THE FUZZY GROUP METHOD OF DATA HANDLING WITH FUZZY INPUT VARIABLES

ZAYCHENKO Yu., professor 

Institute of Applied System Analysis, National Technical University of Ukraine “KPI”

03056, Kiev-56, Peremogy Avenue, 37, Ukraine

Abstract

The problem of forecasting models constructing using experimental data in terms of fuzziness, when input variables are not known exactly and determined as intervals of uncertainty is considered in this paper. The fuzzy group method of data handling is proposed to solve this problem. The theory of this method was suggested and researched in [1-7]. As it is well known, fuzzy GMDH allows to construct fuzzy models and has the following advantages:

1) the problem of optimal model finding is transformed to the problem of linear programming, which is always solvable;

2) the interval regression model is built as the result of method work ; 3) There is a possibility of adaptation of the obtained model.

In this report new fuzzy GMDH was developed for the cases when initial data are not known exactly and are given in a form of uncertainty intervals.

The mathematical model of the problem mentioned above has been built in this work and fuzzy GMDH with fuzzy inputs was elaborated and presented in the paper. The corresponding program, which uses the suggested algorithm, was developed. And the experimental investigations and comparison of the suggested FGMDH with classical GMDH and neural nets in problems of stock prices forecasting was carried out and the results are presented in this paper. Keywords: Group Method of Data Handling, fuzzy, economic indexes, stock prices, forecasting.

1. MATHEMATICAL MODEL OF GMDH WITH FUZZY INPUT DATA

Let’s consider a linear interval regression model: Y = A0Z0+A1Z1+…+AnZn , (1) where Ai are fuzzy numbers, which are described by threes of parameters ),,( iiii AAAA

, where iA

is an interval

center, iA is the upper border of the interval, iA - lower border of the interval,

and Zi are also fuzzy numbers, which are determined by parameters ),,( iii ZZZ

, iZ - lower border, iZ

- center, iZ -

upper border of a fuzzy number. Then Y is an output fuzzy number, which parameters are defined as follows (in accordance with L-R numbers multiplying formulas): Center of interval: ii ZAy

* ,

Deviation in the left part of the membership function: )*)()(*( iiiiii ZAAZZAyy

,

And the lower border of the interval )*)()(**( iiiiiiii ZAAZZAZAy

Deviation in the right part of the membership function: ))(*)(*( iiiiii AAZZZAyy

,

Similarly, the upper border of the interval )*)(*)(*( iiiiiiii ZAAAZZZAy

.

For the interval model to be correct, the real value of input variable Y need lay in the interval obtained by the method workflow. So, the general requirements to estimation linear interval model are the following: to find such values of parameters

),,( iii AAA

of fuzzy coefficients, which allow:

a) Observed values ky to lay in the estimation interval for kY ; b) Total width of estimation interval be minimal. Input data for this task is Zk=[ Zki]i - input training sample, and also yk is known output values, Mk ,1 , M – the number of observation points. There are two cases of fuzzy membership functions described in this work:

- Triangular membership functions - Gaussian membership functions.

Quadratic partial descriptions were chosen:

25

243210),( jijijiji xAxAxxAxAxAAxxf .

1.2. FGMDH with fuzzy input data for triangular membership functions

1.2.1. The form of math model for triangular MF Let’s consider the linear interval regression model: Y = A0Z0+A1Z1+…+AnZn , where Ai – fuzzy number of triangular form, which is described by threes of parameters ),,( iiii AaAA , where

ia is a center of the interval, iA is its upper border, iA - its lower border.

Current task contains the case of symmetrical membership function for parameters Ai, so they can be described via the pair of parameters ( ia , ic ).

iii caA , iii caA , ic – interval width, ic ≥ 0, Zi are also fuzzy numbers of triangular form, which are defined by parameters ),,( iii ZZZ

, iZ - lower border, iZ

-

center, iZ - upper border of fuzzy number. Then Y – fuzzy number, which parameters are defined as follows: Center of the interval: ii Zay

* ,

Deviation in the left part of the membership function: ))(*( iiiii ZcZZayy

,

Lower border of the interval: )*( iiii ZcZay

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 workflow. It can be described in such a way:

MkyZcZa

yZcZa

kikikii

kikiiki

,1,)*(

)*(

,

where Zk=[ Zk]i is an input training sample, yk –known output values, Mk ,1 , M is a number of observation points. So, the general requirements to estimation linear interval model are as follows: to find such values of parameters

),( ii ca of fuzzy coefficients, which enable: a) Observed values ky to lay in an estimation interval for kY ; b) Total width of estimation interval to be minimal. These requirements can be redefined as a linear programming problem:

M

kiiiiiiii

caZcZaZcZa

ii 1,))*()*((min

(2)

under constraints:

MkyZcZa

yZcZa

kikikii

kikiiki

,1,)*(

)*(

(3)

1.2.2. Formalized problem formulation in case of triangular membership functions Let’s consider partial description

25

243210),( jijijiji xAxAxxAxAxAAxxf (4)

Rewriting it in accordance with the model (1) needs such substitution

10 z , ixz 1 ,

jxz 2 , ji xxz 3 , 24 ixz , 2

5 jxz .

Then mathematical model (2)-( 3) will take the form

)2)(22

)(22))()((

2)(2)(2(min

1 1

255

1

24

14

13

13

1 1 1221

110

,

M

k

M

kjkjkjkjk

M

kik

M

kikikik

M

kjkik

M

kikikjkjkjkik

M

k

M

k

M

kjkjkjkik

M

kikik

ca

xcxxxaxc

xxxaxxcxxxxxxa

xcxxaxcxxaMcii

(5)

with the following conditions:

kjkikjkikjk

ikjkjkjkjkikikikik

jkikikikjkjkjkikjkik

yxcxcxxcxc

xccxxxxaxxxxa

xxxxxxxxaxaxaa

25

2432

102

52

4

3210

))(2())(2(

))()((

(6)

kjkik

jkikjkikjkjkjkjkikik

ikikjkikikikjkjkjkikjkik

yxcxc

xxcxcxccxxxxaxx

xxaxxxxxxxxaxaxaa

25

24

32102

52

43210

))(2())

(2())()((

5,0,0 lcl .

As one may see, this is the linear programming problem, but there are no limitations for non-negativity of variables ia , so we need go to the dual problem, introducing dual variables k and Mk . Write down the dual problem:

)max(11

M

kkk

M

kMkk yy (8)

under constraints:

011

M

kk

M

kMk

M

kikik

M

kkik

M

kMkik xxxx

111

)( (9)

M

kjkjk

M

kkjk

M

kMkjk xxxx

111

)(

M

kikikjkjkjkik

M

kkjkikikikjkjkjkik

M

kMkjkikikikjkjkjkik

xxxxxx

xxxxxxxx

xxxxxxxx

1

1

1

))()((

))()((

))()((

M

kikikik

M

kkikikikik

M

kMkikikikik

xxx

xxxxxxxx

1

1

2

1

2

)(

))(2())(2(

M

kjkjkjk

M

kkjkjkjkik

M

kMkjkjkjkjk

xxx

xxxxxxxx

1

1

2

1

2

)(

))(2())(2(

MM

kk

M

kMk 2

11

M

kik

M

kkik

M

kMkik xxx

111

2

M

kjk

M

kkjk

M

kMkjk xxx

111

2

M

kjkik

M

kkjkik

M

kMkjkik xxxxxx

111

2 (10)

M

kik

M

kkik

M

kMkik xxx

1

2

1

2

1

2 2

M

kjk

M

kkjk

M

kMkjk xxx

1

2

1

2

1

2 2

0k

0Mk , Mk ,1 The task (8)-(10) can be solved using simplex-method. After finding the optimal values of dual variables k ,

Mk , we easily obtain the optimal values of desired variables ic , ia , 5,0i , and also a desired fuzzy model for given partial description.

2. FGMDH WITH FUZZY INPUT DATA FOR GAUSSIAN MEMBERSHIP FUNCTIONS 2.1. Math model in the case of Gaussian MF Let’s consider a linear interval regression model: Y = A0Z0+A1Z1+…+AnZn, where Ai – fuzzy numbers of Gaussian shape, which are described by a pair of parameters ( ia , ic ), ia is a center,

ic is a value characterizing the width of the interval, ic ≥ 0, Zi – Gaussian fuzzy numbers, which have membership functions of the following form:

axe

axex

c

ax

c

ax

,)(

21

)(21

22

2

21

2

,)(

Then such fuzzy numbers are described by threes of parameters ),,( 21 cac , where a is a center, 1c is a value characterizing the width of the interval in the left branch of membership function, 2c is its the right branch. So, the task is to find such fuzzy parameters ),( ii ca of fuzzy coefficients iA , that the following conditions would be true:

1) Observation ky should lay in the estimation interval for kY with a degree, which is not less than , 10 .

2) The width of the estimation interval of level be minimal. The width of estimation interval of level equals ( see fig. 1.3): 12 yyd (11)

We can find 1y and 2y from the system:

21

21

22

22

)(

2

1exp

)(

2

1exp

c

ay

c

ay

(12)

So,

acy

acy

ln2

ln2

11

22

and )(ln2 12 ccd Condition 1) can be defined as: )( ky , which can be rewritten as follows:

ln2

ln2

1

2

kkk

kkk

cay

cay

So, the problem is :

to find

M

kkk

M

k

k ccd1

121

ln2)(minmin (13)

under constraints

ln2

ln2

1

2

kkk

kkk

cay

cay (14)

Variables Zi may be defined via parameters ),,( iii ZZZ

, where iZ is a lower border, iZ

is a center, iZ is the upper

border of fuzzy number, thus iii cZZ 1

, iii cZZ 2

. Thus Y is a fuzzy number, whose parameters are defined as follows: a center of the interval: ii Zay

* ,

Deviation in the left part of membership function: ))(*( iiiii ZcZZayy

And the lower border of the interval )*( iiii ZcZay

Deviation in the right part of membership function:

iiiiiiiiiii ZcZaZaZcZZayy

))(*(

So, the upper border of the interval: )*( iiii ZcZay

2.2. Formalized problem formulation in the case of Gaussian MF Partial description

25

243210),( jijijiji xAxAxxAxAxAAxxf

Let’s rewrite it in accordance with the model (1.1). It needs such substitution

10 z , ixz 1 , jxz 2 , ji xxz 3 , 24 ixz , 2

5 jxz .

So the mathematical model (1.10)-(1.11) will take the form:

)2)(22

)(22))()((

2)(2)(2(min

1 1

255

1

24

14

13

13

1 1 1221

110

,

M

k

M

kjkjkjkjk

M

kik

M

kikikik

M

kjkik

M

kikikjkjkjkik

M

k

M

k

M

kjkjkjkik

M

kikik

ca

xcxxxaxc

xxxaxxcxxxxxxa

xcxxaxcxxaMcii

under conditions:

kjkikjkikjk

ikjkjkjkjk

ikikikikikikjkjkjkik

jkikjkjkjkikikik

yxcxcxxcxc

xccxxxxa

xxxxaxxxxxx

xxaxxxaxxxaa

ln2ln2ln2ln2

ln2ln2)))((ln22(

))((ln22()ln2))()((

())(ln2())(ln2(

25

2432

102

5

24

3210

kjkikjkikjk

ikjkjkjkjk

ikikikikikikjkjkjkik

jkikjkjkjkikikik

yxcxcxxcxc

xccxxxxa

xxxxaxxxxxx

xxaxxxaxxxaa

ln2ln2ln2ln2

ln2ln2))((ln22(

))((ln22()ln2))()((

())(ln2())(ln2(

25

2432

102

5

24

3210

5,0,0 lcl .

As we can see, this is the same problem of linear programming as in the case of triangular MF, but there are still no limitations for non-negativity of variables ia , so we need go to dual problem, introducing dual variables k and Mk .

4. EXPERIMENTAL RESULTS OF FGMDH WITH FUZZY INPUTS IN THE PROBLEM OF RTS INDEX FORECASTING

For estimation of efficiency of the suggested FGMDH method with fuzzy inputs the corresponding software kit was elaborated and numerous experiments of forecasting at financial markets were carried out. Some of them are presented below. 4.1. Forecasting of RTS index. 4.1.1. Experiment 1. RTS index forecasting (opening price) In this experiment we used 5 fuzzy input variables, which represent stock prices of leading Russian energetic companies, included to the list for calculations of RTS index: LKOH – shares of “LUKOIL” joint-stock company, EESR – shares of “РАО ЕЭС России“joint-stock company, YUKO – shares of “ЮКОС” joint-stock company, SNGSP – privileged shares of “Сургутнефтегаз” joint-stock company, SNGS – common shares of “Сургутнефтегаз” joint-stock company. Output variable is the RTS (opening price) index value of the same period (03.04.2006 – 18.05.2006). Sample size is 32 values. Training sample size is 18 values (optimal size of training sample for the current experiment). The following results were obtained: 1. For triangular membership function a) For normalized input data (normalizing was done using the following formula:

1;0,minmax

min

i

ii X

XX

XXX )

a criterion value for the current experiment was: MSE = 0.05557

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 4 7 10 13 16 19 22 25 28 31

Left border

Center

Right border

Real values

Fig. 2. Experiment 1 results for triangular membership function and normalized values of input variables b) for non-normalized input data: Criteria values for this experiment were: MSE = 18.48657 MAPE = 0.8%

1200

1300

1400

1500

1600

1700

1800

1900

1 4 7 10 13 16 19 22 25 28 31

Left border

Center

Right border

Real values

Fig. 3. Experiment 1 result for triangular MF and non-normalized values of input variables

2. For the case of Gaussian membership function (optimal level is α=0.8) a) For normalized input data Criterion value for this experiment was: MSE = 0.028013

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 4 7 10 13 16 19 22 25 28 31

Left border

Center

Right border

Real values

Fig. 4. Experiment 1 result for Gaussian MF and normalized input data b) for non-normalized input data:

1200

1300

1400

1500

1600

1700

1800

1900

1 4 7 10 13 16 19 22 25 28 31

Left border

Center

Right border

Real values

Fig. 5. Experiment 1 result with Gaussian MF and non-normalized values of input variables

Criteria values for this experiment were the following: MSE = 9.321461 MAPE = 0.4%

As we can see from the results of experiment 1, forecasting using triangular and Gaussian membership functions gives good

results. Results of experiments with Gaussian MF are better than results of experiments with triangular MF.

For normalized data: Triangular MF Gaussian MF MSE 0.055557 0.028013 For non-normalized data: Triangular MF Gaussian MF MSE 18.48657 9.321461 MAPE 0.8% 0.4% 4.1.2. Experiment 2. Forecasting of RTS index (closing price) In this experiment the same input variables were used as in the experiment 1 .

Output variable is the value of RTS index (closing price) for the same period of time (03.04.2006 – 18.05.2006).

Sample size – 32 values.

Training sample size – 18 values (optimal size of training sample for current experiment).

The following results were obtained:

1. For triangular membership function

a) For normalized input data

Criterion value: MSE = 0.057379

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 4 7 10 13 16 19 22 25 28 31

Left border

Center

Right border

Real values

Fig. 6. Experiment 2 result for triangular MF and normalized values of input variables b) For non-normalized input data Criterion values: MSE = 18.04394 MAPE =0.78%

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

1 4 7 10 13 16 19 22 25 28 31

Left border

Center

Right border

Real values

Fig. 7. Experiment 2 result for triangular MF and non-normalized values of input variables

1. For Gaussian membership function (optimal level α=0.85) a) For normalized input data Criterion value for current experiment was: MSE = 0.029582

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 4 7 10 13 16 19 22 25 28 31

Left border

Center

Right border

Real values

Fig. 8. Experiment 2 result for Gaussian MF and normalized values of input variables b) For non-normalized input data Criterion values for this experiment: MSE = 9.302766; MAPE =0.37%

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Left border

Center

Right border

Real value

Fig. 9. Experiment 2 result for Gaussian MF and non-normalized values of input variables

As we can see from the results of experiment 2, forecasting using both triangular and Gaussian membership functions gives good results. Results of experiments with Gaussian MF are better than results of experiments with triangular MF. For normalized data: Triangular MF Gaussian MF MSE 0.057379 0.029582

For non-normalized data: Triangular MF Gaussian MF MSE 18.04394 9.302766 MAPE 0.78% 0.37%

4.2. Stock prices forecasting

4.2.1 Experiment 4. Stock prices forecasting In the following experiment stock prices of 4 leading energetic companies of Russia were used: EESR – shares of “РАО ЕЭС России” joint-stock company, YUKO – shares of “ЮКОС” joint-stock company, SNGSP – privileged shares of “Сургутнефтегаз” joint-stock company, SNGS – ordinary shares of “Сургутнефтегаз” joint-stock company. Output variable is stock price of other company – “LUKOIL” joint-stock for the same period (03.04.2006 – 18.05.2006) to be forecasted. The sample size – 32 values. Training sample size – 17 values The following results were obtained: 1. For triangular membership function a) For normalized input data: Criterion value: MSE = 0.056481

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

Left border

Center

Right border

Real values

Fig. 10. Experiment 4 results for triangular MF and normalized values of input variables

b) for non-normalized input data: Criteria values: MSE = 0.914998; MAPE = 0.73%

60

65

70

75

80

85

90

95

100

105

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Left border

Center

Right border

Real values

Fig. 11. Experiment 4 results for triangular MF and non-normalized values of input variables

2. For Gaussian membership function (optimal level of α=0.9) a) For normalized input data: Criterion value for this experiment: MSE = 0.030464

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 4 7 10 13 16 19 22 25 28 31

Left border

Center

Right border

Real values

Fig. 12. Experiment 4 results for Gaussian MF and normalized values of input variables b) For non-normalized input data: Criteria values for this experiment were: MSE = 0.493511 MAPE = 0.33% As we can see from the results of experiment 4, forecasting using triangular and Gaussian membership functions is successful and gives good results. Results of experiments with Gaussian MF are better than results of experiments with triangular MF. For normalized data:

Triangular MF Gaussian MF MSE 0.056481 0.030464

For non-normalized data:

Triangular MF Gaussian MF MSE 0.914998 0.493511 MAPE 0.73% 0.33%

5. COMPARISON OF GMDH AND FUZZY GMDH

Experiment 1. Forecasting of RTS index (opening price) RTS index is an official indicator of RTS stock exchange. It is calculated during trade session with every change of price of stocks, included in the list of its calculation. First value of index is the opening price and the last is the closing price. Sample size – 32 values. Training sample size – 18 values (optimal size of the training sample for current experiment).: For normalized inputs when using Gaussian MF in group method of data handling with fuzzy input data the following results were obtained:

MSE for GMDH = 0.1129737 MSE for FGMDH = 0.0536556

-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 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Y Estimation by distinct GMDH Estimation by fuzzy GMDH

Upper border by fuzzy GMDH Lower border by fuzzy GMDH

Fig. 13. Experiment 1 results using GMDH and FGMDH

As the results of experiment 1 show, fuzzy group method of data handling with fuzzy input data with triangular membership function and Gaussian membership function gives more accurate forecasting than GMDH.

In case of triangular MF FGMDH with fuzzy data gives a little worse results than FGMDH. (see table 2) Table 2. MSE comparison for different methods in experiment 1 GMDH FGMDH FGMDH

with fuzzy inputs, Triangular MF

FGMDH with fuzzy inputs, Gaussian MF

MSE 0.1129737 0.0536556 0.055557 0.028013

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

FGMDH with fuzzy inputs, triangular MF FGMDH with fuzzy inputs, Gaussian MF

Real values GMDH

FGMDH

Fig. 14. Comparison of GMDH, FGMDH (center of estimation), and FGMDH with fuzzy inputs (center of estimation) results

Experiment 2. RTS-2 index forecasting (opening price) Sample size – 32 values. Training sample size – 19 values (optimal size of training sample for current experiment). The following results were obtained: For normalized input data when using triangular MF in group method of data handling with fuzzy input data:

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1 4 7 10 13 16 19 22 25 28 31

Left border

Center

Right border

Real values

Fig. 15. Experiment 2 results for triangular MF

For normalized input data using triangular MF in fuzzy group method of data handling with fuzzy input data the results are presented below.

Table 4. MSE of different methods of experiment 2 comparison GMDH FGMDH FGMDH

with fuzzy inputs, triangular MF

FGMDH with fuzzy inputs, Gaussian MF

MSE 0.051121 0.063035 0.061787 0.033097

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

FGMDH with fuzzy inputs, triangular MF

FGMDH with fuzzy inputs, Gaussian MF

Real values

GMDH

FGMDH

Fig. 17. GMDH, FGMDH (center of estimation), and FGMDH with fuzzy inputs (center of estimation) result comparison

6. COMPARISON OF FGMDH WITH FUZZY INPUT DATA AND FUZZY NEURAL NETS

Experiment 3. “LUKOIL” stock price forecasting. 1. Stock price forecasting using fuzzy group method of data handling with triangular MF and Gaussian MF: а) Forecasting based on previous data about stock prices. The following results were obtained:

MSE = 0.215421

55

56

57

58

59

60

61

62

63

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Left border

Center

Right border

Real values

Fig. 18. Experiment 3 results using FGMDH with fuzzy input data and triangular MF

b) Forecasting based on previous data about stock prices of leading Russian energetic companies for the same period: EESR – shares of “РАО ЕЭС России” joint-stock company, YUKO – shares of “ЮКОС” joint-stock company, SNGSP – privileged shares of “Сургутнефтегаз” joint-stock company, SNGS – common shares of “Сургутнефтегаз” joint-stock company. The following results were obtained: MSE = 0.115072

55

56

57

58

59

60

61

62

63

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Left border

Center

Right border

Real values

Fig. 19. Experiment 3 results using FGMDH with fuzzy data and Gaussian MF 2. Stock price forecasting using fuzzy neural nets with Mamdani and Tsukamoto algorithm. 267 everyday indexes of stock prices during period from 1.04.2005 to 30.12.2005 were used for neural net training. The following results were obtained:

Table 5. Experiment 3 results using FNN

Date Real Value

Mamdani with Gaussian MF

Mamdani With Triangular MF

Tsukamoto With Gaussian MF

Tsukamoto With Triangular MF

01.12.2005 58.1 58.23 58.41 58.37 58.48 02.12.2005 58.7 58.54 58.46 58.47 58.37 05.12.2005 59.4 59.14 59.11 59.1 59.05 06.12.2005 59 59.11 59.15 59.25 59.27 07.12.2005 59.85 59.97 60.1 60.19 60.23 08.12.2005 59.6 59.416 59.313 59.37 59.23 09.12.2005 59.9 60.12 60.22 60.27 60.32 12.12.2005 60.65 60.5 60.45 60.48 60.4 13.12.2005 60.65 60.54 60.31 60.42 60.28 14.12.2005 61.15 61.32 61.39 61.4 61.42 15.12.2005 60.25 60.1 59.99 60.06 59.97 16.12.2005 61 61.2 61.25 61.22 61.27 19.12.2005 61.01 61.24 61.34 61.28 61.34 20.12.2005 60.7 60.54 60.44 60.48 60.38 MSE 0.18046 0.28112 0.268013 0. 10093

56

57

58

59

60

61

62

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Real values Mamdani with Gaussian MF

Mamdani with triangular MF Tsukamoto with Gaussian MF

Tsukamoto with triangular MF

Fig. 20. Experiment 3 result using FNN

As we can see from the experiment 3 results, forecasting using FNN with Mamdani controller with Gaussian MF was the best, Tsukamoto controller with Gaussian MF takes the second place. Forecasting using FGMDH with fuzzy input data gives the best results when using Gaussian MF. The results of comparison of FGMDH with fuzzy inputs and FNN are presented below For Gaussian MF: Table 6

Mamdani controller

Tsukamoto Controller

FGMDH With fuzzy inputs (4 input variables)

FGMDH with fuzzy inputs (previous values of output)

MSE 0.18046 0.268013 0.115072 0.094002 For triangular MF: Table 7

Mamdani controller

Tsukamoto Controller

FGMDH With fuzzy inputs (4 input variables)

FGMDH with fuzzy inputs (previous values of output)

MSE 0.28112 0.34443 0.210865 0.215421

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1

Mamdani with GaussianMF

Tsukamoto withGaussian MF

FGMDH with fuzzyinputs and Gaussian MF(4 inp. var)

FGMDH with fuzzyinputs and Gaussian MF(prev. val.)

Mamdani with triangularMF

Tsukamoto withtriangular MF

FGMDH with fuzzyinputs and triangular MF(4 inp. var.)

FGMDH with fuzzyinputs and triangular MF(prev. val.)

Fig. 21. Results of comparison FGMDH and FNN in the experiment 3

Experiment 4. RTS index forecasting (opening price) Forecasting of RTS index (opening price) using fuzzy neural nets with Mamdani and Tsukamoto algorithm. 267 everyday indexes of stock prices from 1.04.2005 to 30.12.2005 Table 8. Experiment 4 results using FNN

Date Real Values

Mamdani With Gaussian MF

Mamdani With Triangular MF

Tsukamoto With Gaussian MF

Tsukamoto With Triangular MF

01.11.2005 935.05 934.2 933.78 933.9 937.8 02.11.2005 943.82 955.28 946.19 945.58 946.29 03.11.2005 965.28 966.6 968.7 966.8 960.9 07.11.2005 972.38 973.3 974.3 974.01 975.41 08.11.2005 969.21 971.08 971.8 946.1 963.92 09.11.2005 972.11 973.42 976 975.11 977.5 10.11.2005 980.51 982.1 983.4 983.1 985.6 11.11.2005 970.92 972.5 974.8 972.89 975.2 14.11.2005 969.86 972 972.9 966.2 963.5 15.11.2005 989.27 991.9 992.7 986.21 995.3 16.11.2005 991.08 993.2 987.6 995.85 996.56 17.11.2005 990.5 992.8 994.67 993.58 996.5 18.11.2005 1013.4 1015.6 1016.9 1011 1019.6 21.11.2005 1014.2 1016.7 1010.4 1010.5 1010.2 MSE 3.692981 3.341179 7.002467 5.119318 MAPE, % 0.256091 0.318056 0.318056 0.419659

880

900

920

940

960

980

1000

1020

1040

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Real values Mamdani with Gaussian MF

Mamdani with triangular MF Tsukamoto with Gaussian MF

Tsukamoto with triangular MF

Fig. 22. Experiment 4 results using FNN

As experiment 3 results show, forecasting using FNN with Mamdani controller with Gaussian MF was the best, Mamdani controller with triangular MF is on the second place.

For Gaussian MF: Table 9

Mamdani Controller

Tsukamoto Controller

FGMDH with fuzzy inputs (5 input variables )

FGMDH with fuzzy inputs (previous values of the input data)

MSE 3.692981 7.002467 2.1151183 2.886697 МАРЕ. %

0.256091 0.318056 0.179447 0.256547

For triangular MF: Table 10

Mamdani Controller

Tsukamoto Controller

FGMDH with fuzzy inputs (5 input variables )

FGMDH with fuzzy inputs (previous values on the input)

MSE 3.34179 5.119318 4.717268 4.977901 МАРЕ. %

0.318056 0.419659

0.40437 0.415434

As current experiment results show, forecasting using FGMDH with fuzzy input data using Gaussian membership function was the best,

fuzzy Mamdani controller with Gaussian and triangular MF is on the second place.

In general, considering the experimental results, we may conclude that both FGMDH with fuzzy input data and fuzzy neural nets give the

results of approximately the same accuracy. Besides, Gaussian MFs have advantage over triangular MFs in both methods.

CONCLUSIONS

In this paper new method of inductive modeling FGMDH with fuzzy inputs was suggested. This method represents the development of fuzzy GMDH when information is fuzzy and given in the form of uncertainty intervals. The mathematical model was constructed and corresponding algorithm was elaborated. The experimental results of application of the suggested method in the forecasting of market indexes and stock prices are presented and discussed.

The comparison of FGMDH with fuzzy inputs, GMDH and fuzzy neural networks in the problem of forecasting in financial sphere was performed.

Experiments has showed that FNN and FGMDH with fuzzy inputs are effective methods for forecasting of social and economic indexes. The accuracy of these methods is high.

FGMDH with fuzzy inputs gives better results in the problem of forecasting at financial markets in comparison with GMDH. The best results were obtained in cases of FGMDH with fuzzy inputs using. Gaussian MF and of fuzzy neural nets.

The main advantages of the suggested method in comparison with classical GMDH are the following:

1. It operates with fuzzy input information and constructs the fuzzy model; 2. The constructed model has minimal possible total width and in this sense it is optimal; 3. For finding optimal model we solve corresponding linear programming problem which is always solvable for

this task as it was proved in preceding papers; 4. We should not a priori set the form of a model the algorithm finds it itself using the ideas of evolution.

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