occurs. The application of orthogonal polynomi-
als as partial descriptions in FGMDH was suggested and investigated in [5, 6].
Chebyshev’s orthogonal polynomials
Chebyshev’s orthogonal polynomials in general case have the following form [5]:
11),arccos(cos)()( ≤ξ≤−ξν=ξ=ξ νν TF .
These polynomials have the following orthogonality property:
⎪
⎪
⎩
⎪⎪
⎨
⎧
=ξ=μπ
≠ξ=μ
π
ξ≠μ
=
−
ξξξ
∫
ξ−
νμ
,0 if
0; if
2
; if 0
1
)()(1
1
2
dTT
(11)
where 21 ξ− is a weighting coefficient )(ξω in the equation (11).
The approximating Chebyshev’s orthogonal polynomial for y is obtained on
the base of function S minimization:
∫ ∑
− =
ξ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ξ−ξξω=
1
1
2
0
)()()( dTbyS
m
i
ii . (12)
From (12) it follows that
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≠ξ
ξ−
ξξ
π
=ξ
ξ−
ξ
π
=
∫
∫
−
−
1
1
2
1
1
2
.0,
1
)()(2
;0,
1
)(1
kd
Ty
kdy
b
k
k (13)
Hence, the approximating equation is obtained in the form:
∑
=
ξ=ξ
m
k
kkTby
0
)()( .
As it may be readily seen from the presented expressions, coefficient kb
(13) does not depend on choice of degree m. Thus, the variable m does not de-
mand recalculation of mjbj ≤∀ while such recalculation is necessary for non-
orthogonal approximation.
The best approximating degree *m may be obtained on the base of hypothe-
sis that investigation results riiy ,,2,1),( K= , have independent Gaussian distri-
bution in the bounds of some polynomial function y of certain degree, e.g.
μ+*m , where
∑
μ+
=
μ+
=
*
*
0
)(
m
j
j
ijim
xbxy ,
and the dispersion 2σ of distribution )( yy − does not depend on μ.
Fuzzy GMDH and its application to forecasting financial processes
Системні дослідження та інформаційні технології, 2019, № 1 99
It is clear that for a very small m ( K,2,1,0=m ) 2
mσ decreases as m grows.
As in accordance with the previously formulated hypothesis dispersion does
not depend on m, the best degree *m is the minimal m for which 1+σ≅σ mm .
For determining *m it is necessary to calculate the approximating polyno-
mials of various degrees. As coefficients jb in the equation do not depend on m ,
the determination of the best degree of polynomial is accelerated.
Let we have the forecasted variable Y and input variables nxxx K,, 21 . Let us
search the relation between them in the following form
)(...)()( 222111 nnn xfAxfAxfAY +++= ,
where iA is a fuzzy number of triangular type given as ),( iii cA α= , functions
if are determined as follows [5, 6]:
∑
=
=
im
j
ijjiii xTbxf
0
)()( .
The degree im of function if is determined using hypothesis defined in the
preceding section. Denoting )( iii xfz = , we get a linear interval model in classi-
cal form.
Investigation of Trigonometric Polynomials as Partial Descriptions
Let a function f(x) be periodic with period π2 defined at the interval ],[ ππ− , and
its derivative f’(x) is also defined at ],[ ππ− . Then the following equality holds
],[)()( ππ−∈∀= xxfxS ,
where
∑
=
++=
1
0 ))(sin)(cos(
2
)(
j
jj jxbjxaaxS .
Coefficients jj ba , are calculated by Euler formulas:
∫
π
π−π
= ;)(cos)(1 dxjxxfa j ∫
π
π−π
= .)(sin)(1 dxjxxfb j
Definition. A trigonometric polynomial of the degree M is called the expres-
sion:
∑
=
++=
M
j
jjM jxbjxaaxT
1
0 ))sin()cos((
2
)( .
The following theorem is true stating that exists such M , NM <2 , which
minimizes the expression [6]:
∑
=
−
N
j
iMi xTxf
1
2))()(( .
Hence the coefficients of corresponding trigonometric polynomial are de-
termined by formulas:
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ISSN 1681–6048 System Research & Information Technologies, 2019, № 1 100
).(sin)(2);(cos)(2
11
i
N
i
ij
N
i
iij jxxf
N
bjxxf
N
a ∑∑
==
==
Let it be the variable Y to be forecasted and input variables nxxx ,,, 21 K .
Let us search the dependence among them in the form:
)(...)()( 222111 nnn xfAxfAxfAY +++= ,
where iA is a fuzzy number of triangular type given as ),( iii cA α= , functions
if are determined in such a way:
)()( iMii xTxf
i
= .
The degree iM of a function if is determined by the theorem described in
the preceding section. Therefore if to assign )( iii xfz = , the linear interval model
will be obtained in its classical form.
ADAPTATION OF FUZZY GMDH MODELS
While forecasting by self-organizing methods (fuzzy GMDH, in particular), the
problem arises regarding the necessity of huge amount of repetitive calculations
in case of the training sample size increase or while forecasting in real time when
it is needed to correct the obtained model in accordance with new available data.
Taking into account new information obtained while forecasting, adaptation may
be done by two approaches. The first one is to correct parameters of a forecasting
model with new data assuming that model structure did not change. The second
approach consists in adaptation of not only model parameters but its optimal
structure as well.
This way demands the repetitive use of full GMDH algorithm and is con-
nected with huge volume of calculations. The second approach is used if adapta-
tion of parameters does not provide good forecast and the new real output values
do not drop in the calculated interval for its estimate.
In our consideration, the first approach is used based on adaptation of
FGMDH model parameters to new available data. Here the recursive identifica-
tion methods are preferably used, especially the recursive least squared method
(LSM). In this method the parameters estimations on the next step are determined
on the base of estimates on the previous step, model error and some information
matrix which is modified during all estimation process and therefore contains data
which may be used at the next steps of adaptation process [5].
Hence, model coefficients adaptation will be simplified substantially. If to
store information matrix obtained while identification of optimal model using
fuzzy GMDH, then for model parameters adaptation it will be enough to fulfill
only one iteration by recursive LSM method.
The Application of Recurrent LSM for Model Coefficients Adaptation
Consider the following model:
)()()( kvkky T +Ψθ= ,
where )(ky is a dependent (output) variable, )(kΨ is a measurements vector,
)(kv are random disturbances, θ is a parameters vector to be estimated.
Fuzzy GMDH and its application to forecasting financial processes
Системні дослідження та інформаційні технології, 2019, № 1 101
The parameters estimate θ at the step N is performed due to such formula
[5, 6]:
)]()1()()[()1()( NNNyNNN T Ψ−θ−γ+−θ=θ
)))
,
where )(Nγ is a coefficients vector which is determined by formula:
)()1()(1
)()1()(
NNPN
NNPN T Ψ−Ψ+
Ψ−
=γ ,
where )1( −NP is so-called “information matrix” determined by formula:
)1()2()1(1
)2()1()1()2()2()1(
−Ψ−−Ψ+
−−Ψ−Ψ−
−−=−
NNPN
NPNNNPNPNP T
T
. (14)
As one can see from (14), the information matrix may be obtained independ-
ently on parameters estimation process and parallel to it. The adaptation of two
parameter vectors ],...,[ 11 m
T αα=θ , ],...,[ 12 m
T CC=θ , is performed using the for-
mulas [35]:
)]()1()()[()1()( 11111 NNNyNNN T Ψ−θ−γ+−θ=θ
)))
;
)]()1()()[()1()( 22222 NNNyNNN T
c Ψ−θ−γ+−θ=θ
)))
;
|)()1()(|)( 11 NNNyNy T
c Ψ−θ−= ,
where |]||,...,[|];,...,[ 1211 m
T
m
T zzzz =Ψ=Ψ .
APPLICATION OF GMDH FOR FORECASTING STOCK EXCHANGE
PROCESSES
Consider the application of GMDH and fuzzy GMDH for forecasting at the stock
exchange NYSE. As input variables, the following stock prices at NYSE were
chosen: close prices of companies Hess Corporation, Repsol YPF, S.A. (ADR),
Eni S.p.A. (ADR), Exxon Mobil Corporation, Chevron Corporation, and Total
S.A. (ADR) [7]. As an output variable, close stock prices of British Petroleum BP
plc (ADR) were chosen.
The second problem was forecasting industrial index Dow-Jones Average.
As the input variables in this problem there were taken close stock prices of the
following companies which form it: American Express Company, Bank of Amer-
ica, Coca-cola, McDonald's, Microsoft Corp., Johnson&Johnson, Intel Corp.
The training sample data were taken in the period since 20 September to 14
November 2011. For the test sample, data of Dow Jones Industrial Average were
taken since 15 November 2011 year to 17 November 2011 year [7].
For the experimental investigations, classical GMDH and fuzzy GMDH
were used. For these experiments, the percentage of training sample variants was
chosen 50%, 70%, 90%. Freedom of choice F was taken 5 and 6.
For fuzzy GMDH, triangular, Gaussian and bell-shaped membership func-
tions were used. For Gaussian and bell-shaped membership functions, the follow-
ing level values were taken: a = 0,3; 0,5; 0,7; 0,9.
Yuriy Zaychenko, Helen Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2019, № 1 102
To construct models, the following four partial description types were used:
• a linear model of the form:
jiji xAxAAxxf 210),( ++= ;
• a squared model:
2
5
2
43210),( jijijiji xAxAxxAxAxAAxxf +++++= ;
• Fourier polynomial of the first degree:
)(cos)(sin
2 11
0
1 xbxaaT ++= ;
• Chebyshev’s polynomial of the second degree:
)1(),( 2
210 −++= jiji xAxAAxxf ,
where jA is a fuzzy number with triangular, Gaussian or bell-shaped membership
function.
For forecasts accuracy analysis, the following criteria of the forecast quality
were used: MSE for test sample, MSE for full sample, MAPE for full and test
sample, Durbin-Watson criterion (DW), R-square, Akaike criterion (AIC), Bayes
information criterion (BIC), and Shwartz criterion (SC). These criteria values
were calculated for each forecast step using the test sample. The results of fore-
cast for BP plc (ADR) shares are presented in the table 1. and for Dow Jones In-
dustrial Average in the table 2.
T a b l e 1 . Forecast quality criteria at each forecast step by fuzzy GMDH for
shares BP plc (ADR) closing prices
Percentage of training sample Step of fore-
cast Criterion
50% 70% 90%
MSE test 1,248355 0,793523 0,317066
MSE 0,748864 0,612827 0,485599
MAPE test 2,041366 2,065155 1,386096
MAPE 1,452656 1,723657 1,505965
DW 0,686478 1,763043 1,839065
R-square 1,064526 0,938058 1,008042
AIC 2,456985 2,228267 2,053752
BIC –3,268440 –3,063 –2,892253
1
SC 2,506693 2,277 2,103459
MSE test 1,256828 1,871883 2,440575
MSE 0,728499 0,989650 0,795699
MAPE test 2,072085 2,367006 2,426847
MAPE 1,468987 1,845798 1,656084
DW 0,686478 1,795021 1,839065
R-square 1,064526 0,874163 1,008042
AIC 2,456985 2,147717 2,053752
BIC –3,268440 –2,986448 –2,892253
2
SC 2,506693 2,197424 2,103459
Fuzzy GMDH and its application to forecasting financial processes
Системні дослідження та інформаційні технології, 2019, № 1 103
The flow charts of forecasts for shares BP plc (ADR) obtained by fuzzy
GMDH are presented on Fig. 1 for 2 steps ahead with 6=F , the training sample size
70% and Gaussian MF, significance level 7,0=a .
T a b l e 2 . Forecast quality criteria by fuzzy GMDH for Dow Jones Industrial
Average
Percentage of training sample Step of forecast Criterion
50% 70% 90%
MSE test 40494,427 33109,754 40286,725
MSE 26900,763 62553,804 26432,081
MAPE test 1,462066 1,363148 1,405372
MAPE 1,149183 1,809130 1,191039
DW 1,917430 1,013536 1,862203
R-square 0,990922 0,804774 0,849329
AIC 12,753417 13,551054 12,729001
BIC –7,360808 –7,482613 –7,356959
1
SC 12,796073 13,593710 12,771656
MSE test 41546,293 31602,995 43907,693
MSE 27793,341 61328,007 32746,387
MAPE test 1,481603 1,374860 1,521950
MAPE 1,167355 1,819287 1,280730
DW 1,917020 1,013536 1,669879
R-square 0,989976 0,804774 0,811488
AIC 12,753849 13,551054 12,935378
BIC –7,360875 –7,482613 –7,389255
2
SC 12,796504 13,593710 12,978034
Fig. 1. Forecast results for BP plc (ADR) shares by FGMDH with quadratic partial de-
scriptions, 3 steps ahead
3 4
2
1
1 –
2 –
3 –
4–
Yuriy Zaychenko, Helen Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2019, № 1 104
Forecasts by fuzzy GMDH for Dow Jones I.A. shares are presented on
Fig. 2, 3 steps ahead, 6=F , the training sample size 70% and Gaussian MF, significance
level 7,0=a .
Further experiments for forecasting share prices of BP plc (ADR) and Dow
Jones I.A. were carried out with application of GMDH and fuzzy GMDH with
different partial descriptions: linear model, squared model, Chebyshev’s polyno-
mials and Fourier polynomials and with application of cascade neo-fuzzy neural
networks as well [7].
The final experimental results of forecasts at 1, 2 and 3 steps ahead with
aforesaid methods for share prices of British Petroleum BP plc (ADR) are pre-
sented in the table 3 and for index Dow Jones Industrial Average in the table 4.
T a b l e 3 . Comparative forecasting results for BP plc (ADR) share prices
Forecast results for GMDH
Partial description (PD)
Forecast results for fuzzy GMDH
Partial description Step
of
fore-
cast
Criteria
Linear Quad-
ratic
Fourier
poly-
nomial
Chebyshev’s
polynomial Linear Quad-
ratic
Fourier
poly
nomial
Chebyshev’s
polynomial
MSE 0,285 1,905 0,859 0,365 0,481 0,130 1,691 0,757 1
MAPE 1,034 1,965 1,624 1,114 1,374 0,813 2,960 1,459
MSE 0,425 3,090 1,094 0,366 0,498 0,150 1,742 1,029 2
MAPE 1,227 2,916 1,814 1,115 1,481 0,818 2,977 1,584
MSE 0,675 4,978 2,144 0,523 0,572 0,308 2,183 1,505 3
MAPE 1,496 4,434 2,050 1,320 1,494 0,908 3,024 1,681
The best results were obtained by fuzzy GMDH with quadratic PD, 70%
training sample and Gaussian MF. The worst results gives Fourier polynomials as
PD. The both GMDH methods, classical and fuzzy, have shown the high forecast
3 4
2
1
2 –
1 –
4 –
3–
Fig. 2. Index Dow Jones I.A. forecast results at 3 steps ahead with FGMDH
Fuzzy GMDH and its application to forecasting financial processes
Системні дослідження та інформаційні технології, 2019, № 1 105
accuracy. If to compare the accuracy of both methods with linear partial descrip-
tions, then linear model by GMDH has shown more accurate results. But with all
used PD most accurate forecasts were obtained using fuzzy GMDH with quad-
ratic partial descriptions.
T a b l e 4 . Comparative forecasting results for index Dow Jones I.A.
Forecast results for GMDH
Partial descriptions
Forecast results for fuzzy GMDH
Partial descriptions Step
of
fore-
cast
Criteria
Linear Quad-
ratic
Fourier
poly-
nomial
Chebyshev’s
polynomial Linear Quad-
ratic
Fourier
polyno-
mial
Chebyshev’s
polynomial
MSE 26900 38225 40142 23818 25176 21332 42205 24464 1
MAPE 1,149 1,298 1,445 1,111 1,137 1,046 1,487 1,125
MSE 27793 39460 40930 23978 25793 223491 59059 24767 2
MAPE 1,167 1,322 1,445 1,119 1,143 1,098 1,614 1,144
MSE 37306 50471 41720 27337 29782 38291 63900 24910 3
MAPE 1,230 1,386 1,460 1,157 1,176 1,099 1,623 1,160
The best results were obtained by fuzzy GMDH with quadratic partial de-
scriptions with bell-shaped membership functions and 50% training sample size.
The worst results were obtained with Fourier polynomial as partial descriptions.
The use of Chebyshev’s polynomial as PD in classical GMDH has shown the best
results. In fuzzy GMDH the most accurate estimates were obtained with linear
and quadratic PD.
FGMDH MODEL WITH FUZZY INPUT DATA
FGMDH Model Construction with Fuzzy Inputs
Let us consider the generalization of fuzzy GMDH for case when input data are
also fuzzy. Then a linear interval regression model takes the following form [6, 8]:
nnZAZAZAY +++= ...1100 ,
where iA is a fuzzy number of triangular shape with parameters ),,( iiii AAAA
(
= ,
where iA
(
is a center of the interval, iA – its upper border, and iA – lower border.
Consider the case of symmetrical membership function for parameters Ai, so
they can be described by the pair of parameters ( iA
(
, ic ), ic – interval width, ic ≥
0: iiiiii cAAcAA +=−=
((
, .
Let iZ be also a fuzzy number of triangular shape defined as ),,( iii ZZZ
(
,
iZ is a lower border, iZ
(
is a center, and iZ is an upper border of the fuzzy
number.
Then Y is a fuzzy number which parameters are defined as follows:
• the center of the interval ii Zay
(( ∑= ;
• the deviation in the left part of the membership function:
∑ +−=− ))(( iiiii ZcZZayy
((( ;
Yuriy Zaychenko, Helen Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2019, № 1 106
• the lower border of the interval
∑ −= )( iiii ZcZay
(
;
• the deviation in the right part of the membership function:
∑ ∑ +−=+−=− iiiiiiiiiii ZcZaZaZcZZayy
((((( ))(( ;
• the upper border of the interval
∑ += )( iiii ZcZay
(
.
For the interval model to be correct, the real value of input variable Y should
lay in the interval obtained by the method FGMDH.
So, the general requirements to a linear interval model are the following: to
find such values of parameters ( iA
(
, ic ) of fuzzy coefficients, which ensure [6, 8]:
1) observed values ky should locate in an estimation interval for kY total
width of the estimation interval should be minimal;
2) these requirements may be redefined as a task of linear programming [6, 8]:
∑ ∑∑
=
−−+
M
k
iiiiiiii
ca
ZcZaZcZa
ii 1,
))()((min
((
,
under conditions
⎪⎩
⎪
⎨
⎧
=≥+
≤−
∑
∑
.,1,)(
,)(
MkyZcZa
yZcZa
kikikii
kikiiki
(
(
Let us consider partial description (1). Then math model takes the form [6]:
∑ ∑ ∑∑
= = ==
++−++−+
M
k
M
k
M
k
jkjkjkik
M
k
ikik
ca
xcxxaxcxxaMc
ii 1 1 1
221
1
10
,
2)(2)(2(min ((
∑∑∑
===
+−++−+−+
M
k
ikikik
M
k
jkik
M
k
ikikjkjkjkik xxxaxxcxxxxxxa
1
4
1
3
1
3 )(22))()(( (((((
)2)(22
1 1
2
55
1
2
4 ∑ ∑∑
= ==
+−++
M
k
M
k
jkjkjkjk
M
k
ik xcxxxaxc ((( ,
with the following constraints
++−−−−+++ ))()((3210 jkikikikjkjkjkikjkik xxxxxxxxaxaxaa ((((((
−+−++−−+ ))(2())(2( 2
5
2
4 jkjkjkjkikikikik xxxxaxxxxa ((((((
kjkikjkikjkik yxcxcxxcxcxcc ≤−−−−−− 2
5
2
43210
(((((( ;
+−−+−+++ ))()((3210 jkikikikjkjkjkikjkik xxxxxxxxaxaxaa ((((((
++−−+−−+ 0
2
5
2
4 ))(2())(2( cxxxxaxxxxa jkjkjkjkikikikik
((((((
kjkikjkikjkik yxcxcxxcxcxc ≥+++++ 2
5
2
4321
(((((( , 5,0,0 =≥ lcl .
Fuzzy GMDH and its application to forecasting financial processes
Системні дослідження та інформаційні технології, 2019, № 1 107
As one can see, this is also the linear programming problem but there are
still no constraints for non-negativity of variables ia , so it is reasonable to pass to
a dual problem introducing dual variables { }kδ and { }Mk+δ .
Investigations of FGMDH with Fuzzy Inputs at Forecasting Problems
The list of securities used for calculation of RTS index consists of the most liquid
shares of Russian companies chosen by Information Committee and based on ex-
pert judgment. The number of securities may not exceed 50.
The experiment contains 5 fuzzy input variables which are the stock prices
of leading Russian energetic companies included into the list of RTS:
• index LKOH – shares of “Lukoil” joint-stock company;
• EESR – shares of “RAO UES of Russia” joint-stock company;
• YUKO – shares of “Yukos” joint-stock company;
• SNGSP – privileged shares of “Surgutneftegas” joint-stock company;
• SNGS – common shares of “Surgutneftegas” joint-stock company;
• Output variable is the value of RTS index (opening price) of the same pe-
riod (03.04.2006 – 18.05.2006).
The whole sample contains 32 instances (points) and training sample size is
18 points (optimal size of the training sample for current experiment). The results
presented below were obtained in [8].
Experiment 1. For normalized input when using Gaussian MF in group
method of data handling with fuzzy input data the results of experiment are pre-
sented (see Fig. 3) and table 5: for GMDH, MSE = 0,1129737, for FGMDH,
MSE = 0,0536556.
T a b l e 5 . MSE comparison for different methods at experiment 1
Error GMDH FGMDH
FGMDH with fuzzy
inputs,
Triangular MF
FGMDH with fuzzy
inputs,
Gaussian MF
MSE 0,1129737 0,0536556 0,055557 0,028013
As the results of experiment 1 show, fuzzy group method of data handling
with fuzzy input data gives more accurate forecast than GMDH and FGMDH. In
-0,2000
0,0000
0,2000
0,4000
0,6000
0,8000
1,0000
1,2000
1,4000
1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132
Y
Estimation by
distinct GMDH
Estimation by
f uzzy GMDH
Upper border by
f uzzy GMDH
Lower border by
f uzzy GMDH
Fig. 3. Results of the Experiment1 using GMDH and FGMDH with fuzzy inputs
Yuriy Zaychenko, Helen Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2019, № 1 108
case of triangular MF FGMDH with fuzzy inputs gives a little worse forecast than
FGMDH with Gaussian MF.
Experiment 2. RTS-2 index forecasting (opening price). The whole sample
size is 32 instances and the training sample size is 18 ones. For normalized input
data using Gaussian MF in FGMDH with fuzzy input data the following experi-
mental results were obtained presented in the table 6 [8]: MSE for GMDH =
0,051121, MSE for FGMDH = 0,063035.
As the results of the experiment 2 show (Table 6), fuzzy group method of
data handling with fuzzy input data gives better result than GMDH and FGMDH
in case of both Gaussian and triangular membership functions.
T a b l e 6 . Comparison of different methods at experiment 2
Error GMDH FGMDH
FGMDH with fuzzy
inputs,
Triangular MF
FGMDH with fuzzy
inputs,
Gaussian MF
MSE 0,051121 0,063035 0,061787 0,033097
CONCLUSION
In this paper, the review of main results dealing with fuzzy inductive modeling
method FGMDH is presented. This method enables to construct models of com-
plex processes using experimental (statistical) data. Two different FGMDH ver-
sions were presented and discussed: with crisp inputs and fuzzy inputs.
The advantage of fuzzy GMDH is that it does not use least square method
for search of unknown model coefficients opposite to classical GMDH and there-
fore the problem of possible ill-conditioned matrix does not exist for it.
Besides, fuzzy GMDH enables to find not point-wise forecast estimates but
interval estimates for forecast values which allow to determine the forecast accu-
racy.
The generalization of fuzzy GMDH with fuzzy inputs was also considered
and analyzed. The experimental investigations of GMDH and fuzzy GMDH in
problems of share prices forecast at NYSE and Russian stock market RTS were
carried out. The comparative results analysis has confirmed the high accuracy of
fuzzy GMDH in problems of forecasting in financial sphere.
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Received 29.01.2019
From the Editorial Board: the article corresponds completely to submitted manuscript.
|
| id | journaliasakpiua-article-168415 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:25:08Z |
| publishDate | 2019 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/50/a05d7f497271ca02d9b94d50376cb850.pdf |
| spelling | journaliasakpiua-article-1684152019-08-07T15:26:27Z Fuzzy GMDH and its application to forecasting financial processes Нечеткий МГУА и его прнменение для прогнозирования финансовых процессов Нечіткий МГУА та його застосування для прогнозування фінансових процесів Zaychenko, Yuriy Zaychenko, Helen нечіткий МГУА функції приналежності адаптація моделей прогнозування нечеткий МГУА функции принадлежности адаптация моделей прогнозирование fuzzy GMDH membership functions models adaptation forecasting This paper is devoted to the investigation and application of the fuzzy inductive modeling method known as Group Method of Data Handling (GMDH) in problems of Data Mining, in particularly its application to solving the forecasting tasks in financial sphere. The advantage of the inductive modeling method GMDH is a possibility of constructing an adequate model directly in the process of algorithm run. The generalization of GMDH in case of uncertainty — a new method fuzzy GMDH is described which enables to construct fuzzy models almost automatically. The algorithm of fuzzy GMDH is considered. Fuzzy GMDH with Gaussian and bell-wise membership functions MF are considered and their similarity with triangular MF is shown. Fuzzy GMDH with different partial descriptions orthogonal polynomials of Chebyshev and Fourier are considered. The problem of adaptation of fuzzy models obtained by FGMDH is considered and the corresponding adaptation algorithm is described. The extension and generalization of fuzzy GMDH in case of fuzzy inputs is considered and its properties are analyzed. The experimental investigations of the suggested FGMDH were carried out. Посвящено исследованиям и применению нечеткого метода индуктивного моделирования известного как нечеткий метод группового учета аргументов (МГУА) в проблемах интеллектуального анализа данных, в частности прогнозирования в финансовой сфере. Преимуществом индуктивного метода моделирования МГУА является возможность конструирования адекватной модели непосредственно в процессе работы алгоритма. Описано обобщение МГУА на случай неопределенности — нечеткий МГУА, позволяющий конструировать нечеткие модели почти автоматически. Рассмотрены алгоритмы нечеткого МГУА для гауссовских и колоколообразных функций принадлежности и показано их сходство с моделью для треугольных функций принадлежности. Приведены варианты НМГУА для ортогональных полиномов Чебышева и Фурье. Рассмотрена проблема адаптации нечетких моделей, полученных по НМГУА, и описан соответствующий алгоритм адаптации. Приведено обобщение нечеткого МГУА на случай нечетких входных переменных. Выполнены экспериментальные исследования НМГУА и приведены их результаты. Присвячено дослідженням та застосуванню нечіткого методу індуктивного моделювання, відомого як нечіткий метод групового урахування аргументів (НМГУА) у проблемах інтелектуального аналізу даних, зокрема прогнозування у фінансовій сфері. Перевагою індуктивного методу моделювання МГУА є можливість конструювання адекватної моделі процесу безпосередньо в процесі роботи алгоритму. Описано узагальнення МГУА на випадок невизначеності — нечіткий МГУА, який дозволяє конструювати нечіткі моделі майже автоматично. Розглянуто алгортми нечіткого МГУА для гаусівських та дзвіноподібних функцій належності і показано їх подібність до моделей з трикутними функціями належності. Наведено варіанти НМГУА для ортогональних поліномів Чебишова та Фур’є. Розглянуто проблему адаптації нечітких моделей, отриманих за допомогою НМГУА, та описано відповідний алгоритм адаптації. Наведено узагальнення нечіткого МГУА на випадок нечітких вхідних змінних. Проведено експериментальні дослідження НМГУА та наведено їх результати. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2019-03-25 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/168415 10.20535/SRIT.2308-8893.2019.1.07 System research and information technologies; No. 1 (2019); 91-109 Системные исследования и информационные технологии; № 1 (2019); 91-109 Системні дослідження та інформаційні технології; № 1 (2019); 91-109 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/168415/168217 Copyright (c) 2021 System research and information technologies |
| spellingShingle | нечіткий МГУА функції приналежності адаптація моделей прогнозування Zaychenko, Yuriy Zaychenko, Helen Нечіткий МГУА та його застосування для прогнозування фінансових процесів |
| title | Нечіткий МГУА та його застосування для прогнозування фінансових процесів |
| title_alt | Fuzzy GMDH and its application to forecasting financial processes Нечеткий МГУА и его прнменение для прогнозирования финансовых процессов |
| title_full | Нечіткий МГУА та його застосування для прогнозування фінансових процесів |
| title_fullStr | Нечіткий МГУА та його застосування для прогнозування фінансових процесів |
| title_full_unstemmed | Нечіткий МГУА та його застосування для прогнозування фінансових процесів |
| title_short | Нечіткий МГУА та його застосування для прогнозування фінансових процесів |
| title_sort | нечіткий мгуа та його застосування для прогнозування фінансових процесів |
| topic | нечіткий МГУА функції приналежності адаптація моделей прогнозування |
| topic_facet | нечіткий МГУА функції приналежності адаптація моделей прогнозування нечеткий МГУА функции принадлежности адаптация моделей прогнозирование fuzzy GMDH membership functions models adaptation forecasting |
| url | https://journal.iasa.kpi.ua/article/view/168415 |
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