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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. . 11 http://sites.google.com/site/ijcsis/ 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 . 12 http://sites.google.com/site/ijcsis/ 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. . 13 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 (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 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 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 . 14 http://sites.google.com/site/ijcsis/ 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 . 15 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 8, 2010 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. . 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 19 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 . 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 . 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. . 19 http://sites.google.com/site/ijcsis/ ISSN 1947-5500

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