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A new approach for determining the length of intervals for fuzzy time series
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  A new approach for determining the length of intervals for fuzzy time series Ufuk Yolcu a, *, Erol Egrioglu a , Vedide R. Uslu a , Murat A. Basaran b , Cagdas H. Aladag c a Department of Statistics, University of Ondokuz Mayıs, Samsun 55139, Turkey b Department of Mathematics, Nigde University, Nigde 51200, Turkey c Department of Statistics, University of Hacettepe, Ankara 06800, Turkey 1. Introduction Fuzzy set theory was introduced by Zadeh [13]. Since then, fuzzy set has been adopted in many applications such asregression, time series forecasting and ext. Fuzzy time serieswas firstly introduced by Song and Chissom [9–11]. All of these studies have been inspired by knowledge presented in the paper[14,15]. Fuzzy time series has been widely studied for recent yearsfor the aim of forecasting. The traditional time series approachesrequire having the linearity assumption and at least 50 observa-tions.Infuzzytimeseriesapproaches,thereisnotonlyalimitationfor the number of observations but also there is no need for thelinearity assumption. The fuzzy time series approaches consist of threesteps.Thefirststepisthefuzzificationofobservations.Inthesecond step, fuzzy relationships are established and the defuzzi-fication is done in the third step. In recent years, researchers havebeen doing many studies to improve and explore all of these threesteps. Chen’s method consists of considerably simple calculations.The method proposed by Chen [1] is easier than the one Song andChissom suggested. The two of other contributions in establishingfuzzy relationships were done by [6,7]. In Hurang’s paper [2], it has been investigated that theidentification of the length of intervals in the fuzzification stageaffects the performance of the fuzzy time series approach inforecasting. However, the length of intervals has been chosenarbitrarily in many research papers [1,3,8,10,11]. In order to approach the problem computationally, Huarng proposed twonovel approaches which are based on the average and thedistribution [2]. Some other studies in the step of the fuzzificationwereconductedby[5,12].Huarng[4]suggestedadifferentmethod which is called ratio-based lengths of intervals. The methodcompared to the others of which the length is chosen arbitrarilyhas generated more accurate forecasts for enrollment, inventorydemandand TAIEXstock price data. In theratio-based approachof Hurang [4], intensive calculations such as relative difference,cumulative distributions of relative differences are done for thecertainnumbersoftheratiovalues.Theratioswereidentifiedonlyfor the considered sample percentiles.We proposed a novel approach to determine the length of interval in order to obtain more accurate forecasts in fuzzy timeseries. The proposed approach is based on a single variableconstrainedoptimization. Weimproved Huarng’s method[4] withour approach. The advantages of the proposed approach can besummarized as follows:   There is no need to do some calculations such as relativedifference, cumulative distributions of relative differences.   This approach is more comprehensive than Huarng’s met-hod because proposed method implementing optimiza-tion determines the ratio, which leads to the best forecasts. Applied Soft Computing 9 (2009) 647–651 A R T I C L E I N F O  Article history: Received 23 July 2007 Received in revised form 12 February 2008 Accepted 11 September 2008 Available online 20 September 2008 Keywords: Fuzzy time seriesForecastingLength of intervalOptimizationFuzzy sets A B S T R A C T In the implementations of fuzzy time series forecasting, the identification of interval lengths has animportant impact on the performance of the procedure. However, the interval length has been chosenarbitrarily in many papers. Huarng developed a new approach which is called ratio-based lengths of intervalsin order to identifythe length ofintervals. Inourpaper, wepropose anew approach which usesasingle-variableconstrainedoptimizationtodeterminetheratioforthelengthofintervals.Theproposedapproach is applied to the two well-known time series, which are enrollment data at The University of Alabama and inventory demand data. The obtained results are compared to those of other methods. Theproposed method produces more accurate predictions for the future values of used time series.   2008 Elsevier B.V. All rights reserved. * Corresponding author. E-mail address:  uyolcu@omu.edu.tr (U. Yolcu). Contents lists available at ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc 1568-4946/$ – see front matter    2008 Elsevier B.V. All rights reserved.doi:10.1016/j.asoc.2008.09.002  However, Huarng’s method employs just a few numbers of selected ratios.   It has been observed that the forecasting accuracies for the twowell-known data sets were significantly improved when theproposed method is employed.TheproposedmethodwasappliedtotheenrollmentdataatTheUniversity of Alabama and the inventory demand used in [4] timeseriesandtheresultswerecomparedtothoseobtainedbyChen[1]and Huarng’s methods [4]. Huarng [4] compared his approach, which is ratio-based lengths of intervals, with other proposedmethod to show how accurate his method produces forecastingresults. Huarng showed that his method outperformed the othermethods. Therefore, we compare our proposed method withHuarng’s method in order to show the efficiency of the proposedmethod. It has been investigated that our proposed methodconsiderably improves the forecasting performance.In Section 2, the general knowledge about fuzzy time series isgiven.InSections3and4Chen’salgorithmandHuarng’smethodarepresented, respectively. Section 5 introduces the proposed method.In Section 6, the resultsobtained byapplying our proposed methodto the enrollment data and the inventory demand data arepresented. In final section, the results are discussed. 2. Fuzzy time series The definition of fuzzy time series was firstly introduced by[9,10]. In contrast to conventional time series procedures, varioustheoretical assumptions do not need to be checked in fuzzy timeseries approach. The most important advantage of fuzzy timeseries approach is to be able to work with a very small set of dataand not to require the linearity assumption. General definitions of fuzzy time series are given as follows:Let  U   be the universe of discourse, where  U   = { u 1 ,  u 2 ,  . . . ,  u b }. Afuzzy set  A i  of   U   is defined as  A i  ¼  f   A i ð u 1 Þ = u 1  þ  f   A i ð u 2 Þ = u 2  þ    þ  f   A i ð u b Þ = u b , where  f   A i  is the membership function of the fuzzy set  A i ;  f   A i  :  U  ! ½ 0 ; 1  . u a isagenericelementoffuzzyset  A i ;  f   A i ð u a Þ isthedegree of belongingness of   u a  to  A i ;  f   A i ð u a Þ 2 ½ 0 ; 1   and 1     b . Definition1.  Fuzzy time series. Let  Y  ( t  ) ( t   = . . . , 0, 1, 2,  . . . ), a subsetof real numbers, be the universe of discourse by which fuzzy sets  f   j ( t  ) are defined. If   F  ( t  ) is a collection of   f  1 ( t  ),  f  2 ( t  ),  . . .  then  F  ( t  ) iscalled a fuzzy time series defined on  Y  ( t  ). Definition 2.  Fuzzy time series relationships assume that  F  ( t  ) iscaused only by  F  ( t     1), then the relationship can be expressed as: F  ( t  ) =  F  ( t     1)* R ( t  ,  t     1), which is the fuzzy relationship between F  ( t  ) and  F  ( t     1), where  *  represents as an operator. To sum up, let F  ( t     1) =  A i  and  F  ( t  ) =  A  j . The fuzzy logical relationship between F  ( t  ) and  F  ( t     1) can be denoted as  A i  !  A  j  where  A i  refers to theleft-hand side and  A  j  refers to the right-hand side of the fuzzylogical relationship. Furthermore, these fuzzy logical relationshipscan be grouped to establish different fuzzy relationships. 3. Related research Many researches on improving procedures for forecasting of fuzzy time series have been recently made. One of those was doneby [1]. The procedure proposed by Chen [1] is simpler than the method proposed by Song and Chissom [9,10]. In Chen’s method, matrix calculations are not necessary. The other considerablestudy was presented by Huarng [2]. Huarng pointed out that it is avery critical to decide on what the interval length will be for theforecasting accuracy. In his latter study, Huarng [4] found thatwhen a time series has an rapidly increasing trend, the increasedinterval length should be used instead of a fixed length. Also,Huarng proposed a ratio-based approach which enables agradually increase interval length. In this section, the proceduresproposed by Chen [1] and Huarng [4] which are used to forecast a fuzzy time series are summarized, respectively.  3.1. Chen’s method Chen[1]hasimprovedtheapproachgivenbySongandChissom[9,10]. The method proposed by Chen [1] which uses a simple operation, instead of complex matrix operations, in the step of establishing fuzzy relationships. The algorithm of Chen’s method[1] is given as follows: Step 1 . Define the universe of discourse and intervals for rulesabstraction.Based on the issue domain, the universe of discourse can bedefined as:  U   = [starting, ending]. As the length of intervals isdetermined,  U   can be partitioned into several equally lengthintervals. Step 2 .Definefuzzysets based on the universe ofdiscourse andfuzzify the historical data. Step 3 . Fuzzify observed rules. Step 4 . Establish fuzzy logical relationships and group thembased on the current states of the data of the fuzzy logicalrelationships.For example,  A 1  !  A 2 ,  A 1  !  A 1 ,  A 1  !  A 3 , can be grouped as:  A 1  !  A 2 ,  A 3  !  A 1 . Step 5 . Forecast.Let  F  ( t     1) =  A i Case 1.  There is only one fuzzy logical relationship in the fuzzylogical relationship sequence. If   A i  !  A  j , then  F  ( t  ), forecast value, isequal to  A  j . Case 2.  If   A i  !  A i ,  A  j ,  . . . ,  A k , then  F  ( t  ), forecast value, is equal to  A i ,  A  j ,  . . . ,  A k . Step 6 . Defuzzify.Apply‘‘Centroid’’methodtogettheresults.Thisprocedure(alsocalled center of area, center of gravity) is the most often adoptedmethod ofdefuzzification.Supposethatthefuzzyforecast of   F  ( t  )is  A k . The defuzzified forecast is equal to the midpoint of the intervalwhich corresponds to  A k .  3.2. Huarng’s method Huarng [2] has pointed out that effective length of intervalsmay improve the forecasting performance. In order to provideconcretesolution, the intervallengthbased on theaverage and thedistribution is introduced by Huarng. Then, Huarng [4] improvedpreviousresultswiththemethodcalledtheratio-basedlengthasanew approach. The algorithm for Huarng’s ratio-based lengthapproach can be summarized as follows: Step 1 . Take the absolute differences between any twoconsecutive observations  j  x i    x i  1 j  for any  x i  and  x t   1 . Step2 .Calculatearelative difference r  t   =  j  x t     x t   1 j /  x t   1 forall t  . Step 3 . Determine the base by mapping MIN( r  1 ,  . . . ,  r  n  1 ) to thebase table in Huarng [4]. Step4 .Plotthecumulativedistributionforall r  t  accordingtothebase determined in step 3. Step 5 . Determine a ratio sample percentile a . Choose the ratioas the smallest relative difference that is larger than at leastproportion of all  r  t  . U. Yolcu et al./Applied Soft Computing 9 (2009) 647–651 648  Step6 .Calculatetheintervals.Theinitialvalueisdeterminedasin[4].Theintervalsarecalculatedbytheformulasgivenbelow:upper 0  ¼ initialFor  j  1lower  j  ¼ upper  j  1 upper  j  ¼ð 1 þ ratio Þ  j  upper 0 interval  j  ¼½ lower  j ; upper  j  : (1) Step7 .Definefuzzysetsbasedontheuniverseof discourseandfuzzify the historical data. Step 8 . Fuzzify observed rules. Step 9 . Establish fuzzy logical relationships and group thembased on the current states of the data of the fuzzy logicalrelationships.For example,  A 1 !  A 2 ,  A 1 !  A 1 ,  A 1 !  A 3 , can be grouped as:  A 1 !  A 2 ,  A 3 ,  A 1 . Step 10 . Forecast.Let  F  ( t   1) =  A i Case 1.  There is only one fuzzy logical relationship in the fuzzylogical relationship sequence. If   A i !  A  j , then  F  ( t  ), forecast value, isequal to  A  j . Case 2.  If   A i !  A i ,  A  j ,  . . . ,  A k , then  F  ( t  ), forecast value, is equal to  A i ,  A  j ,  . . . ,  A k . Step 11 . Defuzzify.Apply‘‘Centroid’’methodtogettheresults.Thisprocedure(alsocalled center of area, center of gravity) is the most often adoptedmethodofdefuzzification.Supposethatthefuzzyforecastof  F  ( t  )is  A k . The defuzzified forecast is equal to the midpoint of the intervalwhich corresponds to  A k . 4. A new method to obtain more efficient forecasting  There will be no fluctuations in the fuzzy time series when aneffective length of intervals is too large. On the other hand, themeaningoffuzzytimeserieswillbediminishedwhenthelengthistoo small [2]. Therefore, the effective lengths of intervals are animportant issue for improving fuzzy time series forecasting. In theratio-based length approach, a ratio is determined with respect tothe sample percentile  a  which is defined by the cumulativedistribution of relative differences. The length of intervals andconsequentlytheperformanceoffuzzytimeserieschange,while a changes the ratio. Furthermore, for very close values of   a , theperformance remains unchanged. For example, Table 6 in [4] shows that the ratio corresponding to the value  a = 0.45 and 0.50wasnotchangedandfoundas0.023.Finally,thesamplepercentile a  was intuitively set to 50%. In our study, we observed that for  a values between 0.40 and 0.60, the ratio values changed between0.02 and 0.04. However, taking the value of the ratio less than 0.02or greater than 0.04 might cause to change the length of intervals.Consequently, the performance of forecasting might change. Thenthe importance of our proposed approach can be summarized asfollows:The proposed method directly optimizes the ratio. It provides aratio in a wide range. Therefore, forecasting performance isimproved. Furthermore, our method allows avoiding calculatingrelative differences, the cumulative distributions and determiningintuitively  a  value. Time series is divided into two parts as in [4].First part is used for estimation (training) purposes while thesecond part is used for testing. The proposed method was appliedto the training data to make estimation and then the predictedvalues were found for the test data. Finally, the roots of meansquareerror(RMSE)valuesareobtainedfromtheactualvaluesandthe predicted values. In order to optimize the ratio which isrequired for identifying intervals, a MATLAB function called‘‘fminbnd’’ is used. This function makes RMSE minimum for0  <  ratio  <  0.2. When the ratio value is greater than 0.2, the lengthof interval will be too large, so that the number of intervals willdecrease. It results in diminishing the fluctuation in fuzzy timeseries [2]. Thus, 0.2 is taken as an upper bound in the optimizationprocedure. Then theproblem of optimizationcan be formulated asfollows:min RMSE ð ratio Þ Subject to  :  ratio 2ð 0 ; 0 : 2   (2)where RMSE is a function of the ratio. To calculate RMSE for theratiovalueobtainedineachstepof theoptimization,thefollowingalgorithm is given below. 4.1. The algorithm of the proposed method Step 1 . Define the universe of discourse and intervals for rulesabstraction.Based on the issue domain, The universe of discourse can bedefined as:  U   = [starting, ending].  U   can be partitioned into severallength intervals using equations in (1). Step2 . Definefuzzysetsbasedontheuniverseofdiscourseandfuzzify the historical data. Step 3 . Fuzzify observed rules. Step 4 . Establish fuzzy logical relationships and group thembased on the current states of the data of the fuzzy logicalrelationships.For example,  A 1 !  A 2 ,  A 1 !  A 1 ,  A 1 !  A 3 , can be grouped as:  A 1 !  A 2 ,  A 3 ,  A 1 . Step 5 . Forecast.Let  F  ( t   1) =  A i Case 1.  There is only one fuzzy logical relationship in the fuzzylogical relationship sequence. If   A i !  A  j , then  F  ( t  ), forecast value, isequal to  A  j . Case 2.  If   A i !  A i ,  A  j ,  . . . ,  A k , then  F  ( t  ), forecast value, is equal to  A i ,  A  j ,  . . . ,  A k . Step 6 . Defuzzify.Apply‘‘Centroid’’methodtogettheresults.Thisprocedure(alsocalled center of area, center of gravity) is the most often adoptedmethodofdefuzzification.Supposethatthefuzzyforecastof  F  ( t  )is  A k . The defuzzified forecast is equal to the midpoint of the intervalwhich corresponds to  A k . Step 7 . Calculate RMSE. 5. Application In order to show what we have achieved from the proposedapproach, we applied it to the enrollment and the inventorydemand data and then have compared the obtained results withthose obtained from Chen [1] and Huarng’s method [4]. The first application was on yearly data on enrollments at the University of Alabama. Years and the corresponding enrollment observationsare listed in the first and the second column of  Table 1,respectively. The enrollment observations from 1971 to 1988were used for the estimation, while the observations from 1989 to1992 were used as a test data.Huarng [4] obtained the forecasts of the test data as he did forthe forecasts of the training data just to be able to increase theforecasting accuracy. That is, the forecast for the second value of the test data was obtained by using the previous observed value U. Yolcu et al./Applied Soft Computing 9 (2009) 647–651  649  (the firstobservationofthetest data)insteadofusingthepreviousforecastvalue.Itwouldbeactuallyconsideredasifwehadnothadthe observations of the test data when RMSE was calculated. Wedid the same analysis as it should be to see the results. The resultsfrom this analysis are presented in Tables 2 and 8. In Huarng’smethod, absolute differences are calculated firstly then thecalculation of relative differences is conducted. According to theminimum value of the relative differences, a base value isdetermined from the base table. The cumulative distributiongraphsforallrelativedifferences,whicharedependentonthebasevalue, are drawn. Then a ratio sample percentile  a  is determinedandtheratioasthesmallestrelativedifferencethatislargerthanatleast proportion of all  r  t   is chosen. All of these calculations taketime. Also, just only one ratio value for the sample percentile  a  isdetermined.According to the results of Huarng’s method, the best RMSEvalue which is 750 is obtained for  a = 0.55 and ratio = 0.032. Ontheotherhand,forChen’smethod,thebestRMSEvalueis500withtheintervallengthof500.Theproposedmethodproduces208forRMSE with ratio = 0.1089. When we compared three methods, itis observed that the proposed method produces the smallestRMSE value. The ratio which gave the minimum RMSE was foundas 0.1089 in the optimization problem given in (2) solved byMATLAB. For this ratio, the universe of discourse was defined as[12000,20120] according to the algorithm given in Section 5. Theintervals were finally defined in Table 3.Based on these intervals, the observations are fuzzified andpresented in Table 4. Table 5 shows the fuzzy relationships. The fuzzified forecasts for the training and the test data arepresented in Table 6.Afterdefuzzification,theobtainedcrispforecastsareshowedinTable 7. For example;thefuzzifiedforecasting value fortheyearof 1972 is  A 2 . In  A 2 , the interval which has got the greatestmembership is  u 2  = [13306.8, 14755.9]. The defuzzified forecastfor the year of 1972 is the midpoint of the interval of   u 2 . It implies14031.35. The forecasts of all years can be similarly found(Table 8).Secondly,weappliedthemethodtotheinventorydemanddatapresented in Table 8. The mentioned data were used in [4]. Data from1to18wereusedfortheestimationwhilethedatafrom19to24 were used for the forecasting purposes. The results from thisapplication are listed in Table 9.As it can be seen from Table 9, the smallest value for RMSE isobtained from the method that weproposed.ForHuarng’smethod,the ratio was calculated as 0.0770 for a = 0.60 and the correspond-ing RMSE value is 32.96. As we look at the results from Chen’smethod, it is seen that the best RMSE value was found as 21.32 atthe interval length 10. When all results obtained from threemethodsarecompared,thebestRMSEvalueis19.95obtainedfromour proposed method. Then the corresponding ratio is 0.1701.  Table 1 The enrollment data at The University of Alabama.Years Actual1971 13,0551972 13,5631973 13,8671974 14,6961975 15,4601976 15,3111977 15,6031978 15,8611979 16,8071980 16,9191981 16,3881982 15,4331983 15,4971984 15,1451985 15,1631986 15,9841987 16,8591988 18,1501989 18,9701990 19,3281991 19,3371992 18,876  Table 2 The results of the analysis for the enrollment data.Chen’s method Huarng’s method Proposed methodInterval length RMSE  a  Ratio RMSE Ratio RMSE100 509 0.40 0.020 1089 0.1089 208200 529 0.45 0.023 815500 500 0.50 0.023 815600 511 0.55 0.032 7501000 575 0.60 0.039 1176  Table 3 Five leveled linguistic values for the enrollment data. u 1  = [12000.0, 13306.8] u 2  = [13306.8, 14755.9] u 3  = [14755.9, 16362.8] u 4  = [16362.8, 18144.7] u 5  = [18144.7, 20120.0]  Table 4 Data set for fuzzy enrollment.Years Actual Fuzzy set1971 13,055 A 1 1972 13,563 A 2 1973 13,867 A 2 1974 14,696 A 2 1975 15,460 A 3 1976 15,311 A 3 1977 15,603 A 3 1978 15,861 A 3 1979 16,807 A 4 1980 16,919 A 4 1981 16,388 A 4 1982 15,433 A 3 1983 15,497 A 3 1984 15,145 A 3 1985 15,163 A 3 1986 15,984 A 3 1987 16,859 A 4 1988 18,150 A 5 1989 18,970 A 5 1990 19,328 A 5 1991 19,337 A 5 1992 18,876 A 5  Table 5 Enrollment fuzzy logical relationships.A 1 ! A 2  A 3 ! A 4 A 2 ! A 2  A 4 ! A 3 A 2 ! A 3  A 4 ! A 4 A 3 ! A 3  A 4 ! A 5  Table 6 The groups of enrollment fuzzy logical relationships.A 1 ! A 2 A 2 ! A 2 , A 3 A 3 ! A 3 , A 4 A 4 ! A 3 , A 4 , A 5 U. Yolcu et al./Applied Soft Computing 9 (2009) 647–651 650  6. Conclusion The approaches of the fuzzy time series are recently gettingquiet popular. Traditional time series analyses assume thatthe number of observations should be greater than 50 and thestructure of time series should be linear. Fuzzy time seriesapproachesdonotrequiresuchassumptions.Althoughthismakesfuzzy approaches very attractive, there are still problems thatneed to be solved. One of these problems is to determine thelengths of intervals. The decision on what the lengths will be isvery important for forecasting accuracy.In order to solve this problem, Huarng proposed a methodin which the lengths of the intervals increases based on a ratio.This approach requires many calculations to determine theratio. Also, it is chosen among only few specified values. Ourproposed method differs than the method proposed by Huarng[4] in terms of determination of ratio. In the proposed method,we use an optimization algorithm to find the ratio which leadsto the smallest RMSE. Calculations in Huarng’s method [4] suchas relative differences, cumulative distributions of relativedifferences and the sample percentile  a  are avoided by usingthe proposed method. Namely, this makes our method com-putationally simpler. Also, the number of the intervals canbe controlled because values of the ratio are restricted in aninterval.In Huarng’s method, the nature of determining ratio relies onspecified a valuessothisrestrictstoexaminethelimitednumberof ratios. In our method, the ratio from the determined interval iscomputed by using constrained optimization. Therefore, therestriction imposed by using the limited numbers of ratios isovercome. In order to show the efficiency of the proposedmethod, two well-known data sets, which are the enrollmentdata at The University of Alabama and the inventory demanddata used by Huarng [4], are examined based on the methodsproposed by Chen [1], Huarng [4] and our method for forecasting accuracy. The proposed method was compared with Huarng’method [4] since Huarng had showed that his method out-performed the other methods [4]. As a result, it is obviouslyobserved that the proposed method produces the lowest RMSEvalues in all two employed data sets. References [1] S.M. Chen, Forecasting enrollments based on fuzzy time-series, Fuzzy Sets andSystems 81 (1996) 311–319.[2] K. Huarng, Effective length of intervals to improve forecasting in fuzzy time-series, Fuzzy Sets and Systems 123 (2001) 387–394.[3] K. Huarng, Heuristic models of fuzzy time series for forecasting, Fuzzy Sets andSystems 123 (3) (2001) 369–386.[4] K. Huarng, Ratio-based lengths of intervals to improve fuzzy time series fore-casting, IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics36 (2006) 328–340.[5] K. Huarng, H. Yu, A dynamic approach to adjusting lengths of intervals in fuzzytime series forecasting, Intelligent Data Analysis 8 (1) (2004) 3–27.[6] K. Huarng, H. Yu, A type 2 fuzzy time series model for stock index forecasting,Physica A 353 (2005) 445–462.[7] K. Huarng, H. Yu, The application of neural networks to forecast fuzzy time series,Physica A 363 (2006) 481–491.[8] J.R.Hwang, S.-M. Chen,C.-H. Lee, Handlingforecastingproblemsusing fuzzytimeseries, Fuzzy Sets and Systems 100 (1998) 217–228.[9] Q.Song,B.S.Chissom,Fuzzytimeseriesanditsmodels,FuzzySetsandSystems54(1993) 269–277.[10] Q.Song,B.S.Chissom,Forecastingenrollmentswithfuzzytimeseries.PartI,FuzzySets and Systems 54 (1993) 1–10.[11] Q. Song, B.S. Chissom, Forecasting enrollments with fuzzy time series. Part II,Fuzzy Sets and Systems 62 (1994) 1–8.[12] H. Yu, A refined fuzzy time series model for forecasting, Physica A 346 (2005)57–681.[13] L.A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) 338–353.[14] L.A. Zadeh, Outline of a new approach to the analysis of complex systems anddecisionsprocesses,IEEETransactionsonSystems,Man,andCybernetics3(1973)28–44.[15] [a] L.A. Zadeh, The concept of a linguistic variable and its application to approx-imate reasoning, Part 1, Information Sciences 8 (1975) 199–249;[b] L.A. Zadeh, The concept of a linguistic variable and its application to approx-imate reasoning, Part 2, Information Sciences 8 (1975) 301–357;[c] L.A. Zadeh, The concept of a linguistic variable and its application to approx-imate reasoning, Part 3, Information Sciences 9 (1975) 43–80.  Table 7 Forecasts for the enrollment data.Years Actual Forecasts1971 13,055 –1972 13,563 14031.351973 13,867 14795.361974 14,696 14795.361975 15,460 14795.361976 15,311 16406.571977 15,603 16406.571978 15,861 16406.571979 16,807 16406.571980 16,919 17315.291981 16,388 17315.291982 15,433 17315.291983 15,497 16406.571984 15,145 16406.571985 15,163 16406.571986 15,984 16406.571987 16,859 16406.571988 18,150 17315.291989 18,970 19132.791990 19,328 19132.791991 19,337 19132.791992 18,876 19132.79  Table 8 The inventory demand data.Time Data1 1432 1523 1614 1395 1376 1747 1428 1419 16210 18011 16412 17113 20614 19315 20716 21817 22918 22519 20420 22721 22322 24223 23924 266  Table 9 The results of the analysis for the inventory demand data.Chen’s method Huarng’s method Proposed methodInterval length RMSE  a  Ratio RMSE Ratio RMSE10 21.32 0.40 0.0670 34.06 0.1701 19.9530 33.49 0.45 0.0670 34.0650 28.26 0.50 0.0670 34.0680 42.18 0.55 0.0670 34.06100 24.57 0.60 0.0770 32.96 U. Yolcu et al./Applied Soft Computing 9 (2009) 647–651  651
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