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An Improved Fuzzy Time Series Model For Forecasting

I.           Introduction

Traditional forecasting methods can deal with many forecasting cases, but they cannot solve forecasting problems in which the historical data are linguistic values. Song and Chissom [12] presented the concept of fuzzy time series based on the historical enrollments of the University of Alabama. They presented the time-invariant fuzzy time series model and the time-variant fuzzy time series model based on the fuzzy set theory for forecasting the enrollments of the University of Alabama.

 

The fuzzy forecasting methods can forecast the data with linguistic values. Fuzzy time series do not need to turn a non-stationary series into a stationary series and do not require more historical data along with some assumptions like normality postulates. Although fuzzy forecasting methods are suitable for incomplete data situations, their performance is not always satisfactory [9,11].

 

Huarng [6] proposed heuristic models; by integrating problem-specific heuristic knowledge to improve forecasting.

Tsaur, et al [14] proposed an analytical approach to find the steady state of fuzzy relation matrix to revise the logic forecasting process. Based on the concept of fuzziness in Information Theory, the concept of entropy is applied to measure the degrees of fuzziness when a time-invariant relation matrix is derived. In order to show the forecasting performance, the best fitted regression equations are applied to compare with the proposed method.

 

Yu [15] proposed weighted models to tackle two issues in fuzzy time series forecasting; namely, recurrence and weighting. Weighted fuzzy time series models appear quite similar to the weight functions in local regression models; however, both are different. The local regression models focus on fitting using a small portion of the data, while the fuzzy relationships in weighted fuzzy time series models are established using the possible data from the whole of the database.

 

Jilani and Burney [7] presented two new multivariate fuzzy time series forecasting methods. These methods assume m-factors with one main factor of interest. Stochastic fuzzy dependence of order k is assumed to define general methods of multivariate fuzzy time series forecasting and control.

 

Cheng et al [4] proposed a novel multiple-attribute fuzzy time series method based on fuzzy clustering. The methods of fuzzy clustering were integrated in the processes of fuzzy time series to partition datasets objectively and enable processing of multiple attributes.

 

Abd Elaal et al [1-2] proposed a novel forecasting fuzzy time series model depend on fuzzy clustering for improving forecasting accuracy. Kai et al [8] proposed a novel forecasting model for fuzzy time series using K-means clustering algorithm for forecasting.

 

In this paper, researchers propose an efficient fuzzy time series forecasting model based on fuzzy clustering to handle forecasting problems and improving forecasting accuracy. Each value (observation) is represented by a fuzzy set. The transition between consecutive values is taken into account in order to model the time series data.

II.         Related works

 

In this section, two related works including: fuzzy clustering and fuzzy time series.

 

A.        Fuzzy clustering (FCMI)

Fuzzy c-means (FCM) is a method of clustering which allows one piece of data to belong to two or more clusters. Fuzzy C-Mean Iterative assume that: the existence of pattern space X={x1, x2,…, xm) and c fuzzy clusters, whose centers have initial values y10, y20,…,yc0. Every iteration the membership function values updated and the cluster centers also. The process terminates when the difference between two consecutive clusters centers do not exceed a given tolerance [5].

 

 

(1)

 

Fuzzy clustering is carried out through an iterative optimization of the objective function , with the update of membership and the cluster centers  by:

 

 

(2)

 

(3)

 

This iteration will stop when

 

(4)

B.       Fuzzy time series

Song and Chissom [13] presented the concept of fuzzy time series based on the historical enrollments of the University of Alabama. Fuzzy time series used to handle forecasting problems. They presented the time-invariant fuzzy time series model and the time-variant fuzzy time series model based on the fuzzy set theory for forecasting the enrollments of the University of Alabama. The definitions and processes of the fuzzy time-series presented by Song and Chissom are described as follows [6,12].

 

Definition 1. (FTS) Assume Y (t) (t = . . 0, 1, 2, . . .) is a subset of a real numbers. Let Y (t) be the universe of discourse defined by the fuzzy set fi (t). If F(t) is a collection of f1(t), f2(t). . . then F(t) is defined as a fuzzy time-series on Y (t) (t = . . . , 0, 1, 2, . . .).

 

Definition 2. (FTSRs) If there exists a fuzzy logical relationship R(t − 1, t), such that F(t) = F(t − 1) × R(t − 1, t), where "×" represents an operation, then F(t) is said to be induced by F(t − 1). The logical relationship between F(t) and F(t − 1) is  F(t − 1) à F(t).

 

Definition 3. (FLR) suppose F(t − 1) = Ai and F(t) = Aj . The relationship between two consecutive observations, F(t) and F(t − 1), referred to as a fuzzy logical relationship, can be denoted by  Ai à Aj , where Ai is called the Left-Hand Side (LHS) and Aj the Right-Hand Side (RHS) of the FLR.

 

Definition 4. (FLRG) All fuzzy logical relationships in the training dataset can be grouped together into different fuzzy logical relationship groups according to the same Left-Hand Sides of the fuzzy logical relationship. For example, there are two fuzzy logical relationships with the same Left-Hand Side (Ai ): Aià Aj1 and Ai à Aj2. These two fuzzy logical relationships can be grouped into a fuzzy logical relationship group Aià Aj1 Aj2.

 

Definition 5. (IFTS & VFTS) Assume that F(t) is a fuzzy time-series and F(t) is caused by F(t − 1) only, and F(t) = F(t − 1) × R(t − 1, t). For any t, if R(t − 1, t) is independent of t, then F(t) is named a time-invariant fuzzy time-series, otherwise a time-variant fuzzy time-series.

 

a)      Song and Chissom model

 

Song and Chissom employed five main steps in time-invariant fuzzy time-series and time-variant fuzzy time series models as follows:

 

Step 1: Define the universe of discourse U. Define the universe of discourse for the observations. According to the issue domain, the universe of discourse for observations is defined as,

U=[Dmin – D1, Dmax + D2]

(5)

 

where, Dmin is the minimum value,

Dmax is the maximum value,

D1, D2 is the positive real numbers.

 

Step 2: Partition universal of discourse U into equal intervals.

 

Step 3: Define the linguistic terms. Each linguistic observation, Ak can be defined by the intervals u1,u2,...,un, as follows:

 

 

(6)

 

Step 4: Fuzzify the historical data. Each historical data can be fuzzified into a fuzzy set.

 

Step 5: Build fuzzy logic relationships. Build fuzzy logic relationships. Two consecutive fuzzy sets Ai(t-1)and Aj(t) can be established into a single FLR as Aià Aj.

 

III.        Proposed model

In this section we proposed an efficient fuzzy time series forecasting model based on fuzzy clustering to handle forecasting problems and improving forecasting accuracy. Most researchers have been taken the same way according to processes of the fuzzy time-series, which are presented by Song and Chissom, but we introduce a novel model based on fuzzy clustering to determine the membership values not as Song and Chissom model, and to increase the performance. Proposed model employed eight main steps in time-invariant fuzzy time-series and time-variant fuzzy time series models as follows:

Step 1: Cluster data into c clusters: Apply fuzzy clustering on a time series Y(t) with n observation  to cluster this time series into c  (2 ≤ c ≤ n) clusters. FCMI is used because it is the most popular one and well known in fuzzy clustering field.

Step 2: Determine membership values for each cluster: In this step, membership values is determining after doing fuzzy cluster. The proposed model selected the maximum membership grade of each value for each cluster which it belong to.

Step 3: Rank each cluster: Proposed model ranking clusters by the center of each cluster, where first cluster has the minimum center, and last cluster has the maximum center.

Step 4: Define the universe of discourse U: In this step, the proposed model defines the universe of discourse as Song and Chissom were defined it as in (5).

Step 5: Partition universal of discourse U into equal intervals: According to this step, the proposed model, partition the universe of discourse into c intervals.

Step 6: Fuzzify the historical data: In this step, proposed model fuzzufy historical data, where the proposed model determine the best fuzzy cluster to each actual data

Step 7: Build fuzzy logic relationships: Proposed model in this step build fuzzy logic relationship as definition 3. if F(t−1) = Ai and F(t) = Aj then the relationship between two consecutive observations:   Ai à Aj

Step 8: Calculate forecasting outputs: The forecasting value for each cluster is calculated by proposed model as:

 

 

(7)

Where dfj is the membership grade,

Xj is the actual value.

 

A.       Evaluating of the proposed model

 

To evaluating the performance of the proposed model, the researchers compare the forecasting values of enrollments of the University of Alabama with some famous models such as Jilani and Burney [7], Tsaur and Yang [14], Yu [15], Kai et al [8], and Cheng, et al [4].

 

The forecasting accuracy is compared by using (NRMSE) Normalized Root Mean Square Error. NRMSE, in statistic is the square root of the sum of the squared deviations between actual and predicted values divided by the sum of the square of actual values.

 

 

(8)

 

In this study, to evaluate the forecasting accuracy of the proposed model, the researchers use the enrollments of the University of Alabama as the forecasting target in the existing forecasting models.

 

Based on the enrollments of the University of Alabama from 1971 to 1992, we can get the universe of discourse U=[13055,19337], partition U into 7 equal intervals, D1=13, and D2=55. Hence, the intervals are u1; u2; u3; u4; u5; u6; u7; where :-

 

u1=[13024.00, 13933.71]

u2=[13933.71, 14843.43],

u3=[14843.43, 15753.14],

u4=[15753.14, 16662.86],

u5=[16662.86, 17572.57],

u6=[17572.57, 18482.29],

u7=[18482.29, 19392.00],

Table I lists the enrollment of the University of Alabama from 1971 to 1992, and membership grades of enrollments for each linguistic. Define the fuzzy set Ai using the linguistic variable "Enrollments of the University of Alabama", let A1 = (very very few), A2 = (very few), A3 = (few), A4 = (moderate), A5 = (many), A6 = (many many), A7 = (too many).The proposed model selected the maximum membership grade for each cluster, the forecasting value for each cluster calculating as in (7):

 

 

 

 

 

 

 

TABLE I.            Data of enrollments of the university of Alabama and membership grades.

Year

Actual

enrollments

A1

A2

A3

A4

A5

A6

A7

1971

13055

0.8

0.1

0

0

0

0

0

1972

13563

1

0

0

0

0

0

0

1973

13867

0.9

0.1

0

0

0

0

0

1974

14696

0.1

0.7

0.2

0.1

0

0

0

1975

15460

0

0

1

0

0

0

0

1976

15311

0

0.1

0.9

0

0

0

0

1977

15603

0

0.1

0.6

0.3

0

0

0

1978

15861

0

0

0

1

0

0

0

1979

16807

0

0

0

0

1

0

0

1980

16919

0

0

0

0

0.9

0

0

1981

16388

0

0

0.1

0.3

0.6

0

0

1982

15433

0

0

1

0

0

0

0

1983

15497

0

0

0.9

0.1

0

0

0

1984

15145

0

0.8

0.2

0

0

0

0

1985

15163

0

0.7

0.2

0

0

0

0

1986

15984

0

0

0

0.9

0

0

0

1987

16859

0

0

0

0

1

0

0

1988

18150

0

0

0

0

0

1

0

1989

18970

0

0

0

0

0

0

1

1990

19328

0

0

0

0

0

0

0.9

1991

19337

0

0

0

0

0

0

0.9

1992

18876

0

0

0

0

0

0.1

0.9

 

 

Figure 1.     Forecasting enrollments of the University of Alabama by the proposed model

 

TABLE II.          Data enrollments the university of Alabama, linguistic values, and forecasted values

Years

Enrollments

Linguistic

Forecasted

1971

13055

A1

13563

1972

13563

A1

13563

1973

13867

A1

13563

1974

14696

A2

15145

1975

15460

A3

15446

1976

15311

A3

15446

1977

15603

A3

15446

1978

15861

A4

15861

1979

16807

A5

16833

1980

16919

A5

16833

1981

16388

A4

15861

1982

15433

A3

15446

1983

15497

A3

15446

1984

15145

A3

15446

1985

15163

A3

15446

1986

15984

A4

15861

1987

16859

A5

16833

1988

18150

A6

18150

1989

18970

A7

18970

1990

19328

A7

18970

1991

19337

A7

18970

1992

18876

A7

18970

 

 

 

Figure 2.     Forecasting results curve of enrollments of the university of Alabama

 

The forecasting value for year 1971 is 13563 while the actual value was 13055. Fig.1 and Table II show linguistic terms and forecasting values deduced by proposed model.

 

 

Figure 3.     NRMSE-chart for the existing models and the proposed model

 

The line-chart comparison in Fig. 2 shows that the proposed model has higher accuracy than the other models. And the empirical comparison among the existing models in Table III also shows that, the proposed model can further improve the forecasting results than the other model.

 

Fig. 3 shows the comparisons among the existing models by using NRMSE, where Jilani and Burney [7] model has 0.02, Tsaur and Yang [14] model has 0.025, Yu [15] model has 0.026, Kai et al [8] model has 0.024, Cheng, et al [4] model has 0.028 and proposed model has 0.015.

 

 

 

 

 

 

TABLE III.        Forecasting enrollments of the university of Alabama

Year

Actual enrollments

Tsaur and Yang

(2005)

Yu

(2005)

Jilani and Burney

(2008)

Cheng et al

(2008)

Kai et al

(2010)

Proposed

1971

13055

13934

13934

13769

 

 

13563

1972

13563

13934

13934

13769

14242

13997

13563

1973

13867

13934

13934

13769

14242

13997

13563

1974

14696

15298

15298

14360

14242

13997

15145

1975

15460

15753

15623

15271

15474.3

15461.2

15446

1976

15311

15753

15623

15271

15474.3

15461.2

15446

1977

15603

15753

15623

15271

15474.3

15461.2

15446

1978

15861

16208

16511

16182

15474.3

15461.2

15861

1979

16807

17118

17269

17094

16146.5

16861.7

16833

1980

16919

17118

17269

17094

16988.3

17394

16833

1981

16388

16208

16511

16182

16988.3

17394

15861

1982

15433

15753

15623

15271

16146.5

15461

15446

1983

15497

15753

15623

15271

15474.3

15461.2

15446

1984

15145

15753

15623

15271

15474.3

15461.2

15446

1985

15163

15753

15623

15271

15474.3

15461.5

15446

1986

15984

16208

16511

16182

15474.3

15461.5

15861

1987

16859

17118

17269

17094

16146.5

16861.7

16833

1988

18150

18937

18937

18004

16988.3

17394

18150

1989

18970

18937

18937

18624

19144

18932.2

18970

1990

19328

18937

18937

18624

19144

18932.2

18970

1991

19337

18937

18937

18624

19144

18932.2

18970

1992

18876

18937

18937

18624

19144

18932.2

18970

NRMSE

0.025

0.026

0.02

0.028

0.024

0.015

 

 

 

 

 

IV.        Empirical study

Based on the data of the iron and steel production witch are provided by the International Iron and Steel Institute in Brussels, Belgium, and publications of the U.S. geological survey from 1975 to 2008 (production values in thousand metric tons), we can get the universe of discourse U=[457000, 954000], partition U into 7 equal intervals, D1=6000, and D2=7000. Hence, the intervals are u1; u2; u3; u4; u5; u6; u7; where :-

 

u1=[ 451000.00, 523857.14]

u2=[ 523857.14, 596714.29],

u3=[ 596714.29, 669571.43],

u4=[ 669571.43, 742428.57],

u5=[ 742428.57, 815285.71],

u6=[ 815285.71, 888142.86],

u7=[ 888142.86, 961000.00],

 

 

Figure 4.     Forecasting of the world production of iron and steel by the proposed model

Table IV lists the World Production of Iron and Steel from 1975 to 2008, and membership grades of enrollments for each linguistic. Define the fuzzy set Ai using the linguistic variable "World Production of Iron and Steel", let A1 = (very very few), A2 = (very few), A3 = (few), A4 = (moderate), A5 = (many), A6 = (many many), A7 = (too many).

 

Fig. 4 and Table V show linguistic terms and forecasting values deduced by proposed model. The forecasting value for year 1975 is 494875 while the actual value was 479000 and the forecasting value for year 2008 is 943000 while the actual value was 932000.

 

TABLE IV.         Data of the world production of iron and steel, and membership grades.

Year

Production

A1

A2

A3

A4

A5

A6

A7

1975

479000

1

0

0

0

0

0

0

1976

498000

1

0

0

0

0

0

0

1977

488000

1

0

0

0

0

0

0

1978

506000

0

0

0

0

0

0

0

1979

532000

0

1

0

0

0

0

0

1980

514000

0

0

0

0

0

0

0

1981

502000

1

0

0

0

0

0

0

1982

457000

0

0

0

0

0

0

0

1983

463000

0

0

0

0

0

0

0

1984

495000

1

0

0

0

0

0

0

1985

499000

1

0

0

0

0

0

0

1986

495000

1

0

0

0

0

0

0

1987

509000

0

0

0

0

0

0

0

1988

539000

0

1

0

0

0

0

0

1989

546000

0

1

0

0

0

0

0

1990

531000

0

1

0

0

0

0

0

1991

509000

0

0

0

0

0

0

0

1992

503000

1

0

0

0

0

0

0

1993

507000

0

0

0

0

0

0

0

1994

516000

0

0

0

0

0

0

0

1995

536000

0

1

0

0

0

0

0

1996

516000

0

0

0

0

0

0

0

1997

540000

0

1

0

0

0

0

0

1998

535000

0

1

0

0

0

0

0

1999

539000

0

1

0

0

0

0

0

2000

573000

0

0

1

0

0

0

0

2001

585000

0

0

1

0

0

0

0

2002

608000

0

0

1

0

0

0

0

2003

673000

0

0

0

1

0

0

0

2004

720000

0

0

0

1

0

0

0

2005

802000

0

0

0

0

1

0

0

2006

881000

0

0

0

0

0

1

0

2007

954000

0

0

0

0

0

0

1

2008

932000

0

0

0

0

0

0

1

 

The proposed model selected the maximum membership grade for each cluster, the forecasting value for each cluster calculating as in (7):

 

 

 

 

 

 

 

 

 

 

 

 

TABLE V.           Data of the world production of iron and steel, linguistic values, and forecasted values

Year

Production

Linguistic

Forecasted

1975

479000

A1

494875

1976

498000

A1

494875

1977

488000

A1

494875

1978

506000

A1

494875

1979

532000

A2

537250

1980

514000

A1

494875

1981

502000

A1

494875

1982

457000

A1

494875

1983

463000

A1

494875

1984

495000

A1

494875

1985

499000

A1

494875

1986

495000

A1

494875

1987

509000

A1

494875

1988

539000

A2

537250

1989

546000

A2

537250

1990

531000

A2

537250

1991

509000

A1

494875

1992

503000

A1

494875

1993

507000

A1

494875

1994

516000

A1

494875

1995

536000

A2

537250

1996

516000

A1

494875

1997

540000

A2

537250

1998

535000

A2

537250

1999

539000

A2

537250

2000

573000

A2

537250

2001

585000

A2

537250

2002

608000

A3

588667

2003

673000

A4

696500

2004

720000

A4

696500

2005

802000

A5

802000

2006

881000

A6

881000

2007

954000

A7

943000

2008

932000

A7

943000

 

The researchers used famous models: Huarng[6], Tsaur and Yang [14], Yu [15], Jilani and Burney [7] to test the proposed model by forecasting of the world production of iron and steel as in Table VI.

 

 

Figure 5.     Forecasting results curve of the world production of iron and steel

 

 

Figure 6.     NRMSE-chart for the existing models and the proposed

 

The line-chart comparison in Fig. 5 shows that the proposed model has higher accuracy than the other models. And the empirical comparison among the existing models in Table VI also shows that, the proposed model can further improve the forecasting results than the other model.

 

 

 

Fig. 6 shows the comparisons among the existing models by using NRMSE, where Huarng[6] model has 0.0496, Tsaur and Yang [14] model has 0.0598, Yu [15] model has 0.0551, Jilani and Burney [7] model has 0.0399, and proposed model has 0.0296.

 

 

 

TABLE VI.         Forecasting of the world production of iron and steel

Year

Actual

Huarng 2001

Tsaur 2005

Yu 2005

Jilani 2008

Proposed

1975

479000

504571

523857

510762

509514

494875

1976

498000

504571

523857

510762

509514

494875

1977

488000

504571

523857

510762

509514

494875

1978

506000

504571

523857

510762

509514

494875

1979

532000

545714

560286

560286

555508

537250

1980

514000

504571

523857

510762

509514

494875

1981

502000

504571

523857

510762

509514

494875

1982

457000

504571

523857

510762

509514

494875

1983

463000

504571

523857

510762

509514

494875

1984

495000

504571

523857

510762

509514

494875

1985

499000

504571

523857

510762

509514

494875

1986

495000

504571

523857

510762

509514

494875

1987

509000

504571

523857

510762

509514

494875

1988

539000

545714

560286

560286

555508

537250

1989

546000

545714

560286

560286

555508

537250

1990

531000

545714

560286

560286

555508

537250

1991

509000

504571

523857

510762

509514

494875

1992

503000

504571

523857

510762

509514

494875

1993

507000

504571

523857

510762

509514

494875

1994

516000

504571

523857

510762

509514

494875

1995

536000

545714

560286

560286

555508

537250

1996

516000

504571

523857

510762

509514

494875

1997

540000

545714

560286

560286

555508

537250

1998

535000

545714

560286

560286

555508

537250

1999

539000

545714

560286

560286

555508

537250

2000

573000

545714

560286

560286

555508

537250

2001

585000

545714

560286

560286

555508

537250

2002

608000

706000

706000

706000

628923

588667

2003

673000

742429

742429

754571

702221

696500

2004

720000

742429

742429

754571

702221

696500

2005

802000

851714

851714

851714

775435

802000

2006

881000

924571

924571

924571

848587

881000

2007

954000

924571

924571

924571

898939

943000

2008

932000

924571

924571

924571

898939

943000

NRMSE

0.0496

0.0598

0.0551

0.0399

0.0296

 

 

 

 

V.          Discussion and conclusion

 

The research proposed an efficient fuzzy time series forecasting model based on fuzzy clustering with high accuracy. The method of FCMI is integrated in the processes of fuzzy time series to partition datasets. Experimental results of enrollments of the University of Alabama, and the comparison between the existing models: Jilani and Burney [7], Tsaur and Yang [14], Yu [15], Kai et al [8], and Cheng, et al [4] and the proposed model show that, the proposed model can further improve the forecasting results than the other models and also the experimental results of the world production of iron and steel, and the comparison between the existing models: Huarng[6], Tsaur and Yang [14], Yu [15], Jilani and Burney[7] and the proposed model show that, the proposed model has higher accuracy than the other models.

 

VI.        References

[1]        A. K. Abd Elaal, H. A. Hefny, and A. H. Abd-Elwahab, "A novel forecasting fuzzy time series model", in: Proceeding of International Conference on Mathematics and Information Security, Sohag Univ., Egypt, 2009.

[2]        A. K. Abd Elaal, H.A. Hefny, and A. H. Abd-Elwahab, "Constructing Fuzzy Time Series Model Based on Fuzzy Clustering for a Forecasting", J. Computer Sci., vol. 7, 2010, pp. 735-739.

[3]        T.-L. Chen, C.-H. Cheng, and H.-J. Teoh, "High-order fuzzy time-series based on multi-period adaptation model for forecasting stock markets", Physica A, vol.387, 2008, pp. 876–888

[4]        C.-H. Cheng, J.-W. Wang, and G.-W. Cheng, "Multi-attribute fuzzy time series method based on fuzzy clustering", Expert Systems with Applications, Vol.34, 2008. pp. 1235–1242.

[5]        M. Friedman and A. Kandel, "Introduction to pattern recognition statistical, structural, neural and fuzzy logic approaches", Imperial college press, London, 1999, p. 329.

[6]        K. Huarng, "Effective lengths of intervals to improve forecasting in fuzzy time series", Fuzzy Sets and Systems, vol.123, 2001, pp. 387–394.

[7]        T.A. Jilani and S. Burney, "Multivariate stochastic fuzzy forecasting models", Expert Systems with Applications, vol.35, 2008, pp. 691–700.

[8]        Kai, F. Fang-Ping, and C. Wen-Gang, "A novel forecasting model of fuzzy time series based on K-means clustering", IWETCS, IEEE, 2010, pp.223–225.

[9]        G. Kirchgässner and J. Wolters, "Introduction to modern time series analysis", Springer-Verlag.Berlin, Germany, 2007, p.153.

[10]     H.-T. Liu, "An improved fuzzy time series forecasting method using trapezoidal fuzzy numbers". Fuzzy Optimization and Decision Making, vol. 6, 2007, pp.63-80.

[11]     A.K. Palit and D. Popovic, "Computational intelligence in time series forecasting theory and engineering applications", Springer-Verlag.London, UK, 2005, p.18.

[12]     Q. Song and B.S. Chissom, "Forecasting enrollments with fuzzy time series. I", Fuzzy sets and systems, vol. 54, 1993, pp. 1-9.

[13]     Q. Song and B.S. Chissom, "New models for forecasting enrollments: fuzzy time series and neural network approaches", ERIC, 1993 p. 27, http://www.eric.ed.gov

[14]     R.-C. Tsaur, J.-C. Yang, and H.-F. Wang, "Fuzzy relation analysis in fuzzy time series model", Computers and Mathematics with Applications, vol.49, 2005, pp. 539-548.

[15]     H.-K. Yu, "Weighted fuzzy time series models for TAIEX forecasting", Physica A, vol.349, 2005, pp.609–624.

 

 

 

 

 

 

About the Author

Ashraf K. Abd-Elaal1

Department of Computer and Information Sciences

The High Institute of Computer Science

Sohag, Egypt

ashrafsohag@yahoo.com

Hesham A. Hefny

Department of Computer and Information Sciences,

Institute of Statistical Studies and Research,

Cairo University, Egypt

Ashraf H. Abd-Elwahab

Department of Computer Sciences

Electronics Research Institute National Center for Research

Cairo, Egypt

 

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