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An Improved Fuzzy Time Series Model For Forecasting

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					                                                                     (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                               Vol. 8, No. 8, 2010

                  An Improved Fuzzy Time Series Model For
                                Forecasting

             Ashraf K. Abd-Elaal1                                 Hesham A. Hefny                              Ashraf H. Abd-Elwahab
        Department of Computer and                   Department of Computer and Information                 Department of Computer Sciences
            Information Sciences                                       Sciences,                              Electronics Research Institute
    The High Institute of Computer Science          Institute of Statistical Studies and Research,            National Center for Research
                Sohag, Egypt                                   Cairo University, Egypt                                Cairo, Egypt
          ashrafsohag@yahoo.com                                   hehefny@ieee.org


Abstract— Researchers introduce in this paper, an efficient fuzzy                    Tsaur, et al [14] proposed an analytical approach to find
time series forecasting model based on fuzzy clustering to handle                the steady state of fuzzy relation matrix to revise the logic
forecasting problems and improving forecasting accuracy. Each                    forecasting process. Based on the concept of fuzziness in
value (observation) is represented by a fuzzy set. The transition                Information Theory, the concept of entropy is applied to
between consecutive values is taken into account in order to
                                                                                 measure the degrees of fuzziness when a time-invariant relation
model the time series data. Proposed model employed eight main
steps in time-invariant fuzzy time-series and time-variant fuzzy                 matrix is derived. In order to show the forecasting
time series models to increase the performance of the proposed                   performance, the best fitted regression equations are applied to
fuzzy time series model. The method of FCMI is integrated in the                 compare with the proposed method.
processes of fuzzy time series to partition datasets. The proposed
model has been implemented to forecast the world production of                        Yu [15] proposed weighted models to tackle two issues in
iron and steel and the enrollments of the University of Alabama.                 fuzzy time series forecasting; namely, recurrence and
The proposed model provide higher accuracy in forecasting. Our                   weighting. Weighted fuzzy time series models appear quite
results show that this approach can lead to satisfactory                         similar to the weight functions in local regression models;
performance for fuzzy time series                                                however, both are different. The local regression models focus
1                                                                                on fitting using a small portion of the data, while the fuzzy
                                                                                 relationships in weighted fuzzy time series models are
        Keywords- forecasting; fuzzy Clustering; fuzzy time series; iron.        established using the possible data from the whole of the
                                                                                 database.
                            I.    INTRODUCTION
                                                                                     Jilani and Burney [7] presented two new multivariate
     Traditional forecasting methods can deal with many                          fuzzy time series forecasting methods. These methods assume
forecasting cases, but they cannot solve forecasting problems in                 m-factors with one main factor of interest. Stochastic fuzzy
which the historical data are linguistic values. Song and                        dependence of order k is assumed to define general methods of
Chissom [12] presented the concept of fuzzy time series based                    multivariate fuzzy time series forecasting and control.
on the historical enrollments of the University of Alabama.
They presented the time-invariant fuzzy time series model and                         Cheng et al [4] proposed a novel multiple-attribute fuzzy
the time-variant fuzzy time series model based on the fuzzy set                  time series method based on fuzzy clustering. The methods of
theory for forecasting the enrollments of the University of                      fuzzy clustering were integrated in the processes of fuzzy time
Alabama.                                                                         series to partition datasets objectively and enable processing of
                                                                                 multiple attributes.
     The fuzzy forecasting methods can forecast the data with
linguistic values. Fuzzy time series do not need to turn a non-                      Abd Elaal et al [1-2] proposed a novel forecasting fuzzy
stationary series into a stationary series and do not require more               time series model depend on fuzzy clustering for improving
historical data along with some assumptions like normality                       forecasting accuracy. Kai et al [8] proposed a novel forecasting
postulates. Although fuzzy forecasting methods are suitable for                  model for fuzzy time series using K-means clustering
incomplete data situations, their performance is not always                      algorithm for forecasting.
satisfactory [9,11].
                                                                                     In this paper, researchers propose an efficient fuzzy time
    Huarng [6] proposed heuristic models; by integrating                         series forecasting model based on fuzzy clustering to handle
problem-specific heuristic knowledge to improve forecasting.                     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
             1
                 Corresponding Author:    Ashraf K. Abd-Elaal
                                                                                 model the time series data.



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                                                                                                            ISSN 1947-5500
                                                                                      (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                                Vol. 8, No. 8, 2010
                                   II.       RELATED WORKS                                       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 = . . . ,
In this section, two related works including: fuzzy clustering                                   0, 1, 2, . . .).
and fuzzy time series.
                                                                                                 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),
A.    Fuzzy clustering (FCMI)                                                                    where “×” represents an operation, then F(t) is said to be
       Fuzzy c-means (FCM) is a method of clustering which                                       induced by F(t − 1). The logical relationship between F(t) and
allows one piece of data to belong to two or more clusters.                                      F(t − 1) is F(t − 1) F(t).
Fuzzy C-Mean Iterative assume that: the existence of pattern
space X={x1, x2,…, xm) and c fuzzy clusters, whose centers                                       Definition 3. (FLR) suppose F(t − 1) = Ai and F(t) = Aj . The
have initial values y10, y20,…,yc0. Every iteration the                                          relationship between two consecutive observations, F(t) and F(t
membership function values updated and the cluster centers                                       − 1), referred to as a fuzzy logical relationship, can be denoted
also. The process terminates when the difference between two                                     by Ai     Aj , where Ai is called the Left-Hand Side (LHS) and
consecutive clusters centers do not exceed a given tolerance                                     Aj the Right-Hand Side (RHS) of the FLR.
[5].
                                                                                                 Definition 4. (FLRG) All fuzzy logical relationships in the
                                                                                                 training dataset can be grouped together into different fuzzy
                                d ijk ) = x j − yi( k )
                                  (
                                                                                      (1)        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
Fuzzy clustering is carried out through an iterative optimization                                (Ai ): Ai      Aj1 and Ai       Aj2. These two fuzzy logical
                                             d ij                                                relationships can be grouped into a fuzzy logical relationship
of the objective function                           , with the update of membership              group Ai Aj1 Aj2.
u ij                                          yi
       and the cluster centers                       by:                                         Definition 5. (IFTS & VFTS) Assume that F(t) is a fuzzy time-
                                                                                 −1
                                                                                                 series and F(t) is caused by F(t − 1) only, and F(t) = F(t − 1) ×
                                 c       d ij k )
                                              (
                                                          
                                                              2 /( β − 1 )
                                                                                                R(t − 1, t). For any t, if R(t − 1, t) is independent of t, then F(t)
               u   (k )
                              = ∑        (k )                                     (2)        is named a time-invariant fuzzy time-series, otherwise a time-
                   ij
                                 l =1   d                                 
                                         lj                                                  variant fuzzy time-series.
                                               m

                                             ∑j =1
                                                         (
                                                      u ij k ) x      j                               a) Song and Chissom model
                               (k +1)
                          y   i          =       m                                    (3)
                                                                                                          Song and Chissom employed five main steps in time-
                                               ∑   j =1
                                                             (
                                                          u ij k )
                                                                                                 invariant fuzzy time-series and time-variant fuzzy time series
                                                                                                 models as follows:

This iteration will stop when                                                                    Step 1: Define the universe of discourse U. Define the universe
                                                              1/ 2                               of discourse for the observations. According to the issue
                           c                         2
                          
                             ∑
                           i=1
                                 (k+1)
                                yi
                                          (k)
                                       − yi               
                                                          
                                                          
                                                                     <ε               (4)        domain, the universe of discourse for observations is defined
                                                                                                 as,
                                                         
                                                                                                               U=[Dmin – D1, Dmax + D2]                     (5)
B. Fuzzy time series                                                                                      where, Dmin is the minimum value,
                                                                                                                Dmax is the maximum value,
         Song and Chissom [13] presented the concept of fuzzy                                                  D1, D2 is the positive real numbers.
time series based on the historical enrollments of the University
of Alabama. Fuzzy time series used to handle forecasting                                         Step 2: Partition universal of discourse U into equal intervals.
problems. They presented the time-invariant fuzzy time series
model and the time-variant fuzzy time series model based on                                      Step 3: Define the linguistic terms. Each linguistic observation,
the fuzzy set theory for forecasting the enrollments of the                                      Ak can be defined by the intervals u1,u2,...,un, as follows:
University of Alabama. The definitions and processes of the
fuzzy time-series presented by Song and Chissom are described                                                         1 0.5
                                                                                                                          +                     k =1
as follows [6,12].                                                                                           0.5 u1 u2 0.5
                                                                                                                        1
                                                                                                       Ak =  uk−1 + uk + uk+1          2 ≤ k ≤ n −1                    (6)
Definition 1. (FTS) Assume Y (t) (t = . . 0, 1, 2, . . .) is a subset                                              0.5 1
                                                                                                                          +                      k =n
of a real numbers. Let Y (t) be the universe of discourse                                                         u n −1 u n




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                                                                                                                                 ISSN 1947-5500
                                                                (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                          Vol. 8, No. 8, 2010

Step 4: Fuzzify the historical data. Each historical data can be
fuzzified into a fuzzy set.                                                A. Evaluating of the proposed model

Step 5: Build fuzzy logic relationships. Build fuzzy logic                     To evaluating the performance of the proposed model, the
relationships. Two consecutive fuzzy sets Ai(t-1)and Aj(t) can             researchers compare the forecasting values of enrollments of
be established into a single FLR as Ai Aj.                                 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].
                           III.   PROPOSED MODEL
    In this section we proposed an efficient fuzzy time series                 The forecasting accuracy is compared by using (NRMSE)
forecasting model based on fuzzy clustering to handle                      Normalized Root Mean Square Error. NRMSE, in statistic is
forecasting problems and improving forecasting accuracy.                   the square root of the sum of the squared deviations between
Most researchers have been taken the same way according to                 actual and predicted values divided by the sum of the square of
processes of the fuzzy time-series, which are presented by Song            actual values.
and Chissom, but we introduce a novel model based on fuzzy
clustering to determine the membership values not as Song and                                          N

Chissom model, and to increase the performance. Proposed                                               ∑ (actual
                                                                                                       i =1
                                                                                                                     i   − predicti ) 2
model employed eight main steps in time-invariant fuzzy time-                            NRMSE =              N                                      (8)
series and time-variant fuzzy time series models as follows:                                                  ∑ (actual )
                                                                                                              i =1
                                                                                                                             i
                                                                                                                                 2



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             In this study, to evaluate the forecasting accuracy of the
most popular one and well known in fuzzy clustering field.                 proposed model, the researchers use the enrollments of the
Step 2: Determine membership values for each cluster: In                   University of Alabama as the forecasting target in the existing
this step, membership values is determining after doing fuzzy              forecasting models.
cluster. The proposed model selected the maximum
membership grade of each value for each cluster which it
belong to.                                                                    Based on the enrollments of the University of Alabama
                                                                           from 1971 to 1992, we can get the universe of discourse
Step 3: Rank each cluster: Proposed model ranking clusters                 U=[13055,19337], partition U into 7 equal intervals, D1=13,
by the center of each cluster, where first cluster has the                 and D2=55. Hence, the intervals are u1; u2; u3; u4; u5; u6; u7;
minimum center, and last cluster has the maximum center.                   where :-
Step 4: Define the universe of discourse U: In this step, the
proposed model defines the universe of discourse as Song and                                               u1=[13024.00, 13933.71]
Chissom were defined it as in (5).                                                                         u2=[13933.71, 14843.43],
Step 5: Partition universal of discourse U into equal                                                      u3=[14843.43, 15753.14],
intervals: According to this step, the proposed model, partition
the universe of discourse into c intervals.                                                                u4=[15753.14, 16662.86],
Step 6: Fuzzify the historical data: In this step, proposed                                                u5=[16662.86, 17572.57],
model fuzzufy historical data, where the proposed model
                                                                                                           u6=[17572.57, 18482.29],
determine the best fuzzy cluster to each actual data
                                                                                                           u7=[18482.29, 19392.00],
Step 7: Build fuzzy logic relationships: Proposed model in
this step build fuzzy logic relationship as definition 3. if F(t−1)            Table I lists the enrollment of the University of Alabama
= Ai and F(t) = Aj then the relationship between two                       from 1971 to 1992, and membership grades of enrollments for
consecutive observations: Ai Aj                                            each linguistic. Define the fuzzy set Ai using the linguistic
                                                                           variable "Enrollments of the University of Alabama", let A1 =
Step 8: Calculate forecasting outputs: The forecasting value
                                                                           (very very few), A2 = (very few), A3 = (few), A4 = (moderate),
for each cluster is calculated by proposed model as:
                                                                           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
                           df x X + df x X + ... + df m x X m
      forecaste ( Ai ) =     1   1 1
                                        m
                                           1                               as in (7):
                                        ∑ df j                  (7)
                                       j =1                                                        1 x (1972)
                                                                               forecaste ( A ) =        1     = 13563
                                                                                            1
     Where dfj is the membership grade,                                                            0 . 8 x (1984)
                                                                               forecaste ( A ) =          0 .8    = 15145
                                                                                            2
              Xj is the actual value.


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                                                                                                              ISSN 1947-5500
                                                                          (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                    Vol. 8, No. 8, 2010
                          1 x (1975) + 1 x (1982)
      forecaste ( A ) =              2            = 15446
                   3

                             1 x (1978)
      forecaste ( A ) =           1     = 15861                                                           TABLE II.              DATA ENROLLMENTS THE UNIVERSITY OF ALABAMA,
                   4                                                                                                         LINGUISTIC VALUES, AND FORECASTED VALUES
                           1 x (1979)
      forecaste ( A ) =         1     = 16807                                                                          Years                    Enrollments                     Linguistic                    Forecasted
                   5
                                                                                                                        1971                         13055                                A1                       13563
                           1 x (1988)
      forecaste ( A ) =         1     = 18150                                                                           1972                         13563                                A1                       13563
                   6
                                                                                                                        1973                         13867                                A1                       13563
                           1 x (1989)
      forecaste ( A ) =         1     = 18970                                                                           1974                         14696                                A2                       15145
                   7
                                                                                                                        1975                         15460                                A3                       15446
TABLE I.            DATA OF ENROLLMENTS OF THE UNIVERSITY OF ALABAMA                                                    1976                         15311                                A3                       15446
                          AND MEMBERSHIP GRADES.                                                                        1977                         15603                                A3                       15446
    Year      Actual         A1      A2      A3      A4     A5    A6     A7                                             1978                         15861                                A4                       15861
             enrollm                                                                                                    1979                         16807                                A5                       16833
    1971      13055         0.8      0.1      0       0     0     0       0
    1972        13563        1       0        0       0     0     0       0                                             1980                         16919                                A5                       16833
    1973        13867       0.9      0.1      0       0     0     0       0                                             1981                         16388                                A4                       15861
    1974        14696       0.1      0.7     0.2     0.1    0     0       0                                             1982                         15433                                A3                       15446
    1975        15460        0       0        1       0     0     0       0                                             1983                         15497                                A3                       15446
    1976        15311        0       0.1     0.9      0     0     0       0                                             1984                         15145                                A3                       15446

    1977        15603        0       0.1     0.6     0.3    0     0       0                                             1985                         15163                                A3                       15446

    1978        15861        0       0        0       1     0     0       0                                             1986                         15984                                A4                       15861
                                                                                                                        1987                         16859                                A5                       16833
    1979        16807        0       0        0       0     1     0       0
                                                                                                                        1988                         18150                                A6                       18150
    1980        16919        0       0        0       0     0.9   0       0
                                                                                                                        1989                         18970                                A7                       18970
    1981        16388        0       0       0.1     0.3    0.6   0       0
                                                                                                                        1990                         19328                                A7                       18970
    1982        15433        0       0        1       0     0     0       0
                                                                                                                        1991                         19337                                A7                       18970
    1983        15497        0       0       0.9     0.1    0     0       0
                                                                                                                        1992                         18876                                A7                       18970
    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
                                                                                                                                            Comparisons of the forecasting results of different models
    1988        18150        0       0        0       0     0     1       0                               19500

    1989        18970        0       0        0       0     0     0       1                               19000
                                                                                                          18500
    1990        19328        0       0        0       0     0     0      0.9                              18000

    1991        19337        0       0        0       0     0     0      0.9                              17500                                                                                                              Actual
                                                                                     Actual enrollments




                                                                                                          17000                                                                                                              Jilani 2008
    1992        18876        0       0        0       0     0     0.1    0.9                              16500
                                                                                                                                                                                                                             Tsaur 2005
                                                                                                                                                                                                                             Yu 2005
                                                                                                          16000                                                                                                              Cheng 2008
                                                                                                          15500                                                                                                              Kai 2010
                                                                                                                                                                                                                             Proposed
                                                                                                          15000

                                                                                                          14500
                                                                                                          14000

                                                                                                          13500

                                                                                                          13000
                                                                                                              71

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                                                                                                                                                                     Years




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




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                                                                                                                                                                ISSN 1947-5500
                                                                                                               (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                                                         Vol. 8, No. 8, 2010
                                                NRMSE for different models                                              also shows that, the proposed model can further improve the
            0.035                                                                                                       forecasting results than the other model.
             0.03



            0.025
                                                                                                                           Fig. 3 shows the comparisons among the existing models
             0.02
                                                                                                                        by using NRMSE, where Jilani and Burney [7] model has 0.02,
                                                                                                                        Tsaur and Yang [14] model has 0.025, Yu [15] model has
    NRMSE




            0.015
                                                                                                                        0.026, Kai et al [8] model has 0.024, Cheng, et al [4] model has
             0.01
                                                                                                                        0.028 and proposed model has 0.015.
            0.005



               0
                    Jilani 2008    Tsaur 2005        Yu 2005            Cheng 2008       Kai 2010   Proposed
                                                               Models




      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

                                                                   TABLE III.                 FORECASTING ENROLLMENTS OF THE UNIVERSITY OF ALABAMA
                                                                            Tsaur and                               Jilani and
                                             Actual                                                   Yu                               Cheng et al        Kai et al
                         Year                                                  Yang                                  Burney                                                Proposed
                                           enrollments                                              (2005)                               (2008)            (2010)
                                                                              (2005)                                  (2008)
                         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




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                                                                                                                                                     ISSN 1947-5500
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                         IV.     EMPIRICAL STUDY                                    TABLE IV.      DATA OF THE WORLD PRODUCTION OF IRON AND STEEL, AND
                                                                                                            MEMBERSHIP GRADES.
   Based on the data of the iron and steel production witch are
                                                                                     Year    Production     A1     A2     A3      A4     A5      A6     A7
provided by the International Iron and Steel Institute in
Brussels, Belgium, and publications of the U.S. geological                           1975       479000      1      0       0      0       0       0     0
survey from 1975 to 2008 (production values in thousand                              1976       498000      1      0       0      0       0       0     0
metric tons), we can get the universe of discourse U=[457000,
954000], partition U into 7 equal intervals, D1=6000, and                            1977       488000      1      0       0      0       0       0     0
D2=7000. Hence, the intervals are u1; u2; u3; u4; u5; u6; u7;                        1978       506000      0      0       0      0       0       0     0
where :-
                                                                                     1979       532000      0      1       0      0       0       0     0
                                                                                     1980       514000      0      0       0      0       0       0     0
                        u1=[ 451000.00, 523857.14]                                   1981       502000      1      0       0      0       0       0     0
                        u2=[ 523857.14, 596714.29],                                  1982       457000      0      0       0      0       0       0     0

                        u3=[ 596714.29, 669571.43],                                  1983       463000      0      0       0      0       0       0     0

                        u4=[ 669571.43, 742428.57],                                  1984       495000      1      0       0      0       0       0     0
                                                                                     1985       499000      1      0       0      0       0       0     0
                        u5=[ 742428.57, 815285.71],
                                                                                     1986       495000      1      0       0      0       0       0     0
                        u6=[ 815285.71, 888142.86],
                                                                                     1987       509000      0      0       0      0       0       0     0
                        u7=[ 888142.86, 961000.00],
                                                                                     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
    Figure 4.   Forecasting of the world production of iron and steel by the
                                                                                     2003       673000      0      0       0      1       0       0     0
                                proposed model
                                                                                     2004       720000      0      0       0      1       0       0     0
    Table IV lists the World Production of Iron and Steel from                       2005       802000      0      0       0      0       1       0     0
1975 to 2008, and membership grades of enrollments for each
linguistic. Define the fuzzy set Ai using the linguistic variable                    2006       881000      0      0       0      0       0       1     0
"World Production of Iron and Steel", let A1 = (very very few),                      2007       954000      0      0       0      0       0       0     1
A2 = (very few), A3 = (few), A4 = (moderate), A5 = (many), A6
                                                                                     2008       932000      0      0       0      0       0       0     1
= (many many), A7 = (too many).

                                                                                        The proposed model selected the maximum membership
    Fig. 4 and Table V show linguistic terms and forecasting                        grade for each cluster, the forecasting value for each cluster
values deduced by proposed model. The forecasting value for                         calculating as in (7):
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.




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                                                                               16                                http://sites.google.com/site/ijcsis/
                                                                                                                 ISSN 1947-5500
                                                                              (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                        Vol. 8, No. 8, 2010
                         1 x (1975) +1 x (1976) +1 x (1977) +1 x (1981) +1 x (1984) +1 x (1985) +1 x (1986) +1 x (1992)
     forecaste ( A ) =                                                  8                                               = 494875
                  1
                         1 x (1979) +1 x (1988) +1 x (1989) +1 x (1990) +1 x (1995) +1 x (1997) +1 x (1998) +1 x (1999)
     forecaste ( A ) =                                                  8                                               = 537250
                  2
                           1 x (2000) +1 x (2001) +1 x (2002)
     forecaste ( A ) =                      3                 = 588667
                  3
                          1 x (2003) +1 x (2004)
     forecaste ( A ) =               2           = 696500
                  4

                            1 x (2005)
     forecaste ( A5 ) =          1     = 802000

                            1 x (2006)
     forecaste ( A ) =           1     = 881000
                  6
                           1 x (2007) +1 x (2008)
     forecaste ( A ) =                2           = 943000
                  7



    TABLE V.       DATA OF THE WORLD PRODUCTION OF IRON AND STEEL,
               LINGUISTIC VALUES, AND FORECASTED VALUES                                      The researchers used famous models: Huarng[6], Tsaur and
                                                                                         Yang [14], Yu [15], Jilani and Burney [7] to test the proposed
            Year         Production      Linguistic      Forecasted                      model by forecasting of the world production of iron and steel
            1975           479000            A1            494875                        as in Table VI.
            1976           498000            A1            494875
            1977           488000            A1            494875
            1978           506000            A1            494875                                                                                    Comparisons of the forecasting results of different models


            1979           532000            A2            537250                                               950000


            1980           514000            A1            494875                                               850000

            1981           502000            A1            494875
                                                                                           Actual enrollments




                                                                                                                                                                                                                           Actual
            1982           457000            A1            494875                                               750000
                                                                                                                                                                                                                           Huarng 2001
                                                                                                                                                                                                                           Tsaur 2005
            1983           463000            A1            494875                                               650000
                                                                                                                                                                                                                           Yu 2005
                                                                                                                                                                                                                           Jilani 2008
            1984           495000            A1            494875                                                                                                                                                          Proposed

                                                                                                                550000
            1985           499000            A1            494875
            1986           495000            A1            494875                                               450000

            1987           509000            A1            494875
                                                                                                                         75

                                                                                                                                77

                                                                                                                                       79

                                                                                                                                              81

                                                                                                                                                     83

                                                                                                                                                     85

                                                                                                                                                            87

                                                                                                                                                                   89

                                                                                                                                                                          91

                                                                                                                                                                                 93

                                                                                                                                                                                 95

                                                                                                                                                                                        97

                                                                                                                                                                                               99

                                                                                                                                                                                                      01

                                                                                                                                                                                                             03

                                                                                                                                                                                                             05

                                                                                                                                                                                                                    07
                                                                                                                       19

                                                                                                                              19

                                                                                                                                     19

                                                                                                                                            19

                                                                                                                                                   19

                                                                                                                                                   19

                                                                                                                                                          19

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                                                                                                                                                                        19

                                                                                                                                                                               19

                                                                                                                                                                               19

                                                                                                                                                                                      19

                                                                                                                                                                                             19

                                                                                                                                                                                                    20

                                                                                                                                                                                                           20

                                                                                                                                                                                                           20

                                                                                                                                                                                                                  20
                                                                                                                                                                           Years
            1988           539000            A2            537250
            1989           546000            A2            537250
            1990           531000            A2            537250                        Figure 5.                                   Forecasting results curve of the world production of iron and steel
            1991           509000            A1            494875
            1992           503000            A1            494875
            1993           507000            A1            494875                                                                                                       NRMSE for different models


            1994           516000            A1            494875                                               0.07


            1995           536000            A2            537250                                               0.06

            1996           516000            A1            494875
                                                                                                                0.05
            1997           540000            A2            537250
            1998           535000            A2            537250                                               0.04
                                                                                          NRMSE




            1999           539000            A2            537250                                               0.03


            2000           573000            A2            537250
                                                                                                                0.02
            2001           585000            A2            537250
            2002           608000            A3            588667                                               0.01



            2003           673000            A4            696500                                                 0
                                                                                                                               Huarng 2001                Tsaur 2005                  Yu 2005                Jilani 2008   Proposed
            2004           720000            A4            696500                                                                                                                     Models


            2005           802000            A5            802000
            2006           881000            A6            881000                                                      Figure 6.              NRMSE-chart for the existing models and the proposed
            2007           954000            A7            943000
            2008           932000            A7            943000



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                                                                                    17                                                                                         http://sites.google.com/site/ijcsis/
                                                                                                                                                                               ISSN 1947-5500
                                                            (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                      Vol. 8, No. 8, 2010
    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                     Fig. 6 shows the comparisons among the existing
also shows that, the proposed model can further improve the            models by using NRMSE, where Huarng[6] model has 0.0496,
forecasting results than the other model.                              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




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                                                                  18                              http://sites.google.com/site/ijcsis/
                                                                                                  ISSN 1947-5500
                                                                          (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                                    Vol. 8, No. 8, 2010
                                                                                       [10]   H.-T. Liu, "An improved fuzzy time series forecasting method using
                  V.      DISCUSSION AND CONCLUSION                                           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
    The research proposed an efficient fuzzy time series                                      forecasting theory and engineering applications", Springer-
forecasting model based on fuzzy clustering with high                                         Verlag.London, UK, 2005, p.18.
accuracy. The method of FCMI is integrated in the processes of                         [12]   Q. Song and B.S. Chissom, "Forecasting enrollments with fuzzy time
fuzzy time series to partition datasets. Experimental results of                              series. I", Fuzzy sets and systems, vol. 54, 1993, pp. 1-9.
                                                                                       [13]   Q. Song and B.S. Chissom, "New models for forecasting enrollments:
enrollments of the University of Alabama, and the comparison                                  fuzzy time series and neural network approaches", ERIC, 1993 p. 27,
between the existing models: Jilani and Burney [7], Tsaur and                                 http://www.eric.ed.gov
Yang [14], Yu [15], Kai et al [8], and Cheng, et al [4] and the                        [14]   R.-C. Tsaur, J.-C. Yang, and H.-F. Wang, "Fuzzy relation analysis in
proposed model show that, the proposed model can further                                      fuzzy time series model", Computers and Mathematics with
improve the forecasting results than the other models and also                                Applications, vol.49, 2005, pp. 539-548.
the experimental results of the world production of iron and                           [15]   H.-K. Yu, "Weighted fuzzy time series models for TAIEX
                                                                                              forecasting", Physica A, vol.349, 2005, pp.609–624.
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                                                         AUTHORS PROFILE
higher accuracy than the other models.
                                                                                                        Mr. Ashraf Khalaf Abd Elaal is a Ph.D. student in
                                                                                                        Computer Sciences Department at the Institute of Statistical
                                                                                                        Studies and Research, Cairo University. His Ph.D. in the
                                                                                                        filed of Computational Intelligence. His research interests
                             VI.    REFERENCES                                                          include fuzzy time series, Fuzzy clustering
    [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.                                                                                Dr. Hesham Ahmed Hefny is an assistant professor and the
    [2]   A. K. Abd Elaal, H.A. Hefny, and A. H. Abd-Elwahab, "Constructing                           head of Computer & Information Sciences Department at
          Fuzzy Time Series Model Based on Fuzzy Clustering for a                                     the Institute of Statistical Studies and research, Cairo
          Forecasting", J. Computer Sci., vol. 7, 2010, pp. 735-739.                                  University. His research interests include Artificial Neural
    [3]   T.-L. Chen, C.-H. Cheng, and H.-J. Teoh, "High-order fuzzy time-                            Networks, Fuzzy Systems, Genetic Algorithms, Swarm
          series based on multi-period adaptation model for forecasting stock                         Intelligence, Pattern Recognition, and Data Mining. Dr.
          markets", Physica A, vol.387, 2008, pp. 876–888                                 Hesham has published over 35 peer refereed papers in academic journals
    [4]   C.-H. Cheng, J.-W. Wang, and G.-W. Cheng, "Multi-attribute fuzzy                and conferences on topics within Artificial Intelligence and related areas.
          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                                Prof. Ashraf Hassan Abdelwahab is a professor of
          statistical, structural, neural and fuzzy logic approaches", Imperial                        computer engineering, Electronics Research Institute,
          college press, London, 1999, p. 329.                                                         Cairo, Egypt. He received his M. Sc. in 1988, Faculty of
    [6]   K. Huarng, "Effective lengths of intervals to improve forecasting in                         Engineering, Cairo University in the area of Artificial
          fuzzy time series", Fuzzy Sets and Systems, vol.123, 2001, pp. 387–                          Intelligence, and in 1992 he received his Ph.D. in Machine
          394.                                                                                         Learning and Evolutionary Algorithms. He has published
    [7]   T.A. Jilani and S. Burney, "Multivariate stochastic fuzzy forecasting           over 60 technical papers in National, and International journals and
          models", Expert Systems with Applications, vol.35, 2008, pp. 691–               conferences in the areas of Evolutionary Algorithms, Machine Learning,
          700.                                                                            and Data Mining.
    [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.




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