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Identification and Forecasting in Management System by Using the GMDH Method in the Case of the Automobile Industry in Japan
Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Kyoto, Japan , 1981
IDENTIFICATION AND FORECASTING IN MANAGEMENT SYSTEM BY USING THE GMDH METHOD IN THE CASE OF THE AUTOMOBILE INDUSTRY IN JAPAN T. Nishikawa and S. Shimizu Department of Management Engineen'ng, Metropolitan College of Technology, Tokyo 6-6, Asahigaoka, Hino-shi, Tokyo , Japan Abstract . This paper outlines the use of linear and nonlinear models in forecasting net car sales of the Japanese automotive industry. In this method, variables representing complex social phenomena can be included in the mathematical models and can be derived by using actual time series data with no additional mathematical complexity. In this paper, we show that it is possible: to derive four satisfactory model structures from a single set of time series data; to estimate coefficients of the structures; and to make accurate forecasts. Reports on the theory and application of the GMDH method have been published previously, but few of them have actually applied the method to management systems . Keywords. GMDH; mathematical modelling; system identification; management forecasting; heuristic filter; multicolinearity; state space model. INTRODUCTION
THE STRUCTURE AND ALGORITHM OF THE GMDH
Several mathematical models such as multiple autoregression, moving-average and Kalmanfilter have been proposed and used in forecasting various management indicators. Their contribution toward accurate forecasting, however, have been somewhat limited. On the other hand, the GMDH method, to which we applied the heuristic filter, make it possible to include the influence of social phenomena on different models. We have selected two examples of the Japanese automotive industry to illustrate the procedures for identifying mathematical models to be used in forecasting net car sales. By applying the GMDH method and including the influence of social phenomena, we can obtain valid forecasts from the models.
The GMDH (Ivakhnenko, 1970) algorithm is an identification and forecasting method based on the principles of heuristic self-organization which was realized by Rosenbalates perception. It was developed to solve complex systems and social phenomena with large dimensions which are difficult to formulate when the data sequence is very short or the relationships between the data have the characteristics of multicolinearity. The structure of GMDH is shown in Fig. 1 and a computational flow of the GMDH method is shown in Fi ~ . 2. As shovm in Fig. 1, the G~IDli is one control technique which oakes it
The Rl'IH:"r a to fs of combinations Id entificat ion of the IJa rtial descriIJtions
2,111 2:H I
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:
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Bl oc k diagr am Ilf thl' ( ;\11)11
713
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- ---I·I
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714
T. Nishikawa and S. Shimizu
sponds to the set Decision of
{xl,n xl,n-l x l ,n_2 ... xl,n-T+l
system output variables the data of input variables descriptions variables layer
no
yes
Use the combinations adopted as the inter mediate variables
Select the complete description from the partial descriptions
x s,n x s,n- 1 x s,n- 2 ... x s,n- T+ I} in that order. Input variables to the 2nd selection layer in the subsequent layer Z( ~ ), 2>2, are the self-selected intermediate variables (expressed by partial descriptions) in the previous layer. And then the m-th input variables to the ( 2+l)-th selection layer, Zm(2+l) can be expressed by the m-th intermediate variables in the £-th selection layer. We divide the actual time series data into two parts; the training sequence and the checking sequence, after using a normalized count of deviations from the mean. In this paper, we used odd for the training sequence and even for the checking sequence. Partial Description In this paper, partial descriptions are identified according to basic functions as given below:
Fig. 2
GMDH Algorithm
possible to combine the variables in the selection processes ana according to the principles of self-organization, and to select the best combination to minimize the heuristic criterion in multilayer selection structure. Hence, we have determined that the basicfunction of the GMDH method includes dividing the time series data after considering the length of previous history(prehistory), determingthe intermediate variables to be introduced, and determining the heuristic criterion. The purpose is essentially to minimize mean square error criterion about using for criterion to select the intermediate variables.
Input Variable Input variables to the 1st selection layer, Z(2),2=1, as shown Fig. 1, are determined according to system variables x ( for h=1,2, .. , h s ) and prehistory T. The relationship ( Nishikawa and Shimizu, 1979 ) between the system variables and the m-th input variables, Z (1), at the time point n can be written m,n as: (1) Zm,n(l) = Xh,n_g m=1,2, .. ,p (P=S'T)' g=1,2"',(T-l), where, xh,n denotes x at the time point n. The numh ber "m" of the input variables is obtained from the following relation: m = (h-l)'T + (g+l) It follows from Eq. (1), that Z (1) correm,n
y
a
O + alx i + a 2x j
y
a
O
(2)
+ alx + a x + a x x i 2 j 3 i j
And the coefficients are for r=O,1,2,3
(3)
are
determined by the training sequence, applying the method of least squares. The m-th partial descriptions at the £-th selection layer can be written from Eqs. (2) and (3) as: F (Z . (£), Z.( 2)) = f ( £) m
= a
m
J
~
o+a l
O,m, x.,
oz . ( £)+a
,rn, x..
1
oZ , ( £) 2 ,m, x., J
F (Z.( £), Z.( £)) = f ( 2) m ~ J m o+a oZ. ( £)+a 2 oZ, ( £) a O,m, x.. l ,m, x.. 1 ,ro, x.. J +a o Z.( 2 )Z . ( ~ ) 3 ,m, '" ~ J
(4)
(5)
where, when £=1, i,j=1,2, .. ,p, i~j, m=1,2, .. pC , when £ ~2, i,j=1,2, .. ,k, i~j, k: the num2 ber of intermediate variable, m=1,2""kC2' The values obtained by substituting actual time series data of Z. ( 2) and Z.( 2) into Eqs. (4) and (5) are the v~lues of ttle m-th intermediate variables in the £-th selection layer. They then become the m-th input variables to the ( £+l)-th selection layer. The complete description of a given system must be replaced by the above mentioned partial description. Selection Criteria of the Intermediate Variables The criteria are used to select combination of variables to minimize the value(£) of the selection criterion, to minimize fitting error, and to identify a more accurate fore-
Identifi cation and Forecasting in Management System
casting model. A r e sultant partial description including this combinatoin of variables represents the optimal description in this selection layer. The number of intermediate variables selected in each selection layer are determined in accordance with the characteristics of the given system, and are selected in ascending order of (£). For our purpose we selected 10 intermediate variables from the 1st selection layer and 5 from the 2nd subseque nt selection layers. We dealt with the following three criteria.
715
partial description in the £-th selection layer and is identified by using a checking sequence of data of Z. ( £) and Z .( £). 1
J
Structure stability criterion. Structure stability criterion (Endo, 1978) is defined as: N' £ =[ l:{y(c)_F(t)(Z~c)( £ ) ,z~c)( £ »}2 3,m T=l T m, £ 1,T J,T N" + l:{y(t)_F(c)(Z~t)( £ ), z~t)( £ »}2 ( N'+N" ) T=l T m, £ 1,T J,T
(8) RMS criterion. RMS criterion (Ivakhnenko, 1972) is defined as:
where, yic)and yit)are the T-th checking se-
Z~C)( £ »}2 ] ,T
(6)
Here, we wish to minimi ze £1 by using a checking sequence from the actual data and then the of £1 becomes a threhold. where, £=1 for i, ~=:, 2 , .. ,p (i#~) . p=s·, . and m=1, 2 , .. ,pC . ££,2, 2 1,J=1, 2 , .. ,k (1#J), k 1S the number of the inter~ediate variables (to be input(~rr~ables at th1s layer) and m=1, 2 " " kC2 ' YT 1S the T-th checking data,T denotes an oider of time s e ri e s and F(t)(.) is the m-th partial desription in mtfie £-th selection layer whose coefficients a n (r=0,1,2,3) are determined r,m, x' by the training sequence of these layers, and subsript (t) denotes the training sequence.
F(t ~ (Z~cT)( £)' Z J, ~cT)( £ » m, )(" 1,
Thus,
is obtained by sustituting .
(t) n(·)· m, x"
Selection
of the Selection Layer.
Now, le~s consider the process yeilding the partial descriptions in the subsequent selection layers. when this proce ss is repeated until the value £ (a =1, 2 ,3) is the threshold which satisfie~ Eq. (9) in any £-th layer that is let through to an y ( £+l)-th layer satisf ying Eq. (9) given below: min{ £ ( £)} f min{ £ ( £+l)} m a,m m a,m
(9)
where, £ (£) and £ ( £+1) are the thresa,m a,m hold of select i on criteria in the £-th and (£+1) -th layers.
represents the
m-th partial description in the £-th layer and is given form the polynomials (Eqs. (4) and (5», the value of intermediate variables 1nto F
quence and training sequence of data respectively, the other symbols are the same as those used in the Unbiasedness criterion.
Z~c~( £ ) andz~ci( £ )
(c)
t~)
~,
J,
],
Determination of the Complete Description To determine a complet e description, we define the absolute error criterion (Nishikawa and Shimizu, 1979) as:
Z. T( £) and Z. T( £) are the
values of the T-th checking sequence in the ~-th layer, and N' is the number of the checking sequence of data. Unbiasedness criterion. Unbiasedness criterion (Ivakhnenko and Koval'chuk, 1972; Nishikawa and Shimi zu, 1979)is defined as:
N
a = min{ l:lYT-F (Z. T(L),Z. T(L)I} m m T= m 1, J, (10) where, L are the last selection layer, YT is the actual data at time point T of the output variables. And then we decide the complete description F (.) that satisfies Eq. (10) from the part~al descriptions selected in L. FORECASTING MODELS
Here, we wish to minimize £2 in obtaining the unbiasedness criterion. £2 is calculated according to the entire sequence and the minimum value of £2 is the threshold. The subscripts i,j,and m are the same as the those for the RMS criterion.
In Eq. (7),
F(t~(.) is the m, "-
same as the RMS criterion Eq.(6). The resultant partial description has been obtained as
F(t~(Z.
T( £)' ZJ T( £» and is given in the m, " 1, , ( ) form of the poly nomial. F c n ( · ) is them-th m, "-
The four forecasting models used to forecast the net car sales of the Japanese automotive industry are discussed below. Model-I This model is a nonlinear model applied as the basic function in Eq. (3). Here the relation between the input and output variables at time point n in data time sequence is defined as follows: Y =aO+alx. +a x . +a x. ·X. n 1,n 2 J,n 3 1,n J,n
(11)
T. Nishikawa and S. Shimizu
716
B
Model-1I This is a linear model applying the foward difference of the output varibles to Eq. (2) as the basic function. The relation between the input and output variables is as follows: (12)
Yn+ l-Y n = aO+alx.~,n +a 2 ,n x.],n
The forecast at dicrete time point n+l is shown by: (13)
Y+ = f (L)+Y m,n n n l where,L is the last selection layer. Model-Ill
This is a nonlinear model which also applies the forward difference to Eq. (3) as the basic function. The relation between inputoutput variables is: . +a x.~,n ·X.],n Yn+ l-Y n =aO+alx.~,n+a 2x ],n 3 (14) Model-IV This is represented by the space state model applying Y =alx . +a x . as the basic func2 ],n n ~,n tion as follows:
xn+ l=A nXn+ BnUn+ Wnwn
(15)
This is called the discrete state equation. Where, X is the s·, dimensional column vector of the object, U is the r dimensional control vector,and w is the v·, dimensional disturbance vector. X =[XT XT n n n-l where,
T
X =[x
l,n
-T -T
wn =[wn wn- 1 where,
-T
w =[w l,n
U =[u u . l,n 2,n n
x
..
2,n
x
s,n
)
••• w
2,n
v,n
u
r,n
)
)T
where, T denotes transpose. A, Band Ware system matrix (Ivakhnenko,Tolkhyanenko and yaremenko, 1974) of dimension s,xs" control matrix sX, and disturbance matrix s,XVT, respectively. A n I An = 0 0 0
n n W -[:"
0
W n-l 0
W n-T+l 0
0
0
where, An-HI ( .Q.;~l) is a matrix of dimension sxs, I is an identity matrix s xs, and 0 is zero matrix s Xs. Application of the GMDH method to the state space model is significant in that the value of putting the method to practical use becomes evident, because the stste variables are self-selected in the identification process by the GMDH and parameters are estimated simultaneously at discrete time point n. Moreover valid forecasts can be realized, because state variables can be obtained by filtering the appropriate heuristics in the computation processes. DETERMINATION OF SYSTEM VARIABLES In this paper, we deal with the net car sales forecasting models in the automotive industry as an object of modelling in the management system. In order to construct valid forecasting models, we used the following set of the variables: xl' annual net car sales in the automobile industry; x ' number of regis2 tered cars annually; x ' number of exported 3 cars annually; x ' GNP(real); x ' annual in4 s come per household; x ' annual savings per 6 household. Hence, the available models from which to select the best forecasting models based on the GMDH depends upon a given set of variables. RESULTS OF IDENTIFICATION AND THE FORECASTS Identification and Forecasts of Models-I, 11, III and IV
-T wn-,+l ) T
w
B = 0 n
A n-l 0 I 0
A n-,+2 0 0 0
An_,+l 0 0 0
0
I 0
0
We simultaneously identified the mathematical model and calculated the forecast. The detailed computational results are summarized in Table 1. As before, the four forecasting models are evaluated and selected by three selection criteria, based on the intermediate variables, and by the absolute error criterion respectively, which we classify by using (0.1), (1.1) and (2,1). The number "I" appearing in the 2nd position denotes the absolute error criterion. We showed the forecasting models obtained by using selection criteria (0.1) in the case of example A as given below. Model-I. The 1st selection became the last selection layer, and then we obtained the following complete description y y = -0.43s23Z
ll
(1)+1.298l6Z
l6
(1)
~n
Identifi ca tion and Forec asting
-0. 885752
(1) 2
11
16
(1)+0 . 17510
Management Syst em
7 17
y = 0 . 225 9424 (3)+0. 84413 2 ( 3)+0.00479 5 2nd s e lection layer
Mode1-II. Both the complete de scription and the partial descriptions are given as : The complete descriptio n y is accomplished by the 2nd partial de script i on at the 3rd s e l e ction laye r. TABLE 1
2 4 (3)=0. 24195 2 2 ( 2 )+0.78385 2 ( 2)+0.0017 6 4 25 (3)=0 . 46039 2 2 ( 2)+0 . 74883 2 ( 2) +0 . 014 29 6
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718
T. Nishikawa and S. Shimizu
CONCLUSION
1st selection layer Z2 (2)=-13.42732Z (1)+15.31448Z (1) 1 16 +0 . 24960 Z4(2)=-4 . 33879Z (1)+4.36569Z (1)-0.61313 2 13 Z6(2)=-23.47082Z (1)+22.06358Z 12 (1) 11 +1.15765 Model-Ill . The 1st selection layer became the last layer, and then the complete description of y is as follows: y =• -9.99913Z (1)+11.70162Z (1) 1 16 -1.62931Z (1)Z16(1)+O.42271 1 Model-IV . The results of identification of the forecasts are shown in Eq. (16) and Table 2. In this model as before, the system variables to be adopted are such that Xl' x 2 ' x 3 ' x 4 and x5 correspond to the variables of forecasting models-I,ll, Ill, ,=3 and w is 1 the disturbance which is year to year increase rate of net car sales.
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-
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(16) TABLE 2
Results in the Analysis of the State Space Model Data
A
Con t ent s
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Moreover, we can obtain good results ( Nishikawa and Shimizu, 1980) by applying forecasting methods so far to the GMDH for forecasting economic and management indicators. REFERENCES
,I
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The advantages of applying models to a complex system depend on the simulation we can make of the system and of the heuristic criteria of the decision-maker by a process no additional mathematical complexity, as a result of which it is useful in clarifying and validating the forecasts made for the system In forecasting, only variables with a strong influence on the forecasts are selected to b, applied in this model, which function as a heuristic filter. In particular, the state space model provides a solution to the nonstationary process acting as a stationary process. Finally, we may say that there are many significant results obtainable from applying the GMDH method. They include obtain ing accurate forecasts and identifying model
X I ,I/ I,
I.
X~. I/
:: .
~ - !::;-'" ----
Endo, A. (1978). Identification of non1inear system by the modified GMDH. J. SIP (in Japanese), 14[2], 24-29. Ivakhnenko, A. G. (1970). Heuristic Se1fOrganization in Problems of Engineering Cybernetics. Automatica, Vol. 6, 207-21 Ivakhnenko, A.G., and P.I. Kova1'chuk (1972) Unique Construction of Regression Curve Using a Small Number of Points. Soviet Autom. Control, Vol. 5, 26-32. Ivakhnenko, A.G., V.A. To10khyanenko and A. G. Yaremenko (1974). Control with Pree iction Optimization Using Discrete Line ar Prediction Models of the Plant. Soviet Autom. Control, Vol. 7, 22-30. Nishikawa, T., and S. Shimizu (1978). The Effectiveness of the GMDH by the Combination of Selection Criterion (in Japanese). MEMOIRS of Metro. Col. of Tech. Tokyo, Vol. 7, 43-52. Nishikawa, T., and S. Shimizu (1981). On th, Effectiveness of Application Ridge Esti· mator to the GMDH Algorithm (in Japanes, J. of JSSE, Vol. 5[2], 36-44.
IDENTIFICATION AND FORECASTING IN MANAGEMENT SYSTEM BY USING THE GMDH METHOD IN THE CASE OF THE AUTOMOBILE INDUSTRY IN JAPAN T. Nishikawa and S. Shimizu Department of Management Engineen'ng, Metropolitan College of Technology, Tokyo 6-6, Asahigaoka, Hino-shi, Tokyo , Japan Abstract . This paper outlines the use of linear and nonlinear models in forecasting net car sales of the Japanese automotive industry. In this method, variables representing complex social phenomena can be included in the mathematical models and can be derived by using actual time series data with no additional mathematical complexity. In this paper, we show that it is possible: to derive four satisfactory model structures from a single set of time series data; to estimate coefficients of the structures; and to make accurate forecasts. Reports on the theory and application of the GMDH method have been published previously, but few of them have actually applied the method to management systems . Keywords. GMDH; mathematical modelling; system identification; management forecasting; heuristic filter; multicolinearity; state space model. INTRODUCTION
THE STRUCTURE AND ALGORITHM OF THE GMDH
Several mathematical models such as multiple autoregression, moving-average and Kalmanfilter have been proposed and used in forecasting various management indicators. Their contribution toward accurate forecasting, however, have been somewhat limited. On the other hand, the GMDH method, to which we applied the heuristic filter, make it possible to include the influence of social phenomena on different models. We have selected two examples of the Japanese automotive industry to illustrate the procedures for identifying mathematical models to be used in forecasting net car sales. By applying the GMDH method and including the influence of social phenomena, we can obtain valid forecasts from the models.
The GMDH (Ivakhnenko, 1970) algorithm is an identification and forecasting method based on the principles of heuristic self-organization which was realized by Rosenbalates perception. It was developed to solve complex systems and social phenomena with large dimensions which are difficult to formulate when the data sequence is very short or the relationships between the data have the characteristics of multicolinearity. The structure of GMDH is shown in Fig. 1 and a computational flow of the GMDH method is shown in Fi ~ . 2. As shovm in Fig. 1, the G~IDli is one control technique which oakes it
The Rl'IH:"r a to fs of combinations Id entificat ion of the IJa rtial descriIJtions
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714
T. Nishikawa and S. Shimizu
sponds to the set Decision of
{xl,n xl,n-l x l ,n_2 ... xl,n-T+l
system output variables the data of input variables descriptions variables layer
no
yes
Use the combinations adopted as the inter mediate variables
Select the complete description from the partial descriptions
x s,n x s,n- 1 x s,n- 2 ... x s,n- T+ I} in that order. Input variables to the 2nd selection layer in the subsequent layer Z( ~ ), 2>2, are the self-selected intermediate variables (expressed by partial descriptions) in the previous layer. And then the m-th input variables to the ( 2+l)-th selection layer, Zm(2+l) can be expressed by the m-th intermediate variables in the £-th selection layer. We divide the actual time series data into two parts; the training sequence and the checking sequence, after using a normalized count of deviations from the mean. In this paper, we used odd for the training sequence and even for the checking sequence. Partial Description In this paper, partial descriptions are identified according to basic functions as given below:
Fig. 2
GMDH Algorithm
possible to combine the variables in the selection processes ana according to the principles of self-organization, and to select the best combination to minimize the heuristic criterion in multilayer selection structure. Hence, we have determined that the basicfunction of the GMDH method includes dividing the time series data after considering the length of previous history(prehistory), determingthe intermediate variables to be introduced, and determining the heuristic criterion. The purpose is essentially to minimize mean square error criterion about using for criterion to select the intermediate variables.
Input Variable Input variables to the 1st selection layer, Z(2),2=1, as shown Fig. 1, are determined according to system variables x ( for h=1,2, .. , h s ) and prehistory T. The relationship ( Nishikawa and Shimizu, 1979 ) between the system variables and the m-th input variables, Z (1), at the time point n can be written m,n as: (1) Zm,n(l) = Xh,n_g m=1,2, .. ,p (P=S'T)' g=1,2"',(T-l), where, xh,n denotes x at the time point n. The numh ber "m" of the input variables is obtained from the following relation: m = (h-l)'T + (g+l) It follows from Eq. (1), that Z (1) correm,n
y
a
O + alx i + a 2x j
y
a
O
(2)
+ alx + a x + a x x i 2 j 3 i j
And the coefficients are for r=O,1,2,3
(3)
are
determined by the training sequence, applying the method of least squares. The m-th partial descriptions at the £-th selection layer can be written from Eqs. (2) and (3) as: F (Z . (£), Z.( 2)) = f ( £) m
= a
m
J
~
o+a l
O,m, x.,
oz . ( £)+a
,rn, x..
1
oZ , ( £) 2 ,m, x., J
F (Z.( £), Z.( £)) = f ( 2) m ~ J m o+a oZ. ( £)+a 2 oZ, ( £) a O,m, x.. l ,m, x.. 1 ,ro, x.. J +a o Z.( 2 )Z . ( ~ ) 3 ,m, '" ~ J
(4)
(5)
where, when £=1, i,j=1,2, .. ,p, i~j, m=1,2, .. pC , when £ ~2, i,j=1,2, .. ,k, i~j, k: the num2 ber of intermediate variable, m=1,2""kC2' The values obtained by substituting actual time series data of Z. ( 2) and Z.( 2) into Eqs. (4) and (5) are the v~lues of ttle m-th intermediate variables in the £-th selection layer. They then become the m-th input variables to the ( £+l)-th selection layer. The complete description of a given system must be replaced by the above mentioned partial description. Selection Criteria of the Intermediate Variables The criteria are used to select combination of variables to minimize the value(£) of the selection criterion, to minimize fitting error, and to identify a more accurate fore-
Identifi cation and Forecasting in Management System
casting model. A r e sultant partial description including this combinatoin of variables represents the optimal description in this selection layer. The number of intermediate variables selected in each selection layer are determined in accordance with the characteristics of the given system, and are selected in ascending order of (£). For our purpose we selected 10 intermediate variables from the 1st selection layer and 5 from the 2nd subseque nt selection layers. We dealt with the following three criteria.
715
partial description in the £-th selection layer and is identified by using a checking sequence of data of Z. ( £) and Z .( £). 1
J
Structure stability criterion. Structure stability criterion (Endo, 1978) is defined as: N' £ =[ l:{y(c)_F(t)(Z~c)( £ ) ,z~c)( £ »}2 3,m T=l T m, £ 1,T J,T N" + l:{y(t)_F(c)(Z~t)( £ ), z~t)( £ »}2 ( N'+N" ) T=l T m, £ 1,T J,T
(8) RMS criterion. RMS criterion (Ivakhnenko, 1972) is defined as:
where, yic)and yit)are the T-th checking se-
Z~C)( £ »}2 ] ,T
(6)
Here, we wish to minimi ze £1 by using a checking sequence from the actual data and then the of £1 becomes a threhold. where, £=1 for i, ~=:, 2 , .. ,p (i#~) . p=s·, . and m=1, 2 , .. ,pC . ££,2, 2 1,J=1, 2 , .. ,k (1#J), k 1S the number of the inter~ediate variables (to be input(~rr~ables at th1s layer) and m=1, 2 " " kC2 ' YT 1S the T-th checking data,T denotes an oider of time s e ri e s and F(t)(.) is the m-th partial desription in mtfie £-th selection layer whose coefficients a n (r=0,1,2,3) are determined r,m, x' by the training sequence of these layers, and subsript (t) denotes the training sequence.
F(t ~ (Z~cT)( £)' Z J, ~cT)( £ » m, )(" 1,
Thus,
is obtained by sustituting .
(t) n(·)· m, x"
Selection
of the Selection Layer.
Now, le~s consider the process yeilding the partial descriptions in the subsequent selection layers. when this proce ss is repeated until the value £ (a =1, 2 ,3) is the threshold which satisfie~ Eq. (9) in any £-th layer that is let through to an y ( £+l)-th layer satisf ying Eq. (9) given below: min{ £ ( £)} f min{ £ ( £+l)} m a,m m a,m
(9)
where, £ (£) and £ ( £+1) are the thresa,m a,m hold of select i on criteria in the £-th and (£+1) -th layers.
represents the
m-th partial description in the £-th layer and is given form the polynomials (Eqs. (4) and (5», the value of intermediate variables 1nto F
quence and training sequence of data respectively, the other symbols are the same as those used in the Unbiasedness criterion.
Z~c~( £ ) andz~ci( £ )
(c)
t~)
~,
J,
],
Determination of the Complete Description To determine a complet e description, we define the absolute error criterion (Nishikawa and Shimizu, 1979) as:
Z. T( £) and Z. T( £) are the
values of the T-th checking sequence in the ~-th layer, and N' is the number of the checking sequence of data. Unbiasedness criterion. Unbiasedness criterion (Ivakhnenko and Koval'chuk, 1972; Nishikawa and Shimi zu, 1979)is defined as:
N
a = min{ l:lYT-F (Z. T(L),Z. T(L)I} m m T= m 1, J, (10) where, L are the last selection layer, YT is the actual data at time point T of the output variables. And then we decide the complete description F (.) that satisfies Eq. (10) from the part~al descriptions selected in L. FORECASTING MODELS
Here, we wish to minimize £2 in obtaining the unbiasedness criterion. £2 is calculated according to the entire sequence and the minimum value of £2 is the threshold. The subscripts i,j,and m are the same as the those for the RMS criterion.
In Eq. (7),
F(t~(.) is the m, "-
same as the RMS criterion Eq.(6). The resultant partial description has been obtained as
F(t~(Z.
T( £)' ZJ T( £» and is given in the m, " 1, , ( ) form of the poly nomial. F c n ( · ) is them-th m, "-
The four forecasting models used to forecast the net car sales of the Japanese automotive industry are discussed below. Model-I This model is a nonlinear model applied as the basic function in Eq. (3). Here the relation between the input and output variables at time point n in data time sequence is defined as follows: Y =aO+alx. +a x . +a x. ·X. n 1,n 2 J,n 3 1,n J,n
(11)
T. Nishikawa and S. Shimizu
716
B
Model-1I This is a linear model applying the foward difference of the output varibles to Eq. (2) as the basic function. The relation between the input and output variables is as follows: (12)
Yn+ l-Y n = aO+alx.~,n +a 2 ,n x.],n
The forecast at dicrete time point n+l is shown by: (13)
Y+ = f (L)+Y m,n n n l where,L is the last selection layer. Model-Ill
This is a nonlinear model which also applies the forward difference to Eq. (3) as the basic function. The relation between inputoutput variables is: . +a x.~,n ·X.],n Yn+ l-Y n =aO+alx.~,n+a 2x ],n 3 (14) Model-IV This is represented by the space state model applying Y =alx . +a x . as the basic func2 ],n n ~,n tion as follows:
xn+ l=A nXn+ BnUn+ Wnwn
(15)
This is called the discrete state equation. Where, X is the s·, dimensional column vector of the object, U is the r dimensional control vector,and w is the v·, dimensional disturbance vector. X =[XT XT n n n-l where,
T
X =[x
l,n
-T -T
wn =[wn wn- 1 where,
-T
w =[w l,n
U =[u u . l,n 2,n n
x
..
2,n
x
s,n
)
••• w
2,n
v,n
u
r,n
)
)T
where, T denotes transpose. A, Band Ware system matrix (Ivakhnenko,Tolkhyanenko and yaremenko, 1974) of dimension s,xs" control matrix sX, and disturbance matrix s,XVT, respectively. A n I An = 0 0 0
n n W -[:"
0
W n-l 0
W n-T+l 0
0
0
where, An-HI ( .Q.;~l) is a matrix of dimension sxs, I is an identity matrix s xs, and 0 is zero matrix s Xs. Application of the GMDH method to the state space model is significant in that the value of putting the method to practical use becomes evident, because the stste variables are self-selected in the identification process by the GMDH and parameters are estimated simultaneously at discrete time point n. Moreover valid forecasts can be realized, because state variables can be obtained by filtering the appropriate heuristics in the computation processes. DETERMINATION OF SYSTEM VARIABLES In this paper, we deal with the net car sales forecasting models in the automotive industry as an object of modelling in the management system. In order to construct valid forecasting models, we used the following set of the variables: xl' annual net car sales in the automobile industry; x ' number of regis2 tered cars annually; x ' number of exported 3 cars annually; x ' GNP(real); x ' annual in4 s come per household; x ' annual savings per 6 household. Hence, the available models from which to select the best forecasting models based on the GMDH depends upon a given set of variables. RESULTS OF IDENTIFICATION AND THE FORECASTS Identification and Forecasts of Models-I, 11, III and IV
-T wn-,+l ) T
w
B = 0 n
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I 0
0
We simultaneously identified the mathematical model and calculated the forecast. The detailed computational results are summarized in Table 1. As before, the four forecasting models are evaluated and selected by three selection criteria, based on the intermediate variables, and by the absolute error criterion respectively, which we classify by using (0.1), (1.1) and (2,1). The number "I" appearing in the 2nd position denotes the absolute error criterion. We showed the forecasting models obtained by using selection criteria (0.1) in the case of example A as given below. Model-I. The 1st selection became the last selection layer, and then we obtained the following complete description y y = -0.43s23Z
ll
(1)+1.298l6Z
l6
(1)
~n
Identifi ca tion and Forec asting
-0. 885752
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Mode1-II. Both the complete de scription and the partial descriptions are given as : The complete descriptio n y is accomplished by the 2nd partial de script i on at the 3rd s e l e ction laye r. TABLE 1
2 4 (3)=0. 24195 2 2 ( 2 )+0.78385 2 ( 2)+0.0017 6 4 25 (3)=0 . 46039 2 2 ( 2)+0 . 74883 2 ( 2) +0 . 014 29 6
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T. Nishikawa and S. Shimizu
CONCLUSION
1st selection layer Z2 (2)=-13.42732Z (1)+15.31448Z (1) 1 16 +0 . 24960 Z4(2)=-4 . 33879Z (1)+4.36569Z (1)-0.61313 2 13 Z6(2)=-23.47082Z (1)+22.06358Z 12 (1) 11 +1.15765 Model-Ill . The 1st selection layer became the last layer, and then the complete description of y is as follows: y =• -9.99913Z (1)+11.70162Z (1) 1 16 -1.62931Z (1)Z16(1)+O.42271 1 Model-IV . The results of identification of the forecasts are shown in Eq. (16) and Table 2. In this model as before, the system variables to be adopted are such that Xl' x 2 ' x 3 ' x 4 and x5 correspond to the variables of forecasting models-I,ll, Ill, ,=3 and w is 1 the disturbance which is year to year increase rate of net car sales.
0 40669 . 0.73 449 0
X :: . ' 1
.1"1,"
-
0.2i 402
X :: . 1/
~097591
.(:: . '1
.1":t , l/
0
.(,1. 11
I
1.0732 1
.(:. , 11
I
0
X :I . II
0
. .(:1 , 11
1.58330
.r" ,11
0
Xl. tI
0
.r1. 1!
1. 41 380 0
0.03114
0.94732
0.15789
I:; , II
I .r .-,.'I
0
I.r;."
0
J
I
.(:: , 11
0.53867
I
X :i , JI
I
X l , /I
I
0
- 0.593 14
Xl . 1I
Xl,l /
- 0 41 097 0.08670
I
,
'I (
0
006557
I
I·I
!
0.03296
0.34i2 4 000560
I
l
U) l . I I
(0\. 1/
I
(1) 1. ',
,
(16) TABLE 2
Results in the Analysis of the State Space Model Data
A
Con t ent s
\l inim uTll ind ex ( f :thre~ h~ l dl of t he c r iterio n in each ~ t't t' c [ i /)11
0. 644518 ' 10
\i ulr i pie correlat ill ll \ "arianCl' of a fitting error in in t e r p lI\at il lll
(19xol
,\ c t ua 1 data
6548. 45700 0 .2425388' 10 '
Il~" " i
Ac tual data - F \J rec a~t!" .
1
3 0 .366029 ' 10 .1 0 .563184 , 10 1
0 . 98782
c()dlicient~
F () r t'ca~ t ~
B
1 ' 0. 14 9237' 10 " I 0 . 76 35 00 - 10 I 2 0. 546526' 10 I ' 2 - O . 39-i35~
layer
0 . 2738868 ' 10 '
0. 99744 1913.0 5000 0 .2930591·1 0' -- ,._--- -- -, 0 . 3310181 . 10'
11. 446
1011
10.8 46
CI..f.U:.!al. data \ ' a r iablt' ~
\\"h ich incl uded in
t he foreca :-; t ing
.1' 1 , 11 . .( -, . 11
I.
X : . JJ
XI , I/ ,
XI , "
Moreover, we can obtain good results ( Nishikawa and Shimizu, 1980) by applying forecasting methods so far to the GMDH for forecasting economic and management indicators. REFERENCES
,I
~153831
0.347 18 0 - 004894 0
The advantages of applying models to a complex system depend on the simulation we can make of the system and of the heuristic criteria of the decision-maker by a process no additional mathematical complexity, as a result of which it is useful in clarifying and validating the forecasts made for the system In forecasting, only variables with a strong influence on the forecasts are selected to b, applied in this model, which function as a heuristic filter. In particular, the state space model provides a solution to the nonstationary process acting as a stationary process. Finally, we may say that there are many significant results obtainable from applying the GMDH method. They include obtain ing accurate forecasts and identifying model
X I ,I/ I,
I.
X~. I/
:: .
~ - !::;-'" ----
Endo, A. (1978). Identification of non1inear system by the modified GMDH. J. SIP (in Japanese), 14[2], 24-29. Ivakhnenko, A. G. (1970). Heuristic Se1fOrganization in Problems of Engineering Cybernetics. Automatica, Vol. 6, 207-21 Ivakhnenko, A.G., and P.I. Kova1'chuk (1972) Unique Construction of Regression Curve Using a Small Number of Points. Soviet Autom. Control, Vol. 5, 26-32. Ivakhnenko, A.G., V.A. To10khyanenko and A. G. Yaremenko (1974). Control with Pree iction Optimization Using Discrete Line ar Prediction Models of the Plant. Soviet Autom. Control, Vol. 7, 22-30. Nishikawa, T., and S. Shimizu (1978). The Effectiveness of the GMDH by the Combination of Selection Criterion (in Japanese). MEMOIRS of Metro. Col. of Tech. Tokyo, Vol. 7, 43-52. Nishikawa, T., and S. Shimizu (1981). On th, Effectiveness of Application Ridge Esti· mator to the GMDH Algorithm (in Japanes, J. of JSSE, Vol. 5[2], 36-44.