Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 927394 11 pageshttpdxdoiorg1011552013927394

Research ArticleHigh-Order Fuzzy Time Series Model Based onGeneralized Fuzzy Logical Relationship

Wangren Qiu12 Xiaodong Liu2 and Hailin Li3

1 Department of Information Engineering Jingdezhen Ceramic Institute Jingdezhen 333001 China2 Research Center of Information and Control Dalian University of Technology Dalian 116024 China3 Institute of Systems Engineering Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Wangren Qiu wangrenqiucngmailcom

Received 22 July 2013 Accepted 17 August 2013

Academic Editor Zhongmei Zhou

Copyright copy 2013 Wangren Qiu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In view of techniques for constructing high-order fuzzy time series models there are three methods which are based on advancedalgorithms computational methods and grouping the fuzzy logical relationships respectivelyThe last kindmodel has been widelyapplied and researched for the reason that it is easy to be understood by the decision makers To improve the fuzzy time seriesforecastingmodel this paper presents a novel high-order fuzzy time series models denoted asGTS(MN) on the basis of generalizedfuzzy logical relationships Firstly the paper introduces some concepts of the generalized fuzzy logical relationship and an operationfor combining the generalized relationshipsThen the proposedmodel is implemented in forecasting enrollments of the Universityof Alabama As an example of in-depth research the proposed approach is also applied to forecast the close price of ShanghaiStock Exchange Composite Index Finally the effects of the number of orders and hierarchies of fuzzy logical relationships on theforecasting results are discussed

1 Introduction

In the last two decades fuzzy time series approach [1ndash3] hasbeen widely used for its power of dealing with impreciseknowledge variables in decision making Many studies havebeen made to propose new methods or improve forecastingaccuracy for fuzzy time series forecasting For simplifying thecomputational process Chen [4] improved Songrsquos methodsand presented a simplified forecasting model in 1996 Sincethe lengths of intervals greatly affect forecasting accuracy infuzzy time series Yu and many others [5ndash10] adjusted thelengths of intervals by the distribution or the optimizationtechnique In view of higher accuracy of forecasting resultsthe weighted models concerned with the various recurrencesand on chronological order had also been improved [11ndash15] In addition many models based on the conventionalfuzzy time series were combined with novel algorithms ortechnologies For example Singh [16ndash18] proposed somemethods to forecast the crop production based on compu-tational method with different parameters Lee et al [19ndash22] presented several models based on the fuzzy time series

genetic algorithm simulated annealing algorithm and type-2 fuzzy set to forecast temperature and TAIFEX Kuo [23ndash25] firstly introduced the particle swarm optimization (PSO)into the fuzzy time series models for forecasting TAIFEXSongrsquos [3] and Aladagrsquos models [26 27] gained more accurateforecasts by employing artificial neural network to determinefuzzy relationships

Although the first-order fuzzy time series models havea simple structure they are easy to encounter troubleon explaining more complex relationships And the first-order models are not able to meet the demand of fore-casting involved in multifactors or longterm time series Ascompared with the alternative forecasting models such asARIMA Hidden Markov and ARCH models there is stillmuch room for higher forecasting accuracy in applying fuzzytime series models For these reasons Chen et al [28ndash32]proposed some new methods which applied a high-orderfuzzy time series model to forecast enrolments Aladag et al[9 26] introduced a high-ordermodel based on feed-forwardneural network Lee et al [20 33] also presented some high-order models based on two-factor and genetic-simulated

2 Mathematical Problems in Engineering

annealing techniques Most of time series researchers [18 2234ndash37] had showed their respectively interest in high-orderfuzzy time series forecasting models

In process of forecasting with fuzzy time series modelsFuzzy Logical Relationship (FLR) is one of the most criticalfactors that influence the forecasting accuracy To obtainhigh forecasting accuracy a lot of efforts have been put intomining the FLRs from fuzzy time series In view of techniquesfor partitioning the universe of discourse and constructingthe fuzzy logic relationships effectively the above high-ordermodels consist of three parts The first one is mining theFLRs by applying some advanced algorithms or theories suchas genetic algorithms rough set neural networks type-2fuzzy set and simulated annealing algorithm [20 22 26 2730 32 34 35] The second one is the class represented bySingh [16ndash18] whosemodels are on the basis of computationalmethod with difference parameters The last but not leastone is the kind of models based on grouping the FLRsrepresented by [9 28 29 31 33 36 37] In general the firstkind of hybrid models can get higher forecasting accuracythan the other two However the forecasting process of thesealgorithms is not easy to be understood Unlike the fuzzy settheory its procedure and forecasts are not understandableand accountable for most of decision makers Although thesecond kind of models has been implemented on a real-lifeproblem of crop production and rice production as well asenrolment forecasts the models have little if anything to dowith FLRs in the procedures of forecastingThemodel obtainshigh forecasting accuracy by dividing the intervals to produceaccurate localizations of the forecasting valuesWith regard tothe third kind of models the procedures of mining FLRs andforecasting principles are based solely on the FLRs sets Theforecasting procedure and principles are obvious and clear tofuzzy time series researchers and easy to be understood by thedecision makers

For these reasons this paper proposes a high-order fuzzytime seriesmodel based on generalized fuzzy logical relation-ships [38]The process of creating relationshipsrsquo matrices andfinding out the patterns of time series fluctuations is carriedout on the basis of understandable fuzzy rules Of the abovethree kinds of models the proposed belongs to the thirdThere are three reasons for Hwangrsquos [28] and Chenrsquos models[29 31] to be chosen as the counterparts for comparing thesingle-factor forecasting results with determinate length ofintervalThe first reason is that the models of Chenrsquos [29] andLirsquos [36] are similar in finding the most appropriate forecast-ing principle with state-transition analysis and backtrackingschemeThe second is that themodels of Li et al [36] and Leeet al [33] aim atmultifactor forecasting problems and the lastis because models [9 37] are improved by finding an optimalinterval length As regards the experiment data sets two datasets were used for the empirical analysis the enrolments ofthe University of Alabama and the close price of ShanghaiStock Exchange Composite Index (SSECI) In view of thethree criteria of evaluations the root mean squared errormean absolute error andmean absolute percentage error theproposed method gets more satisfactory forecasts than thecounterparts

The rest of this paper is organized as follows In Section 2we briefly review the concepts of fuzzy time series InSection 3 a newmodel based onhigh-order generalized fuzzylogical relationships is implemented on the procedure offorecasting enrolments In Section 4 we compare the averageforecasting accuracy rates of the proposed method with themethods presented in [28 29 31] The effects of parameterson forecasting accuracy are also discussed in this sectionConclusions and future works are given in Section 5

2 Preliminaries

In view of making our exposition self-contained this sectionreviews some definitions and the framework of fuzzy timeseries forecasting models Followed with some related defini-tions of generalized fuzzy logical relationship the framework[1ndash3] is summarized in this section

Definition 1 (see [39 40]) A fuzzy set A of the universe ofdiscourse 119880 119880 = 119906

1 1199062 119906

119899 is defined as follows

119860 =119891119860(1199061)

1199061

+119891119860(1199062)

1199062

+ sdot sdot sdot +119891119860(119906119899)

119906119899

(1)

where 119891119860is the membership function of the fuzzy set 119860 119891

119860

119880 rarr [0 1] 119891119860(119906119894) denotes the membership degrees in the

fuzzy set 119860 1 le 119894 le 119899

Definition 2 Let 119884(119905) (119905 = 0 1 2 ) be the universe ofdiscourse in which fuzzy sets 119891

119894(119905) (119894 = 1 2 ) are defined

Let 119865(119905) be a collection of 119891119894(119905) Then 119865(119905) is called a fuzzy

time series on 119884(119905)

Definition 3 Let 119865(119905) be a fuzzy time series If 119865(119905) is causedby 119865(119905 1) 119865(119905119872) then the fuzzy logical relationship isrepresented by 119865(119905119872) 119865(119905119872 minus 1) 119865(119905 1) rarr 119865(119905) andit is called the119872th-order fuzzy time series forecastingmodel

Definition 4 Let119865(119905minus1) = 119860119894and119865(119905) = 119860

119895The relationship

between two consecutive observations 119865(119905) and 119865(119905 minus 1) isreferred to as a fuzzy logical relationship (FLR) and denotedby 119860119894

rarr 119860119895 where 119860

119894is called the left-hand side (LHS)

and 119860119895the right-hand side (RHS) of the FLR

Definition 5 Let 119891119860(119905minus119872) = (120583

1(119905minus119872) 120583

2(119905minus119872) 120583

119899(119905minus

119872)) 119891119860(119905) = (120583

1(119905) 120583

2(119905) 120583

119899(119905)) If 120583

119905minus119872

119894and 120583

119905

119895are

the maximum values of (1205831(119905) 120583

2(119905) 120583

119899(119905)) and (120583

1(119905)

1205832(119905) 120583

119899(119905)) respectively then let 119860119905minus119872

119894rarr 119860119905

119895be called

the 119872-order first principal fuzzy relationship noted as119866119879119878(119872 1) (generalized fuzzy logical relationship) If 120583119905minus119872

119894is

the119873th maximum value of (1205831(119905 minus119872) 120583

2(119905 minus119872) 120583

119899(119905 minus

119872)) then 119860119905

119894rarr 119860119905+1

119895e called the 119872th-order 119873th-principal

fuzzy logical relationship noted as 119866119879119878(119872 119873)

From Definition 5 the fuzzy logical relationship is moregeneral than that of conventional fuzzy time series model Infact the logical relationship is that of conventional modelswhen 119873 = 1 and the forecasting rules are obtained fromgrouping these relationships We then named it generalized

Mathematical Problems in Engineering 3

fuzzy logical relationship According toDefinition 4 all fuzzylogical relationships in the training data set can be furthergrouped together into different fuzzy logical relationshipgroups according to the same left-hand sides of the fuzzylogical relationship For given 119872 and 119873 the fuzzy logicalrelationships can be grouped into119872times119873matrices denoted as119877(119896119897)

(119896 = 1 2 119872 119897 = 1 2 119873) with the groupmethodproposed by Lee et al [13] Here 119903(119894119895) (119894 119895 isin 1 2 119899)the element of matrix 119877

(119896119897) is the number of fuzzy logicalrelationships 119860(119905minus119896119897)

119894rarr 119860119895

Then there are119873 fuzzy logical relationships matrices fora given training data set To forecast time series with thesegeneralized fuzzy logical relationship matrices we definedthe intersection operation as follows

Definition 6 Let 119860(119872119895)119905119895

represent the LHSs of FLRG in the119872-order 119895th-principal fuzzy logical relationship at time 119905Let 119877

(119872119895)(119905119895 119894) be the number of FLRG 119860

119905119895rarr 119860

119894in

the 119872th-order 119895th-principal fuzzy logical relationship (119895 =

1 2 119873 119894 119905119895

= 1 2 119899) To compute the logical rela-tionships between 119873 FLRGs the intersection operator and

119873is

defined as

and119873(119860(1198721)

1199051 119860

(119872119895)

119905119895 119860

(119872119873)

119905119873)

= (min1le119895le119873

119877(119872119895)

(119905119895 1) min

1le119895le119873

119877(119872119895)

(119905119895 119894)

min1le119895le119873

119877(119872119895)

(119905119895 119899))

(2)

Based on the above definitions this paper presents a high-order fuzzy time series model in the following section

3 Proposed Model

31 Procedure of 119866119879119878(119872119873) In this section we present anew forecastingmethod based on high-order and generalizedfuzzy logical relationships Since the proposed model isrelated to the number of orders denoted by 119872 andhierarchiesof principal fuzzy relationship denoted by 119873 we name theproposed model 119866119879119878(119872119873) In other words 119866119879119878(119872119873)

means an 119872-order fuzzy time series model based on 119873-principal fuzzy logical relationships

Step 1 Define the universe of discourse and intervals for rulesof extractionThe universe of discourse can be defined asU =[starting ending] According to equal length of intervals Uis partitioned into several intervals equally For example119880 =

1199061 1199062 119906

119899119898119894is themidpoint of 119906

119894whose corresponding

fuzzy set is 119860119894(119894 = 1 2 119899)

Step 2 Define fuzzy sets based on the universe of discourseand fuzzify the historical data The fuzzy set 119860

119894would be

expressed as 119860119894= (1198861198941 1198861198942 119886

119894119899) where 119886

119894119895isin [0 1] which

indicates the membership degree of 119906119895in 119860119894 The historical

and observed data are fuzzified according to the definition offuzzy sets For example a datum is fuzzified to 119860

119895 when the

maximal membership degree of the datum is the 119895th numberIn other words if 119886

119894119895= max119886

1198941 1198861198942 119886

119894119899 then the data at

time 119905 should be classified into the 119895th class In this paper thefuzzy sets are defined with triangular fuzzy function showedby formula (3)

Consider

1198601=

1

1199061

+05

1199062

+0

1199063

+ sdot sdot sdot +0

119906119899minus1

+0

119906119899

1198602=

05

1199061

+1

1199062

+05

1199063

+0

1199064

+ sdot sdot sdot +0

119906119894

+ sdot sdot sdot +0

119906119899minus1

+0

119906119899

119860119894=

0

1199061

+ sdot sdot sdot +0

119906119894minus2

+05

119906119894minus1

+1

119906119894

+05

119906119894+1

+0

119906119894+2

+ sdot sdot sdot +0

119906119899

119860119899=

0

1199061

+0

1199062

+0

1199063

+ sdot sdot sdot +0

119906119894

+ sdot sdot sdot +0

119906119899minus2

+05

119906119899minus1

+1

119906119899

(3)

Themembership degree of the value 119909119905at time 119905 in119860

119894(119894 =

1 2 119899) is defined by formula (4) Consider

120583119860119894

(119909119905)

=

1 if 119894 = 1 and 119909119905le 1198981

1 if 119894 = 119899 and 119909119905ge 119898119899

max0 1 minus

1003816100381610038161003816119909119894 minus 119898119894

1003816100381610038161003816

2 times 119897in others

(4)

where 119909119905is the observed value at time 119905 and 119897in is the length

of interval

Step 3 Establish the fuzzy logical relationships based on theorders and hierarchies of principal fuzzy logical relationshipGiven the sample data set and the definition of fuzzy setsall fuzzy logical relationships between two consecutive datacan be created To forecast the time series the fuzzy logicalrelationship matrix must be created in this step based on thefuzzy logical relationships

Among many different methods the method proposedby Lee in [13] is chosen in this paper For example the fuzzylogical relationships of a 119866119879119878(119872119873) model can be groupedinto 119872 times 119873 relationship matrices denoted by 119877

(119896119897)(119896 =

1 2 119872 119897 = 1 2 119873)

Step 4 Forecasting model Let 119865(119905 minus 119896) = (1205831(119905 minus 119896) 120583

2(119905 minus

119896) 120583119899(119905minus119896)) and let 120583

119905119897(119905minus119896) be the 119897th maxmembership

degree respectively By Definition 6 we have the intersectionfuzzy logical relationship and

119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873)

4 Mathematical Problems in Engineering

Table 1 Membership degrees of enrollment with respect to fuzzy sets

Year Value 1205831198601

1205831198602

1205831198603

1205831198604

1205831198605

1205831198606

1205831198607

Fuzzified1971 13055 10000 02775 0 0 0 0 0 119860

1 1198602

1972 13563 09685 05315 00315 0 0 0 0 1198601 1198602

1973 13867 08165 06835 01835 0 0 0 0 1198601 1198602

1974 14696 04020 09020 05980 00980 0 0 0 1198602 1198603

1975 15460 00200 05200 09800 04800 0 0 0 1198603 1198602

1976 15311 00945 05945 09055 04055 0 0 0 1198603 1198602

1977 15603 0 04485 09485 05515 00515 0 0 1198603 1198604

1978 15861 0 03195 08195 06805 01805 0 0 1198603 1198604

1979 16807 0 0 03465 08465 06535 01535 0 1198604 1198605

1980 16919 0 0 02905 07905 07095 02095 0 1198604 1198605

1981 16388 0 00560 05560 09440 04440 0 0 1198604 1198603

1982 15433 00335 05335 09665 04665 0 0 0 1198603 1198602

1983 15497 00015 05015 09985 04985 0 0 0 1198603 1198602

1984 15145 01775 06775 08225 03225 0 0 0 1198603 1198602

1985 15163 01685 06685 08315 03315 0 0 0 1198603 1198602

1986 15984 0 02580 07580 07420 02420 0 0 1198603 1198604

1987 16859 0 0 03205 08205 06795 01795 0 1198604 1198605

1988 18150 0 0 0 01750 06750 08250 03250 1198606 1198605

1989 18970 0 0 0 0 02650 07650 07350 1198606 1198607

1990 19328 0 0 0 0 00860 05860 09140 1198607 1198606

1991 19337 0 0 0 0 00815 05815 09185 1198607 1198606

1992 18876 0 0 0 0 03120 08120 06880 1198606 1198607

and then the 119896th forecasts 119865119896(119905) is conducted by formula (5)Consider

119865119896(119905)

= and119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873) ∘ (119898

1 1198982 119898

119899)119879

(5)

where ldquo∘rdquo is a composition operation for forecasting with thefollowing principles

(1) if the sum of and119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873) equals to

0 then the forecasted value is 1198981199051 and the midpoint

of the interval corresponding to 1198601199051

(2) otherwise the forecasted value is theweighting aggre-gate of119898

1199051 119898

119905119873

Then for a given 119872 there are 119872 forecasts for time 119905 Theconclusive forecasting value for time 119905 can be obtained by thefollowing formula

119865 (119905) =

119872

sum

119896=1

119865119896(119905) lowast 119908

119896 (6)

where 119908119896(119896 = 1 2 119872) is the adjustment parameter

for the 119896th forecast the parameter also can be obtained byminimizing the RMSE or other criteria of evaluations for thetraining data set

32 Computation of GTS(119872119873) on Forecasting EnrollmentsSince most of the conventional models have been presentedfor forecasting the historical enrolments of the Universityof Alabama in this section we present stepwise proceduresof the proposed method for forecasting the time series datawith 119872 = 3 and 119873 = 2 The historical enrolments ofthe University of Alabama from 1971 to 1992 are shown inTable 1

Step 1 Partitioning the universe of discourse 119880 into sevenintervals 119906

1 1199062 119906

7 where 119906

1= [13000 14000] 119906

2=

[14000 15000] 1199063

= [15000 16000] 1199064

= [16000 17000]1199065= [17000 18000] 119906

6= [18000 19000] and 119906

7= [19000

20000] their midpoints are 13500 14500 15500 16500 1750018500 and 19500 respectively

Step 2 Let 1198601= ldquonot manyrdquo 119860

2= ldquonot too manyrdquo 119860

3=

ldquomanyrdquo 1198604= ldquomany manyrdquo 119860

5= ldquovery manyrdquo 119860

6= ldquotoo

manyrdquo and 1198607= ldquotoo many manyrdquo 119860

1 1198602 119860

7are fuzzy

sets corresponding to linguistic values for ldquoenrolmentsrdquo

With triangular-shaped membership function defined byformula (4) the fuzzy sets and all observations are definedby the last column of Table 1 by formula (3) Thus the fuzzylogical relationships corresponding to given 119872 and 119873 arelisted in Table 2

Step 3 Divide the derived fuzzy logical relationships intogroups based on the states of the enrollments of fuzzy logicalrelationships

Mathematical Problems in Engineering 5

Table 2 Fuzzy logical relationships with119872 = 3 and119873 = 2

Year fuzzified 119896 = 1 119896 = 2 119896 = 3

119897 = 1 119897 = 2 119897 = 1 119897 = 2 119897 = 1 119897 = 2

1971 1198601 1198602

1972 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

1

1973 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

11198601997888rarr 119860

1 1198602997888rarr 119860

1

1974 1198602 1198603

1198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

2

1975 1198603 1198602

1198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1976 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1977 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

3

1978 1198603 1198604

1198603997888rarr 119860

3 1198604997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1979 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1980 1198604 1198605

1198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1981 1198604 1198603

1198604997888rarr 119860

4 1198605997888rarr 119860

41198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1982 1198603 1198602

1198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1983 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1984 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

3

1985 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1986 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1987 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1988 1198606 1198605

1198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

61198603997888rarr 119860

6 1198602997888rarr 119860

6

1989 1198606 1198607

1198606997888rarr 119860

6 1198605997888rarr 119860

61198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

6

1990 1198607 1198606

1198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

71198604997888rarr 119860

7 1198605997888rarr 119860

7

1991 1198607 1198606

1198607997888rarr 119860

7 1198606997888rarr 119860

71198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

7

1992 1198606 1198607

1198607997888rarr 119860

6 1198606997888rarr 119860

61198607997888rarr 119860

6 1198606997888rarr 119860

61198606997888rarr 119860

6 1198607997888rarr 119860

6

In this paper we use the method of Lee [13] to constructthe fuzzy logical relationships matrix For example let k = 1and l = 1 The set of fuzzy logical relationships is listed as

1198601rarr 11986011198601rarr 11986011198601rarr 11986021198602rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198604

1198604rarr 11986041198604rarr 11986031198603rarr 11986031198603rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198606

1198606rarr 1198606 1198606rarr 1198607 1198607rarr 1198607 1198607rarr 1198606 which

are all of FLRGs in the third column of Table 2

And they would be grouped and weighted by the recur-rent fuzzy relationships as follows

Group 1 1198601rarr 1198601with weight 2 119860

1rarr 1198602with weight 1

Group 2 1198602rarr 1198603with weight 1

Group 3 1198603rarr 1198603with weight 7 119860

3rarr 1198604with weight 2

Group 4 1198604

rarr 1198604with weight 2 119860

4rarr 119860

3with weight 1

1198604rarr 1198606with weight 1

Group 5 1198606rarr 1198606with weight 1 119860

6rarr 1198607with weight 1

Group 6 1198607rarr 1198607with weight 1 119860

7rarr 1198606with weight 1

The fuzzy logical relationship matrix then is 119877(11) Basedon the fourth fifth sixth seventh and eighth columnof Table 2 119877(12) 119877(21) 119877(22) 119877(31) and 119877

(32) can also be

obtained in the similar way These six fuzzy logical relation-ship matrices have been listed as follows

119877(11)

=

[[[[[[[[[[[[

[

2 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 7 2 0 0 0

0 0 1 2 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 1 1

]]]]]]]]]]]]

]

119877(12)

=

[[[[[[[[[[[[

[

0 0 0 0 0 0 0

2 1 6 0 0 0 0

0 0 2 0 0 0 0

0 0 2 2 0 0 0

0 0 0 2 0 2 0

0 0 0 0 0 1 1

0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(7)

119877(21)

=

[[[[[[[[[[

[

1 1 1 0 0 0 0

0 0 1 0 0 0 0

0 0 5 3 0 1 0

0 0 2 1 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 0 2

0 0 0 0 0 1 0

]]]]]]]]]]

]

6 Mathematical Problems in Engineering

119877(22)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

1 1 6 1 0 0 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 1 1 0 1 1

0 0 0 0 0 1 0

0 0 0 0 0 0 1

]]]]]]]]]

]

(8)

119877(31)

=

[[[[[[[[[

[

0 1 2 0 0 0 0

0 0 1 0 0 0 0

0 0 3 4 0 2 0

0 0 3 2 0 0 1

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 0 0

]]]]]]]]]

]

119877(32)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

0 1 5 2 0 1 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 2 0 0 0 2

0 0 0 0 0 0 0

0 0 0 0 0 1 0

]]]]]]]]]

]

(9)

Step 4 Calculate the forecasts To carry out the calculationsby the proposed principles we illustrate the forecastingprocess of enrolment of 1976 as follows From Table 1 thefuzzy set of enrolment of 1975 is (002 052 098 048 0 0 0)The two maximum membership attributes are the third andthe second Then 119896 = 1 the third row of 119877(11) is (0 0 7 20 0 0) and the second row of 119877(12) is (2 1 6 0 0 0 0) from(7) According to Definition 6 we have and

2(119860(11)

19751 119860(12)

19752) =

(0 0 1 0 0 0 0)

According to the second principle listed in Section 311198651(1976) then is119898

3 that is 15500 which is the middle point

of 1198603

When k = 2 the fuzzy set of 1974s is (0402 09020598 0098 0 0 0) shown in Table 1 The two maximummembership attributes are the second and the third Thesecond row of 119877(21) then is (0 0 1 0 0 0 0) and the thirdrow of 119877(22) is (0 0 2 0 0 0 0) from (8) With Definition 6we have and

2(119860(21)

19751 119860(22)

19752) = (0 0 1 0 0 0 0) By the first

principle listed in Section 31 1198652(1976) also is 15500In the similar way fromTable 1 andwith 119896 = 3 we can see

that the fuzzy set of 1973s is (08165 06835 01835 0 0 0 0)The two maximum membership attributes are the first andsecond ones The first row of 119877(31) is (0 1 2 0 0 0 0) andthe second row of 119877(32) is (0 1 5 2 0 0 0) from (9) WithDefinition 6 we have and

2(119860(31)

19751 119860(32)

19752) = (0 1 2 0 0 0 0)

With the second principle listed in Section 31 in mind1198653(1976) equals to (13) times 14500 + (23) times 15500 = 15167At last the forecasted value of 1976 that is 119865(1976) is

15590 which equals to 09130 times 15500 + 00790 times 15500 +

00142times1516 Also (1199081 1199082 1199083) = (09130 00790 00142)

is the regress result of the enrolment data Table 3 shows theactual and the forecasted enrolments of the data set

4 Empirical Analysis

41 Data Description To demonstrate the effectiveness ofthe proposed models amounts of data are needed Here theenrolments and SSECI are used as the illustration data sets forthe empirical analysis

There are some causes for the two time series to be thesubjects in our experiment There are two causes for thechoice of enrolment data at the University of Alabama Thefirst one is that most of the fuzzy time series studies havetaken this well-known time series as their experimentsThusthere are a lot of studies that can be used for our referenceThe other is that it is simple and easy to display the process ofthe proposedmodel Since time series models have been usedto make predictions in the areas of stock price forecasting formany years the daily SSECI covering the period from 1997 to2006 is adopted for further experiment

For the first data the determined length of the sevenintervals is 1000 All of the forecasting results from one orderto ten orders are compared with those of the conventionalfuzzy time series models based on fuzzy logical relationships

To discuss the effects of parameters 119872 and 119873 the ordernumber and hierarchies of fuzzy logical relationships on theforecasting results the ten-order (time-lag periods) fromone order to ten orders models are performed on thesecond data set with ten different lengths of intervals thatis 30 60 90 300 The forecasting results obtained by allof the high-order models are compared in terms of threeevaluation criteria reviewed in following subsection

42 Criteria of Evaluation In statistics the root meansquared error (RMSE) mean absolute error (MAE) andmean absolute percentage error (MAPE) are three typicalways to quantify the difference between values implied byan estimator and the actual values of the quantity beingestimated MSE is a risk function for measuring the averageof the squares of the difference For an unbiased estimatorthe MSE is the variance and the RMSE is the square rootof the variance known as the standard error FurthermoreRMSE is superior to MSE for the reason that its scale isthe same as the forecasts Thus we take RMSE as the firstrepresentative of the size of an average error As an averageof the absolute percent errors MAPE serves as a criterionfor the comparisons of forecasting results in the paper Somecomparisons of accuracy in the forecasted values of ourproposed models with other models are made on the basisof the three criteria

43 Performance Evaluation In Table 4 this study com-pares the RMSE MAE and MAPE forecasting value ofthe proposed method and the counterparts [28 29 31] onthe enrolment experiment Table 4 shows that the proposedmethod gets smaller RMSE MAE and MAPE than Hwangrsquosmodel [28] and also smaller than Chenrsquos model [31] of 2011 inmost cases Although there is a fly in the ointment that is theproposed model not always gets a higher average forecastingaccuracy rate than the model [29] of 2002 these results areimproving as the orders increase In summary the resultssuggest that the proposed model obtains better forecasts as

Mathematical Problems in Engineering 7

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1 Membership function 1

(a)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 2

(b)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 3

(c)

Figure 1 Three membership functions for forecasting SHI of 2003

Table 3 Forecasts of enrollment year actual data

Year Value 119872 = 1 119872 = 2 119872 = 3

119873 = 1 119873 = 2 119873 = 1 119873 = 2 119873 = 1 119873 = 2

1971 13055 mdash mdash mdash mdash mdash mdash1972 13563 13901 13902 mdash mdash mdash mdash1973 13867 13901 13902 13980 13968 mdash mdash1974 14696 13901 13902 13980 13968 14062 139901975 15460 15576 15576 15491 15510 15533 155111976 15311 15799 15576 15792 15590 15760 155901977 15603 15799 15576 15858 15604 15812 156081978 15861 15799 16246 15858 16221 15915 162281979 16807 15799 16246 15858 16342 15915 163471980 16919 16832 17251 16790 17267 16821 172721981 16388 16832 17251 16823 17240 16832 172451982 15433 16832 15576 16823 15698 16821 157191983 15497 15799 15576 15891 15590 15915 156141984 15145 15799 15576 15858 15604 15904 156081985 15163 15799 15576 15858 15604 15915 156201986 15984 15799 15576 15858 15604 15915 156201987 16859 15799 16246 15858 16221 15915 162281988 18150 16832 17251 16790 17267 16821 172601989 18970 19093 18591 18863 18473 18816 184631990 19328 19093 19596 19161 19614 18900 195811991 19337 19093 19094 19161 19151 19132 191631992 18876 19093 19094 19061 19070 19101 19070

MAE 473365 3720021 4713957 375308 5011709 3902802RMSE 6254334 4474897 6316806 4488908 6432859 4597683MAPE 00293 00227 00291 00227 00307 00236

the orders and hierarchies increase Unlike this point this isnot the trend in the three counterparts

Moreover we also apply the proposed method to handleforecasting the close price of Shanghai stock index of 2003with 119897 = 60 The comparison of the three criteria is listedin Table 5 From this table we can see that the proposed119866119879119878(119872119873) gets the best forecasts of the three counterpartswhen 119872 gt 5 and 119873 gt 2 as well as its better performancethan those of Hwangrsquos and Chenrsquos model [28 29] in all cases

and the shortcoming shown in Table 4 is gone On the wholethere is a ldquolawrdquo that forecasting errorswill be reducedwhen119872

or119873 is increased in the proposed model However this trendis not evident for the three counterparts by Tables 4 and 5 InTable 4 the three evaluation criteria of Hwangrsquos and Chenrsquosmodels [28 31] are decreasing while those of model [29] areincreasing In contrast it is obvious that the three evaluationcriteria of Hwangrsquos and Chenrsquos model are increasing whilethose of Chenrsquos model [29] are decreasing from Table 5 The

8 Mathematical Problems in Engineering

Table 4 Comparison of forecasting enrollment

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 625 632 643 641 658 575 562 289 203 191119866119879119878(119872 2) 447 449 460 436 448 456 437 413 391 284119866119879119878(119872 3) 447 449 460 436 448 456 406 382 365 225119866119879119878(119872 4) 447 449 460 434 445 453 423 415 397 253119866119879119878(119872 5) 447 449 460 434 445 453 423 415 397 253

[28] 588 588 578 608 602 620 632 575 554 498[29] 292 298 294 299 307 313 323 320 321 311[31] 543 543 522 420 456 443 452 463 480 498

MAE

119866119879119878(119872 1) 473 471 501 484 501 366 411 212 326 140119866119879119878(119872 2) 372 375 390 365 383 394 406 364 292 218119866119879119878(119872 3) 372 375 390 365 383 393 374 336 333 176119866119879119878(119872 4) 372 375 390 367 384 393 402 386 333 195119866119879119878(119872 5) 372 375 390 367 384 393 402 386 333 195

[28] 494 494 473 526 527 556 582 506 483 424[29] 252 262 256 259 272 277 289 284 282 271[31] 441 441 412 316 347 337 340 346 368 390

MAPE

119866119879119878(119872 1) 00293 00291 00307 00294 00305 00221 00246 00127 00103 00087119866119879119878(119872 2) 00227 00227 00236 00216 00227 00232 00239 00216 00195 00125119866119879119878(119872 3) 00227 00227 00236 00216 00227 00232 00220 00200 00176 00101119866119879119878(119872 4) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111119866119879119878(119872 5) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111

[28] 00299 00299 00283 00317 00313 00329 00345 00299 00285 00253[29] 00153 00159 00153 00154 00162 00164 00170 00166 00165 00158[31] 00268 00268 00248 00188 00203 00195 00196 00198 00211 00223

The bold data means the minimum error of the models with the same order

1300

1350

1400

1450

1500

1550

0 5 10 15 20 25 30 35 40 45 50Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(3 1) with L = 30

Figure 2 Actual values and forecasts of 2003 with the same L

use of different data sets seems to be the best reason toaccount for this contradiction From these discussions we geta conclusion that the proposed modelrsquos performance is morereasonable and robust than the three counterparts

Tables 4 and 5 depict that 119866119879119878(119872 3) 119866119879119878(119872 4) and119866119879119878(119872 5) have the same results although larger 119872 or 119873

is more easy to obtain the smaller forecasting errors In

0 5 10 15 20 25 30 35 40 45 501300

1350

1400

1450

1500

1550

Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(1 1) with L = 90

Figure 3 Actual values and forecasts of 2003 with different L

fact this trend is affected by the definitions of fuzzy setsand membership function As an in-depth analysis Figure 1shows three triangular membership functionsThe RMSEs offorecasts of 2003 by 119866119879119878(119872119873) with 119871 = 30 and the threemembership functions are listed in Table 6 The table tells usthat themore intervals that are concerned in themembership

Mathematical Problems in Engineering 9

Table 5 Comparison of forecasting SSECI

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 2274 2164 2121 2092 2086 2071 2066 2066 2025 2010119866119879119878(119872 2) 1832 1904 1790 1792 1769 1756 1754 1755 1695 1688119866119879119878(119872 3) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 4) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 5) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683

[28] 1841 1841 1942 1943 1951 1939 2005 1971 2000 2026[29] 2107 2102 2087 2073 2008 1988 1979 1969 1818 1821[31] 1756 1756 1781 1778 1838 1831 1867 1925 1947 1960

MAE

119866119879119878(119872 1) 1861 1710 1661 1612 1609 1608 1607 1611 1575 1550119866119879119878(119872 2) 1408 1389 1388 1391 1373 1353 1345 1346 1316 1317119866119879119878(119872 3) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 4) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 5) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306

[28] 1392 1392 1478 1502 1522 1518 1541 1538 1562 1590[29] 1731 1726 1715 1704 1677 1663 1656 1647 1604 1607[31] 1281 1281 1329 1306 1357 1338 1419 1437 1442 1551

MAPE

119866119879119878(119872 1) 00127 00117 00113 00110 00110 00110 00109 00110 00107 00105119866119879119878(119872 2) 00096 00095 00095 00095 00094 00092 00092 00092 00090 00090119866119879119878(119872 3) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 4) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 5) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089

[28] 00095 00095 00101 00102 00104 00103 00105 00105 00106 00108[29] 00118 00118 00117 00116 00114 00113 00113 00112 00109 00109[31] 00087 00087 00091 00089 00092 00091 00097 00098 00098 00099

The bold data means the minimum error of the models with the same order

Table 6 Effects of membership function

Model 119896 = 3 119896 = 6 119896 = 9

Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3119866119879119878(119896 1) 175133 173290 173755 171512 169790 170653 164968 162923 164014119866119879119878(119896 2) 158071 157476 159365 154880 154686 157182 147028 146918 147719119866119879119878(119896 3) 158071 156926 158736 154880 153137 154631 147028 145116 144828119866119879119878(119896 4) 158071 156902 158670 154880 153119 154813 147028 145218 144558119866119879119878(119896 5) 158071 156902 159157 154880 153119 155422 147028 145218 145082119866119879119878(119896 6) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 7) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 8) 158071 156902 159157 154880 153119 155424 147028 145218 145068The bold data means the minimum error of the models with the same order

function the more different forecasting accuracy obtainedIt is more obvious when 119872 is increasing There is stillanother conclusion that the forecasts of the thirdmembershipfunction are not always better than those of the first function

To further investigate the relation between the parameters119872 and the length of interval the proposed model has beenapplied to forecasting stock index close prices covering tenyears from 1997 to 2006 with 119872 = 1 2 50 and 119871 =

30 60 300 Since it has been affirmed by Tables 4 and5 that the characters of MAE and MAPE are similar to

those of RMSE MAE and MAPE we will only list theRMSE comparison of these experiments as follows Someexamples of actual values and forecasts of 2003 are depicted inFigures 2 and 3 Figure 2 shows us that 119866119879119878(3 1) has a betterperformance than 119866119879119878(1 1) in the same length of intervalsthat is 119871 = 30 Figure 3 illustrates that 119866119879119878(1 1) getsthe better forecasts when the lengths of intervals are smallFurthermore some properties will be further described inFigures 4 and 5 which depict the mean forecasting errorsof the ten years Figure 4 shows the relation between the

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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2 Mathematical Problems in Engineering

annealing techniques Most of time series researchers [18 2234ndash37] had showed their respectively interest in high-orderfuzzy time series forecasting models

In process of forecasting with fuzzy time series modelsFuzzy Logical Relationship (FLR) is one of the most criticalfactors that influence the forecasting accuracy To obtainhigh forecasting accuracy a lot of efforts have been put intomining the FLRs from fuzzy time series In view of techniquesfor partitioning the universe of discourse and constructingthe fuzzy logic relationships effectively the above high-ordermodels consist of three parts The first one is mining theFLRs by applying some advanced algorithms or theories suchas genetic algorithms rough set neural networks type-2fuzzy set and simulated annealing algorithm [20 22 26 2730 32 34 35] The second one is the class represented bySingh [16ndash18] whosemodels are on the basis of computationalmethod with difference parameters The last but not leastone is the kind of models based on grouping the FLRsrepresented by [9 28 29 31 33 36 37] In general the firstkind of hybrid models can get higher forecasting accuracythan the other two However the forecasting process of thesealgorithms is not easy to be understood Unlike the fuzzy settheory its procedure and forecasts are not understandableand accountable for most of decision makers Although thesecond kind of models has been implemented on a real-lifeproblem of crop production and rice production as well asenrolment forecasts the models have little if anything to dowith FLRs in the procedures of forecastingThemodel obtainshigh forecasting accuracy by dividing the intervals to produceaccurate localizations of the forecasting valuesWith regard tothe third kind of models the procedures of mining FLRs andforecasting principles are based solely on the FLRs sets Theforecasting procedure and principles are obvious and clear tofuzzy time series researchers and easy to be understood by thedecision makers

For these reasons this paper proposes a high-order fuzzytime seriesmodel based on generalized fuzzy logical relation-ships [38]The process of creating relationshipsrsquo matrices andfinding out the patterns of time series fluctuations is carriedout on the basis of understandable fuzzy rules Of the abovethree kinds of models the proposed belongs to the thirdThere are three reasons for Hwangrsquos [28] and Chenrsquos models[29 31] to be chosen as the counterparts for comparing thesingle-factor forecasting results with determinate length ofintervalThe first reason is that the models of Chenrsquos [29] andLirsquos [36] are similar in finding the most appropriate forecast-ing principle with state-transition analysis and backtrackingschemeThe second is that themodels of Li et al [36] and Leeet al [33] aim atmultifactor forecasting problems and the lastis because models [9 37] are improved by finding an optimalinterval length As regards the experiment data sets two datasets were used for the empirical analysis the enrolments ofthe University of Alabama and the close price of ShanghaiStock Exchange Composite Index (SSECI) In view of thethree criteria of evaluations the root mean squared errormean absolute error andmean absolute percentage error theproposed method gets more satisfactory forecasts than thecounterparts

The rest of this paper is organized as follows In Section 2we briefly review the concepts of fuzzy time series InSection 3 a newmodel based onhigh-order generalized fuzzylogical relationships is implemented on the procedure offorecasting enrolments In Section 4 we compare the averageforecasting accuracy rates of the proposed method with themethods presented in [28 29 31] The effects of parameterson forecasting accuracy are also discussed in this sectionConclusions and future works are given in Section 5

2 Preliminaries

In view of making our exposition self-contained this sectionreviews some definitions and the framework of fuzzy timeseries forecasting models Followed with some related defini-tions of generalized fuzzy logical relationship the framework[1ndash3] is summarized in this section

Definition 1 (see [39 40]) A fuzzy set A of the universe ofdiscourse 119880 119880 = 119906

1 1199062 119906

119899 is defined as follows

119860 =119891119860(1199061)

1199061

+119891119860(1199062)

1199062

+ sdot sdot sdot +119891119860(119906119899)

119906119899

(1)

where 119891119860is the membership function of the fuzzy set 119860 119891

119860

119880 rarr [0 1] 119891119860(119906119894) denotes the membership degrees in the

fuzzy set 119860 1 le 119894 le 119899

Definition 2 Let 119884(119905) (119905 = 0 1 2 ) be the universe ofdiscourse in which fuzzy sets 119891

119894(119905) (119894 = 1 2 ) are defined

Let 119865(119905) be a collection of 119891119894(119905) Then 119865(119905) is called a fuzzy

time series on 119884(119905)

Definition 3 Let 119865(119905) be a fuzzy time series If 119865(119905) is causedby 119865(119905 1) 119865(119905119872) then the fuzzy logical relationship isrepresented by 119865(119905119872) 119865(119905119872 minus 1) 119865(119905 1) rarr 119865(119905) andit is called the119872th-order fuzzy time series forecastingmodel

Definition 4 Let119865(119905minus1) = 119860119894and119865(119905) = 119860

119895The relationship

between two consecutive observations 119865(119905) and 119865(119905 minus 1) isreferred to as a fuzzy logical relationship (FLR) and denotedby 119860119894

rarr 119860119895 where 119860

119894is called the left-hand side (LHS)

and 119860119895the right-hand side (RHS) of the FLR

Definition 5 Let 119891119860(119905minus119872) = (120583

1(119905minus119872) 120583

2(119905minus119872) 120583

119899(119905minus

119872)) 119891119860(119905) = (120583

1(119905) 120583

2(119905) 120583

119899(119905)) If 120583

119905minus119872

119894and 120583

119905

119895are

the maximum values of (1205831(119905) 120583

2(119905) 120583

119899(119905)) and (120583

1(119905)

1205832(119905) 120583

119899(119905)) respectively then let 119860119905minus119872

119894rarr 119860119905

119895be called

the 119872-order first principal fuzzy relationship noted as119866119879119878(119872 1) (generalized fuzzy logical relationship) If 120583119905minus119872

119894is

the119873th maximum value of (1205831(119905 minus119872) 120583

2(119905 minus119872) 120583

119899(119905 minus

119872)) then 119860119905

119894rarr 119860119905+1

119895e called the 119872th-order 119873th-principal

fuzzy logical relationship noted as 119866119879119878(119872 119873)

From Definition 5 the fuzzy logical relationship is moregeneral than that of conventional fuzzy time series model Infact the logical relationship is that of conventional modelswhen 119873 = 1 and the forecasting rules are obtained fromgrouping these relationships We then named it generalized

Mathematical Problems in Engineering 3

fuzzy logical relationship According toDefinition 4 all fuzzylogical relationships in the training data set can be furthergrouped together into different fuzzy logical relationshipgroups according to the same left-hand sides of the fuzzylogical relationship For given 119872 and 119873 the fuzzy logicalrelationships can be grouped into119872times119873matrices denoted as119877(119896119897)

(119896 = 1 2 119872 119897 = 1 2 119873) with the groupmethodproposed by Lee et al [13] Here 119903(119894119895) (119894 119895 isin 1 2 119899)the element of matrix 119877

(119896119897) is the number of fuzzy logicalrelationships 119860(119905minus119896119897)

119894rarr 119860119895

Then there are119873 fuzzy logical relationships matrices fora given training data set To forecast time series with thesegeneralized fuzzy logical relationship matrices we definedthe intersection operation as follows

Definition 6 Let 119860(119872119895)119905119895

represent the LHSs of FLRG in the119872-order 119895th-principal fuzzy logical relationship at time 119905Let 119877

(119872119895)(119905119895 119894) be the number of FLRG 119860

119905119895rarr 119860

119894in

the 119872th-order 119895th-principal fuzzy logical relationship (119895 =

1 2 119873 119894 119905119895

= 1 2 119899) To compute the logical rela-tionships between 119873 FLRGs the intersection operator and

119873is

defined as

and119873(119860(1198721)

1199051 119860

(119872119895)

119905119895 119860

(119872119873)

119905119873)

= (min1le119895le119873

119877(119872119895)

(119905119895 1) min

1le119895le119873

119877(119872119895)

(119905119895 119894)

min1le119895le119873

119877(119872119895)

(119905119895 119899))

(2)

Based on the above definitions this paper presents a high-order fuzzy time series model in the following section

3 Proposed Model

31 Procedure of 119866119879119878(119872119873) In this section we present anew forecastingmethod based on high-order and generalizedfuzzy logical relationships Since the proposed model isrelated to the number of orders denoted by 119872 andhierarchiesof principal fuzzy relationship denoted by 119873 we name theproposed model 119866119879119878(119872119873) In other words 119866119879119878(119872119873)

means an 119872-order fuzzy time series model based on 119873-principal fuzzy logical relationships

Step 1 Define the universe of discourse and intervals for rulesof extractionThe universe of discourse can be defined asU =[starting ending] According to equal length of intervals Uis partitioned into several intervals equally For example119880 =

1199061 1199062 119906

119899119898119894is themidpoint of 119906

119894whose corresponding

fuzzy set is 119860119894(119894 = 1 2 119899)

Step 2 Define fuzzy sets based on the universe of discourseand fuzzify the historical data The fuzzy set 119860

119894would be

expressed as 119860119894= (1198861198941 1198861198942 119886

119894119899) where 119886

119894119895isin [0 1] which

indicates the membership degree of 119906119895in 119860119894 The historical

and observed data are fuzzified according to the definition offuzzy sets For example a datum is fuzzified to 119860

119895 when the

maximal membership degree of the datum is the 119895th numberIn other words if 119886

119894119895= max119886

1198941 1198861198942 119886

119894119899 then the data at

time 119905 should be classified into the 119895th class In this paper thefuzzy sets are defined with triangular fuzzy function showedby formula (3)

Consider

1198601=

1

1199061

+05

1199062

+0

1199063

+ sdot sdot sdot +0

119906119899minus1

+0

119906119899

1198602=

05

1199061

+1

1199062

+05

1199063

+0

1199064

+ sdot sdot sdot +0

119906119894

+ sdot sdot sdot +0

119906119899minus1

+0

119906119899

119860119894=

0

1199061

+ sdot sdot sdot +0

119906119894minus2

+05

119906119894minus1

+1

119906119894

+05

119906119894+1

+0

119906119894+2

+ sdot sdot sdot +0

119906119899

119860119899=

0

1199061

+0

1199062

+0

1199063

+ sdot sdot sdot +0

119906119894

+ sdot sdot sdot +0

119906119899minus2

+05

119906119899minus1

+1

119906119899

(3)

Themembership degree of the value 119909119905at time 119905 in119860

119894(119894 =

1 2 119899) is defined by formula (4) Consider

120583119860119894

(119909119905)

=

1 if 119894 = 1 and 119909119905le 1198981

1 if 119894 = 119899 and 119909119905ge 119898119899

max0 1 minus

1003816100381610038161003816119909119894 minus 119898119894

1003816100381610038161003816

2 times 119897in others

(4)

where 119909119905is the observed value at time 119905 and 119897in is the length

of interval

Step 3 Establish the fuzzy logical relationships based on theorders and hierarchies of principal fuzzy logical relationshipGiven the sample data set and the definition of fuzzy setsall fuzzy logical relationships between two consecutive datacan be created To forecast the time series the fuzzy logicalrelationship matrix must be created in this step based on thefuzzy logical relationships

Among many different methods the method proposedby Lee in [13] is chosen in this paper For example the fuzzylogical relationships of a 119866119879119878(119872119873) model can be groupedinto 119872 times 119873 relationship matrices denoted by 119877

(119896119897)(119896 =

1 2 119872 119897 = 1 2 119873)

Step 4 Forecasting model Let 119865(119905 minus 119896) = (1205831(119905 minus 119896) 120583

2(119905 minus

119896) 120583119899(119905minus119896)) and let 120583

119905119897(119905minus119896) be the 119897th maxmembership

degree respectively By Definition 6 we have the intersectionfuzzy logical relationship and

119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873)

4 Mathematical Problems in Engineering

Table 1 Membership degrees of enrollment with respect to fuzzy sets

Year Value 1205831198601

1205831198602

1205831198603

1205831198604

1205831198605

1205831198606

1205831198607

Fuzzified1971 13055 10000 02775 0 0 0 0 0 119860

1 1198602

1972 13563 09685 05315 00315 0 0 0 0 1198601 1198602

1973 13867 08165 06835 01835 0 0 0 0 1198601 1198602

1974 14696 04020 09020 05980 00980 0 0 0 1198602 1198603

1975 15460 00200 05200 09800 04800 0 0 0 1198603 1198602

1976 15311 00945 05945 09055 04055 0 0 0 1198603 1198602

1977 15603 0 04485 09485 05515 00515 0 0 1198603 1198604

1978 15861 0 03195 08195 06805 01805 0 0 1198603 1198604

1979 16807 0 0 03465 08465 06535 01535 0 1198604 1198605

1980 16919 0 0 02905 07905 07095 02095 0 1198604 1198605

1981 16388 0 00560 05560 09440 04440 0 0 1198604 1198603

1982 15433 00335 05335 09665 04665 0 0 0 1198603 1198602

1983 15497 00015 05015 09985 04985 0 0 0 1198603 1198602

1984 15145 01775 06775 08225 03225 0 0 0 1198603 1198602

1985 15163 01685 06685 08315 03315 0 0 0 1198603 1198602

1986 15984 0 02580 07580 07420 02420 0 0 1198603 1198604

1987 16859 0 0 03205 08205 06795 01795 0 1198604 1198605

1988 18150 0 0 0 01750 06750 08250 03250 1198606 1198605

1989 18970 0 0 0 0 02650 07650 07350 1198606 1198607

1990 19328 0 0 0 0 00860 05860 09140 1198607 1198606

1991 19337 0 0 0 0 00815 05815 09185 1198607 1198606

1992 18876 0 0 0 0 03120 08120 06880 1198606 1198607

and then the 119896th forecasts 119865119896(119905) is conducted by formula (5)Consider

119865119896(119905)

= and119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873) ∘ (119898

1 1198982 119898

119899)119879

(5)

where ldquo∘rdquo is a composition operation for forecasting with thefollowing principles

(1) if the sum of and119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873) equals to

0 then the forecasted value is 1198981199051 and the midpoint

of the interval corresponding to 1198601199051

(2) otherwise the forecasted value is theweighting aggre-gate of119898

1199051 119898

119905119873

Then for a given 119872 there are 119872 forecasts for time 119905 Theconclusive forecasting value for time 119905 can be obtained by thefollowing formula

119865 (119905) =

119872

sum

119896=1

119865119896(119905) lowast 119908

119896 (6)

where 119908119896(119896 = 1 2 119872) is the adjustment parameter

for the 119896th forecast the parameter also can be obtained byminimizing the RMSE or other criteria of evaluations for thetraining data set

32 Computation of GTS(119872119873) on Forecasting EnrollmentsSince most of the conventional models have been presentedfor forecasting the historical enrolments of the Universityof Alabama in this section we present stepwise proceduresof the proposed method for forecasting the time series datawith 119872 = 3 and 119873 = 2 The historical enrolments ofthe University of Alabama from 1971 to 1992 are shown inTable 1

Step 1 Partitioning the universe of discourse 119880 into sevenintervals 119906

1 1199062 119906

7 where 119906

1= [13000 14000] 119906

2=

[14000 15000] 1199063

= [15000 16000] 1199064

= [16000 17000]1199065= [17000 18000] 119906

6= [18000 19000] and 119906

7= [19000

20000] their midpoints are 13500 14500 15500 16500 1750018500 and 19500 respectively

Step 2 Let 1198601= ldquonot manyrdquo 119860

2= ldquonot too manyrdquo 119860

3=

ldquomanyrdquo 1198604= ldquomany manyrdquo 119860

5= ldquovery manyrdquo 119860

6= ldquotoo

manyrdquo and 1198607= ldquotoo many manyrdquo 119860

1 1198602 119860

7are fuzzy

sets corresponding to linguistic values for ldquoenrolmentsrdquo

With triangular-shaped membership function defined byformula (4) the fuzzy sets and all observations are definedby the last column of Table 1 by formula (3) Thus the fuzzylogical relationships corresponding to given 119872 and 119873 arelisted in Table 2

Step 3 Divide the derived fuzzy logical relationships intogroups based on the states of the enrollments of fuzzy logicalrelationships

Mathematical Problems in Engineering 5

Table 2 Fuzzy logical relationships with119872 = 3 and119873 = 2

Year fuzzified 119896 = 1 119896 = 2 119896 = 3

119897 = 1 119897 = 2 119897 = 1 119897 = 2 119897 = 1 119897 = 2

1971 1198601 1198602

1972 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

1

1973 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

11198601997888rarr 119860

1 1198602997888rarr 119860

1

1974 1198602 1198603

1198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

2

1975 1198603 1198602

1198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1976 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1977 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

3

1978 1198603 1198604

1198603997888rarr 119860

3 1198604997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1979 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1980 1198604 1198605

1198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1981 1198604 1198603

1198604997888rarr 119860

4 1198605997888rarr 119860

41198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1982 1198603 1198602

1198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1983 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1984 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

3

1985 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1986 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1987 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1988 1198606 1198605

1198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

61198603997888rarr 119860

6 1198602997888rarr 119860

6

1989 1198606 1198607

1198606997888rarr 119860

6 1198605997888rarr 119860

61198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

6

1990 1198607 1198606

1198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

71198604997888rarr 119860

7 1198605997888rarr 119860

7

1991 1198607 1198606

1198607997888rarr 119860

7 1198606997888rarr 119860

71198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

7

1992 1198606 1198607

1198607997888rarr 119860

6 1198606997888rarr 119860

61198607997888rarr 119860

6 1198606997888rarr 119860

61198606997888rarr 119860

6 1198607997888rarr 119860

6

In this paper we use the method of Lee [13] to constructthe fuzzy logical relationships matrix For example let k = 1and l = 1 The set of fuzzy logical relationships is listed as

1198601rarr 11986011198601rarr 11986011198601rarr 11986021198602rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198604

1198604rarr 11986041198604rarr 11986031198603rarr 11986031198603rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198606

1198606rarr 1198606 1198606rarr 1198607 1198607rarr 1198607 1198607rarr 1198606 which

are all of FLRGs in the third column of Table 2

And they would be grouped and weighted by the recur-rent fuzzy relationships as follows

Group 1 1198601rarr 1198601with weight 2 119860

1rarr 1198602with weight 1

Group 2 1198602rarr 1198603with weight 1

Group 3 1198603rarr 1198603with weight 7 119860

3rarr 1198604with weight 2

Group 4 1198604

rarr 1198604with weight 2 119860

4rarr 119860

3with weight 1

1198604rarr 1198606with weight 1

Group 5 1198606rarr 1198606with weight 1 119860

6rarr 1198607with weight 1

Group 6 1198607rarr 1198607with weight 1 119860

7rarr 1198606with weight 1

The fuzzy logical relationship matrix then is 119877(11) Basedon the fourth fifth sixth seventh and eighth columnof Table 2 119877(12) 119877(21) 119877(22) 119877(31) and 119877

(32) can also be

obtained in the similar way These six fuzzy logical relation-ship matrices have been listed as follows

119877(11)

=

[[[[[[[[[[[[

[

2 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 7 2 0 0 0

0 0 1 2 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 1 1

]]]]]]]]]]]]

]

119877(12)

=

[[[[[[[[[[[[

[

0 0 0 0 0 0 0

2 1 6 0 0 0 0

0 0 2 0 0 0 0

0 0 2 2 0 0 0

0 0 0 2 0 2 0

0 0 0 0 0 1 1

0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(7)

119877(21)

=

[[[[[[[[[[

[

1 1 1 0 0 0 0

0 0 1 0 0 0 0

0 0 5 3 0 1 0

0 0 2 1 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 0 2

0 0 0 0 0 1 0

]]]]]]]]]]

]

6 Mathematical Problems in Engineering

119877(22)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

1 1 6 1 0 0 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 1 1 0 1 1

0 0 0 0 0 1 0

0 0 0 0 0 0 1

]]]]]]]]]

]

(8)

119877(31)

=

[[[[[[[[[

[

0 1 2 0 0 0 0

0 0 1 0 0 0 0

0 0 3 4 0 2 0

0 0 3 2 0 0 1

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 0 0

]]]]]]]]]

]

119877(32)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

0 1 5 2 0 1 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 2 0 0 0 2

0 0 0 0 0 0 0

0 0 0 0 0 1 0

]]]]]]]]]

]

(9)

Step 4 Calculate the forecasts To carry out the calculationsby the proposed principles we illustrate the forecastingprocess of enrolment of 1976 as follows From Table 1 thefuzzy set of enrolment of 1975 is (002 052 098 048 0 0 0)The two maximum membership attributes are the third andthe second Then 119896 = 1 the third row of 119877(11) is (0 0 7 20 0 0) and the second row of 119877(12) is (2 1 6 0 0 0 0) from(7) According to Definition 6 we have and

2(119860(11)

19751 119860(12)

19752) =

(0 0 1 0 0 0 0)

According to the second principle listed in Section 311198651(1976) then is119898

3 that is 15500 which is the middle point

of 1198603

When k = 2 the fuzzy set of 1974s is (0402 09020598 0098 0 0 0) shown in Table 1 The two maximummembership attributes are the second and the third Thesecond row of 119877(21) then is (0 0 1 0 0 0 0) and the thirdrow of 119877(22) is (0 0 2 0 0 0 0) from (8) With Definition 6we have and

2(119860(21)

19751 119860(22)

19752) = (0 0 1 0 0 0 0) By the first

principle listed in Section 31 1198652(1976) also is 15500In the similar way fromTable 1 andwith 119896 = 3 we can see

that the fuzzy set of 1973s is (08165 06835 01835 0 0 0 0)The two maximum membership attributes are the first andsecond ones The first row of 119877(31) is (0 1 2 0 0 0 0) andthe second row of 119877(32) is (0 1 5 2 0 0 0) from (9) WithDefinition 6 we have and

2(119860(31)

19751 119860(32)

19752) = (0 1 2 0 0 0 0)

With the second principle listed in Section 31 in mind1198653(1976) equals to (13) times 14500 + (23) times 15500 = 15167At last the forecasted value of 1976 that is 119865(1976) is

15590 which equals to 09130 times 15500 + 00790 times 15500 +

00142times1516 Also (1199081 1199082 1199083) = (09130 00790 00142)

is the regress result of the enrolment data Table 3 shows theactual and the forecasted enrolments of the data set

4 Empirical Analysis

41 Data Description To demonstrate the effectiveness ofthe proposed models amounts of data are needed Here theenrolments and SSECI are used as the illustration data sets forthe empirical analysis

There are some causes for the two time series to be thesubjects in our experiment There are two causes for thechoice of enrolment data at the University of Alabama Thefirst one is that most of the fuzzy time series studies havetaken this well-known time series as their experimentsThusthere are a lot of studies that can be used for our referenceThe other is that it is simple and easy to display the process ofthe proposedmodel Since time series models have been usedto make predictions in the areas of stock price forecasting formany years the daily SSECI covering the period from 1997 to2006 is adopted for further experiment

For the first data the determined length of the sevenintervals is 1000 All of the forecasting results from one orderto ten orders are compared with those of the conventionalfuzzy time series models based on fuzzy logical relationships

To discuss the effects of parameters 119872 and 119873 the ordernumber and hierarchies of fuzzy logical relationships on theforecasting results the ten-order (time-lag periods) fromone order to ten orders models are performed on thesecond data set with ten different lengths of intervals thatis 30 60 90 300 The forecasting results obtained by allof the high-order models are compared in terms of threeevaluation criteria reviewed in following subsection

42 Criteria of Evaluation In statistics the root meansquared error (RMSE) mean absolute error (MAE) andmean absolute percentage error (MAPE) are three typicalways to quantify the difference between values implied byan estimator and the actual values of the quantity beingestimated MSE is a risk function for measuring the averageof the squares of the difference For an unbiased estimatorthe MSE is the variance and the RMSE is the square rootof the variance known as the standard error FurthermoreRMSE is superior to MSE for the reason that its scale isthe same as the forecasts Thus we take RMSE as the firstrepresentative of the size of an average error As an averageof the absolute percent errors MAPE serves as a criterionfor the comparisons of forecasting results in the paper Somecomparisons of accuracy in the forecasted values of ourproposed models with other models are made on the basisof the three criteria

43 Performance Evaluation In Table 4 this study com-pares the RMSE MAE and MAPE forecasting value ofthe proposed method and the counterparts [28 29 31] onthe enrolment experiment Table 4 shows that the proposedmethod gets smaller RMSE MAE and MAPE than Hwangrsquosmodel [28] and also smaller than Chenrsquos model [31] of 2011 inmost cases Although there is a fly in the ointment that is theproposed model not always gets a higher average forecastingaccuracy rate than the model [29] of 2002 these results areimproving as the orders increase In summary the resultssuggest that the proposed model obtains better forecasts as

Mathematical Problems in Engineering 7

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1 Membership function 1

(a)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 2

(b)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 3

(c)

Figure 1 Three membership functions for forecasting SHI of 2003

Table 3 Forecasts of enrollment year actual data

Year Value 119872 = 1 119872 = 2 119872 = 3

119873 = 1 119873 = 2 119873 = 1 119873 = 2 119873 = 1 119873 = 2

1971 13055 mdash mdash mdash mdash mdash mdash1972 13563 13901 13902 mdash mdash mdash mdash1973 13867 13901 13902 13980 13968 mdash mdash1974 14696 13901 13902 13980 13968 14062 139901975 15460 15576 15576 15491 15510 15533 155111976 15311 15799 15576 15792 15590 15760 155901977 15603 15799 15576 15858 15604 15812 156081978 15861 15799 16246 15858 16221 15915 162281979 16807 15799 16246 15858 16342 15915 163471980 16919 16832 17251 16790 17267 16821 172721981 16388 16832 17251 16823 17240 16832 172451982 15433 16832 15576 16823 15698 16821 157191983 15497 15799 15576 15891 15590 15915 156141984 15145 15799 15576 15858 15604 15904 156081985 15163 15799 15576 15858 15604 15915 156201986 15984 15799 15576 15858 15604 15915 156201987 16859 15799 16246 15858 16221 15915 162281988 18150 16832 17251 16790 17267 16821 172601989 18970 19093 18591 18863 18473 18816 184631990 19328 19093 19596 19161 19614 18900 195811991 19337 19093 19094 19161 19151 19132 191631992 18876 19093 19094 19061 19070 19101 19070

MAE 473365 3720021 4713957 375308 5011709 3902802RMSE 6254334 4474897 6316806 4488908 6432859 4597683MAPE 00293 00227 00291 00227 00307 00236

the orders and hierarchies increase Unlike this point this isnot the trend in the three counterparts

Moreover we also apply the proposed method to handleforecasting the close price of Shanghai stock index of 2003with 119897 = 60 The comparison of the three criteria is listedin Table 5 From this table we can see that the proposed119866119879119878(119872119873) gets the best forecasts of the three counterpartswhen 119872 gt 5 and 119873 gt 2 as well as its better performancethan those of Hwangrsquos and Chenrsquos model [28 29] in all cases

and the shortcoming shown in Table 4 is gone On the wholethere is a ldquolawrdquo that forecasting errorswill be reducedwhen119872

or119873 is increased in the proposed model However this trendis not evident for the three counterparts by Tables 4 and 5 InTable 4 the three evaluation criteria of Hwangrsquos and Chenrsquosmodels [28 31] are decreasing while those of model [29] areincreasing In contrast it is obvious that the three evaluationcriteria of Hwangrsquos and Chenrsquos model are increasing whilethose of Chenrsquos model [29] are decreasing from Table 5 The

8 Mathematical Problems in Engineering

Table 4 Comparison of forecasting enrollment

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 625 632 643 641 658 575 562 289 203 191119866119879119878(119872 2) 447 449 460 436 448 456 437 413 391 284119866119879119878(119872 3) 447 449 460 436 448 456 406 382 365 225119866119879119878(119872 4) 447 449 460 434 445 453 423 415 397 253119866119879119878(119872 5) 447 449 460 434 445 453 423 415 397 253

[28] 588 588 578 608 602 620 632 575 554 498[29] 292 298 294 299 307 313 323 320 321 311[31] 543 543 522 420 456 443 452 463 480 498

MAE

119866119879119878(119872 1) 473 471 501 484 501 366 411 212 326 140119866119879119878(119872 2) 372 375 390 365 383 394 406 364 292 218119866119879119878(119872 3) 372 375 390 365 383 393 374 336 333 176119866119879119878(119872 4) 372 375 390 367 384 393 402 386 333 195119866119879119878(119872 5) 372 375 390 367 384 393 402 386 333 195

[28] 494 494 473 526 527 556 582 506 483 424[29] 252 262 256 259 272 277 289 284 282 271[31] 441 441 412 316 347 337 340 346 368 390

MAPE

119866119879119878(119872 1) 00293 00291 00307 00294 00305 00221 00246 00127 00103 00087119866119879119878(119872 2) 00227 00227 00236 00216 00227 00232 00239 00216 00195 00125119866119879119878(119872 3) 00227 00227 00236 00216 00227 00232 00220 00200 00176 00101119866119879119878(119872 4) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111119866119879119878(119872 5) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111

[28] 00299 00299 00283 00317 00313 00329 00345 00299 00285 00253[29] 00153 00159 00153 00154 00162 00164 00170 00166 00165 00158[31] 00268 00268 00248 00188 00203 00195 00196 00198 00211 00223

The bold data means the minimum error of the models with the same order

1300

1350

1400

1450

1500

1550

0 5 10 15 20 25 30 35 40 45 50Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(3 1) with L = 30

Figure 2 Actual values and forecasts of 2003 with the same L

use of different data sets seems to be the best reason toaccount for this contradiction From these discussions we geta conclusion that the proposed modelrsquos performance is morereasonable and robust than the three counterparts

Tables 4 and 5 depict that 119866119879119878(119872 3) 119866119879119878(119872 4) and119866119879119878(119872 5) have the same results although larger 119872 or 119873

is more easy to obtain the smaller forecasting errors In

0 5 10 15 20 25 30 35 40 45 501300

1350

1400

1450

1500

1550

Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(1 1) with L = 90

Figure 3 Actual values and forecasts of 2003 with different L

fact this trend is affected by the definitions of fuzzy setsand membership function As an in-depth analysis Figure 1shows three triangular membership functionsThe RMSEs offorecasts of 2003 by 119866119879119878(119872119873) with 119871 = 30 and the threemembership functions are listed in Table 6 The table tells usthat themore intervals that are concerned in themembership

Mathematical Problems in Engineering 9

Table 5 Comparison of forecasting SSECI

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 2274 2164 2121 2092 2086 2071 2066 2066 2025 2010119866119879119878(119872 2) 1832 1904 1790 1792 1769 1756 1754 1755 1695 1688119866119879119878(119872 3) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 4) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 5) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683

[28] 1841 1841 1942 1943 1951 1939 2005 1971 2000 2026[29] 2107 2102 2087 2073 2008 1988 1979 1969 1818 1821[31] 1756 1756 1781 1778 1838 1831 1867 1925 1947 1960

MAE

119866119879119878(119872 1) 1861 1710 1661 1612 1609 1608 1607 1611 1575 1550119866119879119878(119872 2) 1408 1389 1388 1391 1373 1353 1345 1346 1316 1317119866119879119878(119872 3) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 4) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 5) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306

[28] 1392 1392 1478 1502 1522 1518 1541 1538 1562 1590[29] 1731 1726 1715 1704 1677 1663 1656 1647 1604 1607[31] 1281 1281 1329 1306 1357 1338 1419 1437 1442 1551

MAPE

119866119879119878(119872 1) 00127 00117 00113 00110 00110 00110 00109 00110 00107 00105119866119879119878(119872 2) 00096 00095 00095 00095 00094 00092 00092 00092 00090 00090119866119879119878(119872 3) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 4) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 5) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089

[28] 00095 00095 00101 00102 00104 00103 00105 00105 00106 00108[29] 00118 00118 00117 00116 00114 00113 00113 00112 00109 00109[31] 00087 00087 00091 00089 00092 00091 00097 00098 00098 00099

The bold data means the minimum error of the models with the same order

Table 6 Effects of membership function

Model 119896 = 3 119896 = 6 119896 = 9

Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3119866119879119878(119896 1) 175133 173290 173755 171512 169790 170653 164968 162923 164014119866119879119878(119896 2) 158071 157476 159365 154880 154686 157182 147028 146918 147719119866119879119878(119896 3) 158071 156926 158736 154880 153137 154631 147028 145116 144828119866119879119878(119896 4) 158071 156902 158670 154880 153119 154813 147028 145218 144558119866119879119878(119896 5) 158071 156902 159157 154880 153119 155422 147028 145218 145082119866119879119878(119896 6) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 7) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 8) 158071 156902 159157 154880 153119 155424 147028 145218 145068The bold data means the minimum error of the models with the same order

function the more different forecasting accuracy obtainedIt is more obvious when 119872 is increasing There is stillanother conclusion that the forecasts of the thirdmembershipfunction are not always better than those of the first function

To further investigate the relation between the parameters119872 and the length of interval the proposed model has beenapplied to forecasting stock index close prices covering tenyears from 1997 to 2006 with 119872 = 1 2 50 and 119871 =

30 60 300 Since it has been affirmed by Tables 4 and5 that the characters of MAE and MAPE are similar to

those of RMSE MAE and MAPE we will only list theRMSE comparison of these experiments as follows Someexamples of actual values and forecasts of 2003 are depicted inFigures 2 and 3 Figure 2 shows us that 119866119879119878(3 1) has a betterperformance than 119866119879119878(1 1) in the same length of intervalsthat is 119871 = 30 Figure 3 illustrates that 119866119879119878(1 1) getsthe better forecasts when the lengths of intervals are smallFurthermore some properties will be further described inFigures 4 and 5 which depict the mean forecasting errorsof the ten years Figure 4 shows the relation between the

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

fuzzy logical relationship According toDefinition 4 all fuzzylogical relationships in the training data set can be furthergrouped together into different fuzzy logical relationshipgroups according to the same left-hand sides of the fuzzylogical relationship For given 119872 and 119873 the fuzzy logicalrelationships can be grouped into119872times119873matrices denoted as119877(119896119897)

(119896 = 1 2 119872 119897 = 1 2 119873) with the groupmethodproposed by Lee et al [13] Here 119903(119894119895) (119894 119895 isin 1 2 119899)the element of matrix 119877

(119896119897) is the number of fuzzy logicalrelationships 119860(119905minus119896119897)

119894rarr 119860119895

Then there are119873 fuzzy logical relationships matrices fora given training data set To forecast time series with thesegeneralized fuzzy logical relationship matrices we definedthe intersection operation as follows

Definition 6 Let 119860(119872119895)119905119895

represent the LHSs of FLRG in the119872-order 119895th-principal fuzzy logical relationship at time 119905Let 119877

(119872119895)(119905119895 119894) be the number of FLRG 119860

119905119895rarr 119860

119894in

the 119872th-order 119895th-principal fuzzy logical relationship (119895 =

1 2 119873 119894 119905119895

= 1 2 119899) To compute the logical rela-tionships between 119873 FLRGs the intersection operator and

119873is

defined as

and119873(119860(1198721)

1199051 119860

(119872119895)

119905119895 119860

(119872119873)

119905119873)

= (min1le119895le119873

119877(119872119895)

(119905119895 1) min

1le119895le119873

119877(119872119895)

(119905119895 119894)

min1le119895le119873

119877(119872119895)

(119905119895 119899))

(2)

Based on the above definitions this paper presents a high-order fuzzy time series model in the following section

3 Proposed Model

31 Procedure of 119866119879119878(119872119873) In this section we present anew forecastingmethod based on high-order and generalizedfuzzy logical relationships Since the proposed model isrelated to the number of orders denoted by 119872 andhierarchiesof principal fuzzy relationship denoted by 119873 we name theproposed model 119866119879119878(119872119873) In other words 119866119879119878(119872119873)

means an 119872-order fuzzy time series model based on 119873-principal fuzzy logical relationships

Step 1 Define the universe of discourse and intervals for rulesof extractionThe universe of discourse can be defined asU =[starting ending] According to equal length of intervals Uis partitioned into several intervals equally For example119880 =

1199061 1199062 119906

119899119898119894is themidpoint of 119906

119894whose corresponding

fuzzy set is 119860119894(119894 = 1 2 119899)

Step 2 Define fuzzy sets based on the universe of discourseand fuzzify the historical data The fuzzy set 119860

119894would be

expressed as 119860119894= (1198861198941 1198861198942 119886

119894119899) where 119886

119894119895isin [0 1] which

indicates the membership degree of 119906119895in 119860119894 The historical

and observed data are fuzzified according to the definition offuzzy sets For example a datum is fuzzified to 119860

119895 when the

maximal membership degree of the datum is the 119895th numberIn other words if 119886

119894119895= max119886

1198941 1198861198942 119886

119894119899 then the data at

time 119905 should be classified into the 119895th class In this paper thefuzzy sets are defined with triangular fuzzy function showedby formula (3)

Consider

1198601=

1

1199061

+05

1199062

+0

1199063

+ sdot sdot sdot +0

119906119899minus1

+0

119906119899

1198602=

05

1199061

+1

1199062

+05

1199063

+0

1199064

+ sdot sdot sdot +0

119906119894

+ sdot sdot sdot +0

119906119899minus1

+0

119906119899

119860119894=

0

1199061

+ sdot sdot sdot +0

119906119894minus2

+05

119906119894minus1

+1

119906119894

+05

119906119894+1

+0

119906119894+2

+ sdot sdot sdot +0

119906119899

119860119899=

0

1199061

+0

1199062

+0

1199063

+ sdot sdot sdot +0

119906119894

+ sdot sdot sdot +0

119906119899minus2

+05

119906119899minus1

+1

119906119899

(3)

Themembership degree of the value 119909119905at time 119905 in119860

119894(119894 =

1 2 119899) is defined by formula (4) Consider

120583119860119894

(119909119905)

=

1 if 119894 = 1 and 119909119905le 1198981

1 if 119894 = 119899 and 119909119905ge 119898119899

max0 1 minus

1003816100381610038161003816119909119894 minus 119898119894

1003816100381610038161003816

2 times 119897in others

(4)

where 119909119905is the observed value at time 119905 and 119897in is the length

of interval

Step 3 Establish the fuzzy logical relationships based on theorders and hierarchies of principal fuzzy logical relationshipGiven the sample data set and the definition of fuzzy setsall fuzzy logical relationships between two consecutive datacan be created To forecast the time series the fuzzy logicalrelationship matrix must be created in this step based on thefuzzy logical relationships

Among many different methods the method proposedby Lee in [13] is chosen in this paper For example the fuzzylogical relationships of a 119866119879119878(119872119873) model can be groupedinto 119872 times 119873 relationship matrices denoted by 119877

(119896119897)(119896 =

1 2 119872 119897 = 1 2 119873)

Step 4 Forecasting model Let 119865(119905 minus 119896) = (1205831(119905 minus 119896) 120583

2(119905 minus

119896) 120583119899(119905minus119896)) and let 120583

119905119897(119905minus119896) be the 119897th maxmembership

degree respectively By Definition 6 we have the intersectionfuzzy logical relationship and

119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873)

4 Mathematical Problems in Engineering

Table 1 Membership degrees of enrollment with respect to fuzzy sets

Year Value 1205831198601

1205831198602

1205831198603

1205831198604

1205831198605

1205831198606

1205831198607

Fuzzified1971 13055 10000 02775 0 0 0 0 0 119860

1 1198602

1972 13563 09685 05315 00315 0 0 0 0 1198601 1198602

1973 13867 08165 06835 01835 0 0 0 0 1198601 1198602

1974 14696 04020 09020 05980 00980 0 0 0 1198602 1198603

1975 15460 00200 05200 09800 04800 0 0 0 1198603 1198602

1976 15311 00945 05945 09055 04055 0 0 0 1198603 1198602

1977 15603 0 04485 09485 05515 00515 0 0 1198603 1198604

1978 15861 0 03195 08195 06805 01805 0 0 1198603 1198604

1979 16807 0 0 03465 08465 06535 01535 0 1198604 1198605

1980 16919 0 0 02905 07905 07095 02095 0 1198604 1198605

1981 16388 0 00560 05560 09440 04440 0 0 1198604 1198603

1982 15433 00335 05335 09665 04665 0 0 0 1198603 1198602

1983 15497 00015 05015 09985 04985 0 0 0 1198603 1198602

1984 15145 01775 06775 08225 03225 0 0 0 1198603 1198602

1985 15163 01685 06685 08315 03315 0 0 0 1198603 1198602

1986 15984 0 02580 07580 07420 02420 0 0 1198603 1198604

1987 16859 0 0 03205 08205 06795 01795 0 1198604 1198605

1988 18150 0 0 0 01750 06750 08250 03250 1198606 1198605

1989 18970 0 0 0 0 02650 07650 07350 1198606 1198607

1990 19328 0 0 0 0 00860 05860 09140 1198607 1198606

1991 19337 0 0 0 0 00815 05815 09185 1198607 1198606

1992 18876 0 0 0 0 03120 08120 06880 1198606 1198607

and then the 119896th forecasts 119865119896(119905) is conducted by formula (5)Consider

119865119896(119905)

= and119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873) ∘ (119898

1 1198982 119898

119899)119879

(5)

where ldquo∘rdquo is a composition operation for forecasting with thefollowing principles

(1) if the sum of and119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873) equals to

0 then the forecasted value is 1198981199051 and the midpoint

of the interval corresponding to 1198601199051

(2) otherwise the forecasted value is theweighting aggre-gate of119898

1199051 119898

119905119873

Then for a given 119872 there are 119872 forecasts for time 119905 Theconclusive forecasting value for time 119905 can be obtained by thefollowing formula

119865 (119905) =

119872

sum

119896=1

119865119896(119905) lowast 119908

119896 (6)

where 119908119896(119896 = 1 2 119872) is the adjustment parameter

for the 119896th forecast the parameter also can be obtained byminimizing the RMSE or other criteria of evaluations for thetraining data set

32 Computation of GTS(119872119873) on Forecasting EnrollmentsSince most of the conventional models have been presentedfor forecasting the historical enrolments of the Universityof Alabama in this section we present stepwise proceduresof the proposed method for forecasting the time series datawith 119872 = 3 and 119873 = 2 The historical enrolments ofthe University of Alabama from 1971 to 1992 are shown inTable 1

Step 1 Partitioning the universe of discourse 119880 into sevenintervals 119906

1 1199062 119906

7 where 119906

1= [13000 14000] 119906

2=

[14000 15000] 1199063

= [15000 16000] 1199064

= [16000 17000]1199065= [17000 18000] 119906

6= [18000 19000] and 119906

7= [19000

20000] their midpoints are 13500 14500 15500 16500 1750018500 and 19500 respectively

Step 2 Let 1198601= ldquonot manyrdquo 119860

2= ldquonot too manyrdquo 119860

3=

ldquomanyrdquo 1198604= ldquomany manyrdquo 119860

5= ldquovery manyrdquo 119860

6= ldquotoo

manyrdquo and 1198607= ldquotoo many manyrdquo 119860

1 1198602 119860

7are fuzzy

sets corresponding to linguistic values for ldquoenrolmentsrdquo

With triangular-shaped membership function defined byformula (4) the fuzzy sets and all observations are definedby the last column of Table 1 by formula (3) Thus the fuzzylogical relationships corresponding to given 119872 and 119873 arelisted in Table 2

Step 3 Divide the derived fuzzy logical relationships intogroups based on the states of the enrollments of fuzzy logicalrelationships

Mathematical Problems in Engineering 5

Table 2 Fuzzy logical relationships with119872 = 3 and119873 = 2

Year fuzzified 119896 = 1 119896 = 2 119896 = 3

119897 = 1 119897 = 2 119897 = 1 119897 = 2 119897 = 1 119897 = 2

1971 1198601 1198602

1972 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

1

1973 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

11198601997888rarr 119860

1 1198602997888rarr 119860

1

1974 1198602 1198603

1198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

2

1975 1198603 1198602

1198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1976 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1977 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

3

1978 1198603 1198604

1198603997888rarr 119860

3 1198604997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1979 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1980 1198604 1198605

1198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1981 1198604 1198603

1198604997888rarr 119860

4 1198605997888rarr 119860

41198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1982 1198603 1198602

1198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1983 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1984 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

3

1985 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1986 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1987 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1988 1198606 1198605

1198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

61198603997888rarr 119860

6 1198602997888rarr 119860

6

1989 1198606 1198607

1198606997888rarr 119860

6 1198605997888rarr 119860

61198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

6

1990 1198607 1198606

1198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

71198604997888rarr 119860

7 1198605997888rarr 119860

7

1991 1198607 1198606

1198607997888rarr 119860

7 1198606997888rarr 119860

71198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

7

1992 1198606 1198607

1198607997888rarr 119860

6 1198606997888rarr 119860

61198607997888rarr 119860

6 1198606997888rarr 119860

61198606997888rarr 119860

6 1198607997888rarr 119860

6

In this paper we use the method of Lee [13] to constructthe fuzzy logical relationships matrix For example let k = 1and l = 1 The set of fuzzy logical relationships is listed as

1198601rarr 11986011198601rarr 11986011198601rarr 11986021198602rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198604

1198604rarr 11986041198604rarr 11986031198603rarr 11986031198603rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198606

1198606rarr 1198606 1198606rarr 1198607 1198607rarr 1198607 1198607rarr 1198606 which

are all of FLRGs in the third column of Table 2

And they would be grouped and weighted by the recur-rent fuzzy relationships as follows

Group 1 1198601rarr 1198601with weight 2 119860

1rarr 1198602with weight 1

Group 2 1198602rarr 1198603with weight 1

Group 3 1198603rarr 1198603with weight 7 119860

3rarr 1198604with weight 2

Group 4 1198604

rarr 1198604with weight 2 119860

4rarr 119860

3with weight 1

1198604rarr 1198606with weight 1

Group 5 1198606rarr 1198606with weight 1 119860

6rarr 1198607with weight 1

Group 6 1198607rarr 1198607with weight 1 119860

7rarr 1198606with weight 1

The fuzzy logical relationship matrix then is 119877(11) Basedon the fourth fifth sixth seventh and eighth columnof Table 2 119877(12) 119877(21) 119877(22) 119877(31) and 119877

(32) can also be

obtained in the similar way These six fuzzy logical relation-ship matrices have been listed as follows

119877(11)

=

[[[[[[[[[[[[

[

2 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 7 2 0 0 0

0 0 1 2 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 1 1

]]]]]]]]]]]]

]

119877(12)

=

[[[[[[[[[[[[

[

0 0 0 0 0 0 0

2 1 6 0 0 0 0

0 0 2 0 0 0 0

0 0 2 2 0 0 0

0 0 0 2 0 2 0

0 0 0 0 0 1 1

0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(7)

119877(21)

=

[[[[[[[[[[

[

1 1 1 0 0 0 0

0 0 1 0 0 0 0

0 0 5 3 0 1 0

0 0 2 1 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 0 2

0 0 0 0 0 1 0

]]]]]]]]]]

]

6 Mathematical Problems in Engineering

119877(22)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

1 1 6 1 0 0 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 1 1 0 1 1

0 0 0 0 0 1 0

0 0 0 0 0 0 1

]]]]]]]]]

]

(8)

119877(31)

=

[[[[[[[[[

[

0 1 2 0 0 0 0

0 0 1 0 0 0 0

0 0 3 4 0 2 0

0 0 3 2 0 0 1

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 0 0

]]]]]]]]]

]

119877(32)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

0 1 5 2 0 1 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 2 0 0 0 2

0 0 0 0 0 0 0

0 0 0 0 0 1 0

]]]]]]]]]

]

(9)

Step 4 Calculate the forecasts To carry out the calculationsby the proposed principles we illustrate the forecastingprocess of enrolment of 1976 as follows From Table 1 thefuzzy set of enrolment of 1975 is (002 052 098 048 0 0 0)The two maximum membership attributes are the third andthe second Then 119896 = 1 the third row of 119877(11) is (0 0 7 20 0 0) and the second row of 119877(12) is (2 1 6 0 0 0 0) from(7) According to Definition 6 we have and

2(119860(11)

19751 119860(12)

19752) =

(0 0 1 0 0 0 0)

According to the second principle listed in Section 311198651(1976) then is119898

3 that is 15500 which is the middle point

of 1198603

When k = 2 the fuzzy set of 1974s is (0402 09020598 0098 0 0 0) shown in Table 1 The two maximummembership attributes are the second and the third Thesecond row of 119877(21) then is (0 0 1 0 0 0 0) and the thirdrow of 119877(22) is (0 0 2 0 0 0 0) from (8) With Definition 6we have and

2(119860(21)

19751 119860(22)

19752) = (0 0 1 0 0 0 0) By the first

principle listed in Section 31 1198652(1976) also is 15500In the similar way fromTable 1 andwith 119896 = 3 we can see

that the fuzzy set of 1973s is (08165 06835 01835 0 0 0 0)The two maximum membership attributes are the first andsecond ones The first row of 119877(31) is (0 1 2 0 0 0 0) andthe second row of 119877(32) is (0 1 5 2 0 0 0) from (9) WithDefinition 6 we have and

2(119860(31)

19751 119860(32)

19752) = (0 1 2 0 0 0 0)

With the second principle listed in Section 31 in mind1198653(1976) equals to (13) times 14500 + (23) times 15500 = 15167At last the forecasted value of 1976 that is 119865(1976) is

15590 which equals to 09130 times 15500 + 00790 times 15500 +

00142times1516 Also (1199081 1199082 1199083) = (09130 00790 00142)

is the regress result of the enrolment data Table 3 shows theactual and the forecasted enrolments of the data set

4 Empirical Analysis

41 Data Description To demonstrate the effectiveness ofthe proposed models amounts of data are needed Here theenrolments and SSECI are used as the illustration data sets forthe empirical analysis

There are some causes for the two time series to be thesubjects in our experiment There are two causes for thechoice of enrolment data at the University of Alabama Thefirst one is that most of the fuzzy time series studies havetaken this well-known time series as their experimentsThusthere are a lot of studies that can be used for our referenceThe other is that it is simple and easy to display the process ofthe proposedmodel Since time series models have been usedto make predictions in the areas of stock price forecasting formany years the daily SSECI covering the period from 1997 to2006 is adopted for further experiment

For the first data the determined length of the sevenintervals is 1000 All of the forecasting results from one orderto ten orders are compared with those of the conventionalfuzzy time series models based on fuzzy logical relationships

To discuss the effects of parameters 119872 and 119873 the ordernumber and hierarchies of fuzzy logical relationships on theforecasting results the ten-order (time-lag periods) fromone order to ten orders models are performed on thesecond data set with ten different lengths of intervals thatis 30 60 90 300 The forecasting results obtained by allof the high-order models are compared in terms of threeevaluation criteria reviewed in following subsection

42 Criteria of Evaluation In statistics the root meansquared error (RMSE) mean absolute error (MAE) andmean absolute percentage error (MAPE) are three typicalways to quantify the difference between values implied byan estimator and the actual values of the quantity beingestimated MSE is a risk function for measuring the averageof the squares of the difference For an unbiased estimatorthe MSE is the variance and the RMSE is the square rootof the variance known as the standard error FurthermoreRMSE is superior to MSE for the reason that its scale isthe same as the forecasts Thus we take RMSE as the firstrepresentative of the size of an average error As an averageof the absolute percent errors MAPE serves as a criterionfor the comparisons of forecasting results in the paper Somecomparisons of accuracy in the forecasted values of ourproposed models with other models are made on the basisof the three criteria

43 Performance Evaluation In Table 4 this study com-pares the RMSE MAE and MAPE forecasting value ofthe proposed method and the counterparts [28 29 31] onthe enrolment experiment Table 4 shows that the proposedmethod gets smaller RMSE MAE and MAPE than Hwangrsquosmodel [28] and also smaller than Chenrsquos model [31] of 2011 inmost cases Although there is a fly in the ointment that is theproposed model not always gets a higher average forecastingaccuracy rate than the model [29] of 2002 these results areimproving as the orders increase In summary the resultssuggest that the proposed model obtains better forecasts as

Mathematical Problems in Engineering 7

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1 Membership function 1

(a)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 2

(b)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 3

(c)

Figure 1 Three membership functions for forecasting SHI of 2003

Table 3 Forecasts of enrollment year actual data

Year Value 119872 = 1 119872 = 2 119872 = 3

119873 = 1 119873 = 2 119873 = 1 119873 = 2 119873 = 1 119873 = 2

1971 13055 mdash mdash mdash mdash mdash mdash1972 13563 13901 13902 mdash mdash mdash mdash1973 13867 13901 13902 13980 13968 mdash mdash1974 14696 13901 13902 13980 13968 14062 139901975 15460 15576 15576 15491 15510 15533 155111976 15311 15799 15576 15792 15590 15760 155901977 15603 15799 15576 15858 15604 15812 156081978 15861 15799 16246 15858 16221 15915 162281979 16807 15799 16246 15858 16342 15915 163471980 16919 16832 17251 16790 17267 16821 172721981 16388 16832 17251 16823 17240 16832 172451982 15433 16832 15576 16823 15698 16821 157191983 15497 15799 15576 15891 15590 15915 156141984 15145 15799 15576 15858 15604 15904 156081985 15163 15799 15576 15858 15604 15915 156201986 15984 15799 15576 15858 15604 15915 156201987 16859 15799 16246 15858 16221 15915 162281988 18150 16832 17251 16790 17267 16821 172601989 18970 19093 18591 18863 18473 18816 184631990 19328 19093 19596 19161 19614 18900 195811991 19337 19093 19094 19161 19151 19132 191631992 18876 19093 19094 19061 19070 19101 19070

MAE 473365 3720021 4713957 375308 5011709 3902802RMSE 6254334 4474897 6316806 4488908 6432859 4597683MAPE 00293 00227 00291 00227 00307 00236

the orders and hierarchies increase Unlike this point this isnot the trend in the three counterparts

Moreover we also apply the proposed method to handleforecasting the close price of Shanghai stock index of 2003with 119897 = 60 The comparison of the three criteria is listedin Table 5 From this table we can see that the proposed119866119879119878(119872119873) gets the best forecasts of the three counterpartswhen 119872 gt 5 and 119873 gt 2 as well as its better performancethan those of Hwangrsquos and Chenrsquos model [28 29] in all cases

and the shortcoming shown in Table 4 is gone On the wholethere is a ldquolawrdquo that forecasting errorswill be reducedwhen119872

or119873 is increased in the proposed model However this trendis not evident for the three counterparts by Tables 4 and 5 InTable 4 the three evaluation criteria of Hwangrsquos and Chenrsquosmodels [28 31] are decreasing while those of model [29] areincreasing In contrast it is obvious that the three evaluationcriteria of Hwangrsquos and Chenrsquos model are increasing whilethose of Chenrsquos model [29] are decreasing from Table 5 The

8 Mathematical Problems in Engineering

Table 4 Comparison of forecasting enrollment

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 625 632 643 641 658 575 562 289 203 191119866119879119878(119872 2) 447 449 460 436 448 456 437 413 391 284119866119879119878(119872 3) 447 449 460 436 448 456 406 382 365 225119866119879119878(119872 4) 447 449 460 434 445 453 423 415 397 253119866119879119878(119872 5) 447 449 460 434 445 453 423 415 397 253

[28] 588 588 578 608 602 620 632 575 554 498[29] 292 298 294 299 307 313 323 320 321 311[31] 543 543 522 420 456 443 452 463 480 498

MAE

119866119879119878(119872 1) 473 471 501 484 501 366 411 212 326 140119866119879119878(119872 2) 372 375 390 365 383 394 406 364 292 218119866119879119878(119872 3) 372 375 390 365 383 393 374 336 333 176119866119879119878(119872 4) 372 375 390 367 384 393 402 386 333 195119866119879119878(119872 5) 372 375 390 367 384 393 402 386 333 195

[28] 494 494 473 526 527 556 582 506 483 424[29] 252 262 256 259 272 277 289 284 282 271[31] 441 441 412 316 347 337 340 346 368 390

MAPE

119866119879119878(119872 1) 00293 00291 00307 00294 00305 00221 00246 00127 00103 00087119866119879119878(119872 2) 00227 00227 00236 00216 00227 00232 00239 00216 00195 00125119866119879119878(119872 3) 00227 00227 00236 00216 00227 00232 00220 00200 00176 00101119866119879119878(119872 4) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111119866119879119878(119872 5) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111

[28] 00299 00299 00283 00317 00313 00329 00345 00299 00285 00253[29] 00153 00159 00153 00154 00162 00164 00170 00166 00165 00158[31] 00268 00268 00248 00188 00203 00195 00196 00198 00211 00223

The bold data means the minimum error of the models with the same order

1300

1350

1400

1450

1500

1550

0 5 10 15 20 25 30 35 40 45 50Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(3 1) with L = 30

Figure 2 Actual values and forecasts of 2003 with the same L

use of different data sets seems to be the best reason toaccount for this contradiction From these discussions we geta conclusion that the proposed modelrsquos performance is morereasonable and robust than the three counterparts

Tables 4 and 5 depict that 119866119879119878(119872 3) 119866119879119878(119872 4) and119866119879119878(119872 5) have the same results although larger 119872 or 119873

is more easy to obtain the smaller forecasting errors In

0 5 10 15 20 25 30 35 40 45 501300

1350

1400

1450

1500

1550

Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(1 1) with L = 90

Figure 3 Actual values and forecasts of 2003 with different L

fact this trend is affected by the definitions of fuzzy setsand membership function As an in-depth analysis Figure 1shows three triangular membership functionsThe RMSEs offorecasts of 2003 by 119866119879119878(119872119873) with 119871 = 30 and the threemembership functions are listed in Table 6 The table tells usthat themore intervals that are concerned in themembership

Mathematical Problems in Engineering 9

Table 5 Comparison of forecasting SSECI

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 2274 2164 2121 2092 2086 2071 2066 2066 2025 2010119866119879119878(119872 2) 1832 1904 1790 1792 1769 1756 1754 1755 1695 1688119866119879119878(119872 3) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 4) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 5) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683

[28] 1841 1841 1942 1943 1951 1939 2005 1971 2000 2026[29] 2107 2102 2087 2073 2008 1988 1979 1969 1818 1821[31] 1756 1756 1781 1778 1838 1831 1867 1925 1947 1960

MAE

119866119879119878(119872 1) 1861 1710 1661 1612 1609 1608 1607 1611 1575 1550119866119879119878(119872 2) 1408 1389 1388 1391 1373 1353 1345 1346 1316 1317119866119879119878(119872 3) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 4) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 5) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306

[28] 1392 1392 1478 1502 1522 1518 1541 1538 1562 1590[29] 1731 1726 1715 1704 1677 1663 1656 1647 1604 1607[31] 1281 1281 1329 1306 1357 1338 1419 1437 1442 1551

MAPE

119866119879119878(119872 1) 00127 00117 00113 00110 00110 00110 00109 00110 00107 00105119866119879119878(119872 2) 00096 00095 00095 00095 00094 00092 00092 00092 00090 00090119866119879119878(119872 3) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 4) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 5) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089

[28] 00095 00095 00101 00102 00104 00103 00105 00105 00106 00108[29] 00118 00118 00117 00116 00114 00113 00113 00112 00109 00109[31] 00087 00087 00091 00089 00092 00091 00097 00098 00098 00099

The bold data means the minimum error of the models with the same order

Table 6 Effects of membership function

Model 119896 = 3 119896 = 6 119896 = 9

Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3119866119879119878(119896 1) 175133 173290 173755 171512 169790 170653 164968 162923 164014119866119879119878(119896 2) 158071 157476 159365 154880 154686 157182 147028 146918 147719119866119879119878(119896 3) 158071 156926 158736 154880 153137 154631 147028 145116 144828119866119879119878(119896 4) 158071 156902 158670 154880 153119 154813 147028 145218 144558119866119879119878(119896 5) 158071 156902 159157 154880 153119 155422 147028 145218 145082119866119879119878(119896 6) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 7) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 8) 158071 156902 159157 154880 153119 155424 147028 145218 145068The bold data means the minimum error of the models with the same order

function the more different forecasting accuracy obtainedIt is more obvious when 119872 is increasing There is stillanother conclusion that the forecasts of the thirdmembershipfunction are not always better than those of the first function

To further investigate the relation between the parameters119872 and the length of interval the proposed model has beenapplied to forecasting stock index close prices covering tenyears from 1997 to 2006 with 119872 = 1 2 50 and 119871 =

30 60 300 Since it has been affirmed by Tables 4 and5 that the characters of MAE and MAPE are similar to

those of RMSE MAE and MAPE we will only list theRMSE comparison of these experiments as follows Someexamples of actual values and forecasts of 2003 are depicted inFigures 2 and 3 Figure 2 shows us that 119866119879119878(3 1) has a betterperformance than 119866119879119878(1 1) in the same length of intervalsthat is 119871 = 30 Figure 3 illustrates that 119866119879119878(1 1) getsthe better forecasts when the lengths of intervals are smallFurthermore some properties will be further described inFigures 4 and 5 which depict the mean forecasting errorsof the ten years Figure 4 shows the relation between the

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Table 1 Membership degrees of enrollment with respect to fuzzy sets

Year Value 1205831198601

1205831198602

1205831198603

1205831198604

1205831198605

1205831198606

1205831198607

Fuzzified1971 13055 10000 02775 0 0 0 0 0 119860

1 1198602

1972 13563 09685 05315 00315 0 0 0 0 1198601 1198602

1973 13867 08165 06835 01835 0 0 0 0 1198601 1198602

1974 14696 04020 09020 05980 00980 0 0 0 1198602 1198603

1975 15460 00200 05200 09800 04800 0 0 0 1198603 1198602

1976 15311 00945 05945 09055 04055 0 0 0 1198603 1198602

1977 15603 0 04485 09485 05515 00515 0 0 1198603 1198604

1978 15861 0 03195 08195 06805 01805 0 0 1198603 1198604

1979 16807 0 0 03465 08465 06535 01535 0 1198604 1198605

1980 16919 0 0 02905 07905 07095 02095 0 1198604 1198605

1981 16388 0 00560 05560 09440 04440 0 0 1198604 1198603

1982 15433 00335 05335 09665 04665 0 0 0 1198603 1198602

1983 15497 00015 05015 09985 04985 0 0 0 1198603 1198602

1984 15145 01775 06775 08225 03225 0 0 0 1198603 1198602

1985 15163 01685 06685 08315 03315 0 0 0 1198603 1198602

1986 15984 0 02580 07580 07420 02420 0 0 1198603 1198604

1987 16859 0 0 03205 08205 06795 01795 0 1198604 1198605

1988 18150 0 0 0 01750 06750 08250 03250 1198606 1198605

1989 18970 0 0 0 0 02650 07650 07350 1198606 1198607

1990 19328 0 0 0 0 00860 05860 09140 1198607 1198606

1991 19337 0 0 0 0 00815 05815 09185 1198607 1198606

1992 18876 0 0 0 0 03120 08120 06880 1198606 1198607

and then the 119896th forecasts 119865119896(119905) is conducted by formula (5)Consider

119865119896(119905)

= and119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873) ∘ (119898

1 1198982 119898

119899)119879

(5)

where ldquo∘rdquo is a composition operation for forecasting with thefollowing principles

(1) if the sum of and119873(119860(1198961)

1199051 119860

(119896119897)

119905119897 119860

(119896119873)

119905119873) equals to

0 then the forecasted value is 1198981199051 and the midpoint

of the interval corresponding to 1198601199051

(2) otherwise the forecasted value is theweighting aggre-gate of119898

1199051 119898

119905119873

Then for a given 119872 there are 119872 forecasts for time 119905 Theconclusive forecasting value for time 119905 can be obtained by thefollowing formula

119865 (119905) =

119872

sum

119896=1

119865119896(119905) lowast 119908

119896 (6)

where 119908119896(119896 = 1 2 119872) is the adjustment parameter

for the 119896th forecast the parameter also can be obtained byminimizing the RMSE or other criteria of evaluations for thetraining data set

32 Computation of GTS(119872119873) on Forecasting EnrollmentsSince most of the conventional models have been presentedfor forecasting the historical enrolments of the Universityof Alabama in this section we present stepwise proceduresof the proposed method for forecasting the time series datawith 119872 = 3 and 119873 = 2 The historical enrolments ofthe University of Alabama from 1971 to 1992 are shown inTable 1

Step 1 Partitioning the universe of discourse 119880 into sevenintervals 119906

1 1199062 119906

7 where 119906

1= [13000 14000] 119906

2=

[14000 15000] 1199063

= [15000 16000] 1199064

= [16000 17000]1199065= [17000 18000] 119906

6= [18000 19000] and 119906

7= [19000

20000] their midpoints are 13500 14500 15500 16500 1750018500 and 19500 respectively

Step 2 Let 1198601= ldquonot manyrdquo 119860

2= ldquonot too manyrdquo 119860

3=

ldquomanyrdquo 1198604= ldquomany manyrdquo 119860

5= ldquovery manyrdquo 119860

6= ldquotoo

manyrdquo and 1198607= ldquotoo many manyrdquo 119860

1 1198602 119860

7are fuzzy

sets corresponding to linguistic values for ldquoenrolmentsrdquo

With triangular-shaped membership function defined byformula (4) the fuzzy sets and all observations are definedby the last column of Table 1 by formula (3) Thus the fuzzylogical relationships corresponding to given 119872 and 119873 arelisted in Table 2

Step 3 Divide the derived fuzzy logical relationships intogroups based on the states of the enrollments of fuzzy logicalrelationships

Mathematical Problems in Engineering 5

Table 2 Fuzzy logical relationships with119872 = 3 and119873 = 2

Year fuzzified 119896 = 1 119896 = 2 119896 = 3

119897 = 1 119897 = 2 119897 = 1 119897 = 2 119897 = 1 119897 = 2

1971 1198601 1198602

1972 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

1

1973 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

11198601997888rarr 119860

1 1198602997888rarr 119860

1

1974 1198602 1198603

1198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

2

1975 1198603 1198602

1198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1976 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1977 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

3

1978 1198603 1198604

1198603997888rarr 119860

3 1198604997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1979 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1980 1198604 1198605

1198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1981 1198604 1198603

1198604997888rarr 119860

4 1198605997888rarr 119860

41198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1982 1198603 1198602

1198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1983 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1984 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

3

1985 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1986 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1987 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1988 1198606 1198605

1198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

61198603997888rarr 119860

6 1198602997888rarr 119860

6

1989 1198606 1198607

1198606997888rarr 119860

6 1198605997888rarr 119860

61198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

6

1990 1198607 1198606

1198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

71198604997888rarr 119860

7 1198605997888rarr 119860

7

1991 1198607 1198606

1198607997888rarr 119860

7 1198606997888rarr 119860

71198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

7

1992 1198606 1198607

1198607997888rarr 119860

6 1198606997888rarr 119860

61198607997888rarr 119860

6 1198606997888rarr 119860

61198606997888rarr 119860

6 1198607997888rarr 119860

6

In this paper we use the method of Lee [13] to constructthe fuzzy logical relationships matrix For example let k = 1and l = 1 The set of fuzzy logical relationships is listed as

1198601rarr 11986011198601rarr 11986011198601rarr 11986021198602rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198604

1198604rarr 11986041198604rarr 11986031198603rarr 11986031198603rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198606

1198606rarr 1198606 1198606rarr 1198607 1198607rarr 1198607 1198607rarr 1198606 which

are all of FLRGs in the third column of Table 2

And they would be grouped and weighted by the recur-rent fuzzy relationships as follows

Group 1 1198601rarr 1198601with weight 2 119860

1rarr 1198602with weight 1

Group 2 1198602rarr 1198603with weight 1

Group 3 1198603rarr 1198603with weight 7 119860

3rarr 1198604with weight 2

Group 4 1198604

rarr 1198604with weight 2 119860

4rarr 119860

3with weight 1

1198604rarr 1198606with weight 1

Group 5 1198606rarr 1198606with weight 1 119860

6rarr 1198607with weight 1

Group 6 1198607rarr 1198607with weight 1 119860

7rarr 1198606with weight 1

The fuzzy logical relationship matrix then is 119877(11) Basedon the fourth fifth sixth seventh and eighth columnof Table 2 119877(12) 119877(21) 119877(22) 119877(31) and 119877

(32) can also be

obtained in the similar way These six fuzzy logical relation-ship matrices have been listed as follows

119877(11)

=

[[[[[[[[[[[[

[

2 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 7 2 0 0 0

0 0 1 2 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 1 1

]]]]]]]]]]]]

]

119877(12)

=

[[[[[[[[[[[[

[

0 0 0 0 0 0 0

2 1 6 0 0 0 0

0 0 2 0 0 0 0

0 0 2 2 0 0 0

0 0 0 2 0 2 0

0 0 0 0 0 1 1

0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(7)

119877(21)

=

[[[[[[[[[[

[

1 1 1 0 0 0 0

0 0 1 0 0 0 0

0 0 5 3 0 1 0

0 0 2 1 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 0 2

0 0 0 0 0 1 0

]]]]]]]]]]

]

6 Mathematical Problems in Engineering

119877(22)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

1 1 6 1 0 0 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 1 1 0 1 1

0 0 0 0 0 1 0

0 0 0 0 0 0 1

]]]]]]]]]

]

(8)

119877(31)

=

[[[[[[[[[

[

0 1 2 0 0 0 0

0 0 1 0 0 0 0

0 0 3 4 0 2 0

0 0 3 2 0 0 1

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 0 0

]]]]]]]]]

]

119877(32)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

0 1 5 2 0 1 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 2 0 0 0 2

0 0 0 0 0 0 0

0 0 0 0 0 1 0

]]]]]]]]]

]

(9)

Step 4 Calculate the forecasts To carry out the calculationsby the proposed principles we illustrate the forecastingprocess of enrolment of 1976 as follows From Table 1 thefuzzy set of enrolment of 1975 is (002 052 098 048 0 0 0)The two maximum membership attributes are the third andthe second Then 119896 = 1 the third row of 119877(11) is (0 0 7 20 0 0) and the second row of 119877(12) is (2 1 6 0 0 0 0) from(7) According to Definition 6 we have and

2(119860(11)

19751 119860(12)

19752) =

(0 0 1 0 0 0 0)

According to the second principle listed in Section 311198651(1976) then is119898

3 that is 15500 which is the middle point

of 1198603

When k = 2 the fuzzy set of 1974s is (0402 09020598 0098 0 0 0) shown in Table 1 The two maximummembership attributes are the second and the third Thesecond row of 119877(21) then is (0 0 1 0 0 0 0) and the thirdrow of 119877(22) is (0 0 2 0 0 0 0) from (8) With Definition 6we have and

2(119860(21)

19751 119860(22)

19752) = (0 0 1 0 0 0 0) By the first

principle listed in Section 31 1198652(1976) also is 15500In the similar way fromTable 1 andwith 119896 = 3 we can see

that the fuzzy set of 1973s is (08165 06835 01835 0 0 0 0)The two maximum membership attributes are the first andsecond ones The first row of 119877(31) is (0 1 2 0 0 0 0) andthe second row of 119877(32) is (0 1 5 2 0 0 0) from (9) WithDefinition 6 we have and

2(119860(31)

19751 119860(32)

19752) = (0 1 2 0 0 0 0)

With the second principle listed in Section 31 in mind1198653(1976) equals to (13) times 14500 + (23) times 15500 = 15167At last the forecasted value of 1976 that is 119865(1976) is

15590 which equals to 09130 times 15500 + 00790 times 15500 +

00142times1516 Also (1199081 1199082 1199083) = (09130 00790 00142)

is the regress result of the enrolment data Table 3 shows theactual and the forecasted enrolments of the data set

4 Empirical Analysis

41 Data Description To demonstrate the effectiveness ofthe proposed models amounts of data are needed Here theenrolments and SSECI are used as the illustration data sets forthe empirical analysis

There are some causes for the two time series to be thesubjects in our experiment There are two causes for thechoice of enrolment data at the University of Alabama Thefirst one is that most of the fuzzy time series studies havetaken this well-known time series as their experimentsThusthere are a lot of studies that can be used for our referenceThe other is that it is simple and easy to display the process ofthe proposedmodel Since time series models have been usedto make predictions in the areas of stock price forecasting formany years the daily SSECI covering the period from 1997 to2006 is adopted for further experiment

For the first data the determined length of the sevenintervals is 1000 All of the forecasting results from one orderto ten orders are compared with those of the conventionalfuzzy time series models based on fuzzy logical relationships

To discuss the effects of parameters 119872 and 119873 the ordernumber and hierarchies of fuzzy logical relationships on theforecasting results the ten-order (time-lag periods) fromone order to ten orders models are performed on thesecond data set with ten different lengths of intervals thatis 30 60 90 300 The forecasting results obtained by allof the high-order models are compared in terms of threeevaluation criteria reviewed in following subsection

42 Criteria of Evaluation In statistics the root meansquared error (RMSE) mean absolute error (MAE) andmean absolute percentage error (MAPE) are three typicalways to quantify the difference between values implied byan estimator and the actual values of the quantity beingestimated MSE is a risk function for measuring the averageof the squares of the difference For an unbiased estimatorthe MSE is the variance and the RMSE is the square rootof the variance known as the standard error FurthermoreRMSE is superior to MSE for the reason that its scale isthe same as the forecasts Thus we take RMSE as the firstrepresentative of the size of an average error As an averageof the absolute percent errors MAPE serves as a criterionfor the comparisons of forecasting results in the paper Somecomparisons of accuracy in the forecasted values of ourproposed models with other models are made on the basisof the three criteria

43 Performance Evaluation In Table 4 this study com-pares the RMSE MAE and MAPE forecasting value ofthe proposed method and the counterparts [28 29 31] onthe enrolment experiment Table 4 shows that the proposedmethod gets smaller RMSE MAE and MAPE than Hwangrsquosmodel [28] and also smaller than Chenrsquos model [31] of 2011 inmost cases Although there is a fly in the ointment that is theproposed model not always gets a higher average forecastingaccuracy rate than the model [29] of 2002 these results areimproving as the orders increase In summary the resultssuggest that the proposed model obtains better forecasts as

Mathematical Problems in Engineering 7

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1 Membership function 1

(a)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 2

(b)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 3

(c)

Figure 1 Three membership functions for forecasting SHI of 2003

Table 3 Forecasts of enrollment year actual data

Year Value 119872 = 1 119872 = 2 119872 = 3

119873 = 1 119873 = 2 119873 = 1 119873 = 2 119873 = 1 119873 = 2

1971 13055 mdash mdash mdash mdash mdash mdash1972 13563 13901 13902 mdash mdash mdash mdash1973 13867 13901 13902 13980 13968 mdash mdash1974 14696 13901 13902 13980 13968 14062 139901975 15460 15576 15576 15491 15510 15533 155111976 15311 15799 15576 15792 15590 15760 155901977 15603 15799 15576 15858 15604 15812 156081978 15861 15799 16246 15858 16221 15915 162281979 16807 15799 16246 15858 16342 15915 163471980 16919 16832 17251 16790 17267 16821 172721981 16388 16832 17251 16823 17240 16832 172451982 15433 16832 15576 16823 15698 16821 157191983 15497 15799 15576 15891 15590 15915 156141984 15145 15799 15576 15858 15604 15904 156081985 15163 15799 15576 15858 15604 15915 156201986 15984 15799 15576 15858 15604 15915 156201987 16859 15799 16246 15858 16221 15915 162281988 18150 16832 17251 16790 17267 16821 172601989 18970 19093 18591 18863 18473 18816 184631990 19328 19093 19596 19161 19614 18900 195811991 19337 19093 19094 19161 19151 19132 191631992 18876 19093 19094 19061 19070 19101 19070

MAE 473365 3720021 4713957 375308 5011709 3902802RMSE 6254334 4474897 6316806 4488908 6432859 4597683MAPE 00293 00227 00291 00227 00307 00236

the orders and hierarchies increase Unlike this point this isnot the trend in the three counterparts

Moreover we also apply the proposed method to handleforecasting the close price of Shanghai stock index of 2003with 119897 = 60 The comparison of the three criteria is listedin Table 5 From this table we can see that the proposed119866119879119878(119872119873) gets the best forecasts of the three counterpartswhen 119872 gt 5 and 119873 gt 2 as well as its better performancethan those of Hwangrsquos and Chenrsquos model [28 29] in all cases

and the shortcoming shown in Table 4 is gone On the wholethere is a ldquolawrdquo that forecasting errorswill be reducedwhen119872

or119873 is increased in the proposed model However this trendis not evident for the three counterparts by Tables 4 and 5 InTable 4 the three evaluation criteria of Hwangrsquos and Chenrsquosmodels [28 31] are decreasing while those of model [29] areincreasing In contrast it is obvious that the three evaluationcriteria of Hwangrsquos and Chenrsquos model are increasing whilethose of Chenrsquos model [29] are decreasing from Table 5 The

8 Mathematical Problems in Engineering

Table 4 Comparison of forecasting enrollment

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 625 632 643 641 658 575 562 289 203 191119866119879119878(119872 2) 447 449 460 436 448 456 437 413 391 284119866119879119878(119872 3) 447 449 460 436 448 456 406 382 365 225119866119879119878(119872 4) 447 449 460 434 445 453 423 415 397 253119866119879119878(119872 5) 447 449 460 434 445 453 423 415 397 253

[28] 588 588 578 608 602 620 632 575 554 498[29] 292 298 294 299 307 313 323 320 321 311[31] 543 543 522 420 456 443 452 463 480 498

MAE

119866119879119878(119872 1) 473 471 501 484 501 366 411 212 326 140119866119879119878(119872 2) 372 375 390 365 383 394 406 364 292 218119866119879119878(119872 3) 372 375 390 365 383 393 374 336 333 176119866119879119878(119872 4) 372 375 390 367 384 393 402 386 333 195119866119879119878(119872 5) 372 375 390 367 384 393 402 386 333 195

[28] 494 494 473 526 527 556 582 506 483 424[29] 252 262 256 259 272 277 289 284 282 271[31] 441 441 412 316 347 337 340 346 368 390

MAPE

119866119879119878(119872 1) 00293 00291 00307 00294 00305 00221 00246 00127 00103 00087119866119879119878(119872 2) 00227 00227 00236 00216 00227 00232 00239 00216 00195 00125119866119879119878(119872 3) 00227 00227 00236 00216 00227 00232 00220 00200 00176 00101119866119879119878(119872 4) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111119866119879119878(119872 5) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111

[28] 00299 00299 00283 00317 00313 00329 00345 00299 00285 00253[29] 00153 00159 00153 00154 00162 00164 00170 00166 00165 00158[31] 00268 00268 00248 00188 00203 00195 00196 00198 00211 00223

The bold data means the minimum error of the models with the same order

1300

1350

1400

1450

1500

1550

0 5 10 15 20 25 30 35 40 45 50Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(3 1) with L = 30

Figure 2 Actual values and forecasts of 2003 with the same L

use of different data sets seems to be the best reason toaccount for this contradiction From these discussions we geta conclusion that the proposed modelrsquos performance is morereasonable and robust than the three counterparts

Tables 4 and 5 depict that 119866119879119878(119872 3) 119866119879119878(119872 4) and119866119879119878(119872 5) have the same results although larger 119872 or 119873

is more easy to obtain the smaller forecasting errors In

0 5 10 15 20 25 30 35 40 45 501300

1350

1400

1450

1500

1550

Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(1 1) with L = 90

Figure 3 Actual values and forecasts of 2003 with different L

fact this trend is affected by the definitions of fuzzy setsand membership function As an in-depth analysis Figure 1shows three triangular membership functionsThe RMSEs offorecasts of 2003 by 119866119879119878(119872119873) with 119871 = 30 and the threemembership functions are listed in Table 6 The table tells usthat themore intervals that are concerned in themembership

Mathematical Problems in Engineering 9

Table 5 Comparison of forecasting SSECI

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 2274 2164 2121 2092 2086 2071 2066 2066 2025 2010119866119879119878(119872 2) 1832 1904 1790 1792 1769 1756 1754 1755 1695 1688119866119879119878(119872 3) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 4) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 5) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683

[28] 1841 1841 1942 1943 1951 1939 2005 1971 2000 2026[29] 2107 2102 2087 2073 2008 1988 1979 1969 1818 1821[31] 1756 1756 1781 1778 1838 1831 1867 1925 1947 1960

MAE

119866119879119878(119872 1) 1861 1710 1661 1612 1609 1608 1607 1611 1575 1550119866119879119878(119872 2) 1408 1389 1388 1391 1373 1353 1345 1346 1316 1317119866119879119878(119872 3) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 4) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 5) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306

[28] 1392 1392 1478 1502 1522 1518 1541 1538 1562 1590[29] 1731 1726 1715 1704 1677 1663 1656 1647 1604 1607[31] 1281 1281 1329 1306 1357 1338 1419 1437 1442 1551

MAPE

119866119879119878(119872 1) 00127 00117 00113 00110 00110 00110 00109 00110 00107 00105119866119879119878(119872 2) 00096 00095 00095 00095 00094 00092 00092 00092 00090 00090119866119879119878(119872 3) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 4) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 5) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089

[28] 00095 00095 00101 00102 00104 00103 00105 00105 00106 00108[29] 00118 00118 00117 00116 00114 00113 00113 00112 00109 00109[31] 00087 00087 00091 00089 00092 00091 00097 00098 00098 00099

The bold data means the minimum error of the models with the same order

Table 6 Effects of membership function

Model 119896 = 3 119896 = 6 119896 = 9

Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3119866119879119878(119896 1) 175133 173290 173755 171512 169790 170653 164968 162923 164014119866119879119878(119896 2) 158071 157476 159365 154880 154686 157182 147028 146918 147719119866119879119878(119896 3) 158071 156926 158736 154880 153137 154631 147028 145116 144828119866119879119878(119896 4) 158071 156902 158670 154880 153119 154813 147028 145218 144558119866119879119878(119896 5) 158071 156902 159157 154880 153119 155422 147028 145218 145082119866119879119878(119896 6) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 7) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 8) 158071 156902 159157 154880 153119 155424 147028 145218 145068The bold data means the minimum error of the models with the same order

function the more different forecasting accuracy obtainedIt is more obvious when 119872 is increasing There is stillanother conclusion that the forecasts of the thirdmembershipfunction are not always better than those of the first function

To further investigate the relation between the parameters119872 and the length of interval the proposed model has beenapplied to forecasting stock index close prices covering tenyears from 1997 to 2006 with 119872 = 1 2 50 and 119871 =

30 60 300 Since it has been affirmed by Tables 4 and5 that the characters of MAE and MAPE are similar to

those of RMSE MAE and MAPE we will only list theRMSE comparison of these experiments as follows Someexamples of actual values and forecasts of 2003 are depicted inFigures 2 and 3 Figure 2 shows us that 119866119879119878(3 1) has a betterperformance than 119866119879119878(1 1) in the same length of intervalsthat is 119871 = 30 Figure 3 illustrates that 119866119879119878(1 1) getsthe better forecasts when the lengths of intervals are smallFurthermore some properties will be further described inFigures 4 and 5 which depict the mean forecasting errorsof the ten years Figure 4 shows the relation between the

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 2 Fuzzy logical relationships with119872 = 3 and119873 = 2

Year fuzzified 119896 = 1 119896 = 2 119896 = 3

119897 = 1 119897 = 2 119897 = 1 119897 = 2 119897 = 1 119897 = 2

1971 1198601 1198602

1972 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

1

1973 1198601 1198602

1198601997888rarr 119860

1 1198602997888rarr 119860

11198601997888rarr 119860

1 1198602997888rarr 119860

1

1974 1198602 1198603

1198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

21198601997888rarr 119860

2 1198602997888rarr 119860

2

1975 1198603 1198602

1198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1976 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

31198601997888rarr 119860

3 1198602997888rarr 119860

3

1977 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198602997888rarr 119860

3 1198603997888rarr 119860

3

1978 1198603 1198604

1198603997888rarr 119860

3 1198604997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1979 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1980 1198604 1198605

1198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1981 1198604 1198603

1198604997888rarr 119860

4 1198605997888rarr 119860

41198604997888rarr 119860

4 1198605997888rarr 119860

41198603997888rarr 119860

4 1198604997888rarr 119860

4

1982 1198603 1198602

1198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1983 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

31198604997888rarr 119860

3 1198605997888rarr 119860

3

1984 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198604997888rarr 119860

3 1198603997888rarr 119860

3

1985 1198603 1198602

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1986 1198603 1198604

1198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

31198603997888rarr 119860

3 1198602997888rarr 119860

3

1987 1198604 1198605

1198603997888rarr 119860

4 1198604997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

41198603997888rarr 119860

4 1198602997888rarr 119860

4

1988 1198606 1198605

1198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

61198603997888rarr 119860

6 1198602997888rarr 119860

6

1989 1198606 1198607

1198606997888rarr 119860

6 1198605997888rarr 119860

61198604997888rarr 119860

6 1198605997888rarr 119860

61198603997888rarr 119860

6 1198604997888rarr 119860

6

1990 1198607 1198606

1198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

71198604997888rarr 119860

7 1198605997888rarr 119860

7

1991 1198607 1198606

1198607997888rarr 119860

7 1198606997888rarr 119860

71198606997888rarr 119860

7 1198607997888rarr 119860

71198606997888rarr 119860

7 1198605997888rarr 119860

7

1992 1198606 1198607

1198607997888rarr 119860

6 1198606997888rarr 119860

61198607997888rarr 119860

6 1198606997888rarr 119860

61198606997888rarr 119860

6 1198607997888rarr 119860

6

In this paper we use the method of Lee [13] to constructthe fuzzy logical relationships matrix For example let k = 1and l = 1 The set of fuzzy logical relationships is listed as

1198601rarr 11986011198601rarr 11986011198601rarr 11986021198602rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198604

1198604rarr 11986041198604rarr 11986031198603rarr 11986031198603rarr 11986031198603rarr

1198603 1198603rarr 1198603 1198603rarr 1198604 1198604rarr 1198606

1198606rarr 1198606 1198606rarr 1198607 1198607rarr 1198607 1198607rarr 1198606 which

are all of FLRGs in the third column of Table 2

And they would be grouped and weighted by the recur-rent fuzzy relationships as follows

Group 1 1198601rarr 1198601with weight 2 119860

1rarr 1198602with weight 1

Group 2 1198602rarr 1198603with weight 1

Group 3 1198603rarr 1198603with weight 7 119860

3rarr 1198604with weight 2

Group 4 1198604

rarr 1198604with weight 2 119860

4rarr 119860

3with weight 1

1198604rarr 1198606with weight 1

Group 5 1198606rarr 1198606with weight 1 119860

6rarr 1198607with weight 1

Group 6 1198607rarr 1198607with weight 1 119860

7rarr 1198606with weight 1

The fuzzy logical relationship matrix then is 119877(11) Basedon the fourth fifth sixth seventh and eighth columnof Table 2 119877(12) 119877(21) 119877(22) 119877(31) and 119877

(32) can also be

obtained in the similar way These six fuzzy logical relation-ship matrices have been listed as follows

119877(11)

=

[[[[[[[[[[[[

[

2 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 7 2 0 0 0

0 0 1 2 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 1 1

]]]]]]]]]]]]

]

119877(12)

=

[[[[[[[[[[[[

[

0 0 0 0 0 0 0

2 1 6 0 0 0 0

0 0 2 0 0 0 0

0 0 2 2 0 0 0

0 0 0 2 0 2 0

0 0 0 0 0 1 1

0 0 0 0 0 0 1

]]]]]]]]]]]]

]

(7)

119877(21)

=

[[[[[[[[[[

[

1 1 1 0 0 0 0

0 0 1 0 0 0 0

0 0 5 3 0 1 0

0 0 2 1 0 1 0

0 0 0 0 0 0 0

0 0 0 0 0 0 2

0 0 0 0 0 1 0

]]]]]]]]]]

]

6 Mathematical Problems in Engineering

119877(22)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

1 1 6 1 0 0 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 1 1 0 1 1

0 0 0 0 0 1 0

0 0 0 0 0 0 1

]]]]]]]]]

]

(8)

119877(31)

=

[[[[[[[[[

[

0 1 2 0 0 0 0

0 0 1 0 0 0 0

0 0 3 4 0 2 0

0 0 3 2 0 0 1

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 0 0

]]]]]]]]]

]

119877(32)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

0 1 5 2 0 1 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 2 0 0 0 2

0 0 0 0 0 0 0

0 0 0 0 0 1 0

]]]]]]]]]

]

(9)

Step 4 Calculate the forecasts To carry out the calculationsby the proposed principles we illustrate the forecastingprocess of enrolment of 1976 as follows From Table 1 thefuzzy set of enrolment of 1975 is (002 052 098 048 0 0 0)The two maximum membership attributes are the third andthe second Then 119896 = 1 the third row of 119877(11) is (0 0 7 20 0 0) and the second row of 119877(12) is (2 1 6 0 0 0 0) from(7) According to Definition 6 we have and

2(119860(11)

19751 119860(12)

19752) =

(0 0 1 0 0 0 0)

According to the second principle listed in Section 311198651(1976) then is119898

3 that is 15500 which is the middle point

of 1198603

When k = 2 the fuzzy set of 1974s is (0402 09020598 0098 0 0 0) shown in Table 1 The two maximummembership attributes are the second and the third Thesecond row of 119877(21) then is (0 0 1 0 0 0 0) and the thirdrow of 119877(22) is (0 0 2 0 0 0 0) from (8) With Definition 6we have and

2(119860(21)

19751 119860(22)

19752) = (0 0 1 0 0 0 0) By the first

principle listed in Section 31 1198652(1976) also is 15500In the similar way fromTable 1 andwith 119896 = 3 we can see

that the fuzzy set of 1973s is (08165 06835 01835 0 0 0 0)The two maximum membership attributes are the first andsecond ones The first row of 119877(31) is (0 1 2 0 0 0 0) andthe second row of 119877(32) is (0 1 5 2 0 0 0) from (9) WithDefinition 6 we have and

2(119860(31)

19751 119860(32)

19752) = (0 1 2 0 0 0 0)

With the second principle listed in Section 31 in mind1198653(1976) equals to (13) times 14500 + (23) times 15500 = 15167At last the forecasted value of 1976 that is 119865(1976) is

15590 which equals to 09130 times 15500 + 00790 times 15500 +

00142times1516 Also (1199081 1199082 1199083) = (09130 00790 00142)

is the regress result of the enrolment data Table 3 shows theactual and the forecasted enrolments of the data set

4 Empirical Analysis

41 Data Description To demonstrate the effectiveness ofthe proposed models amounts of data are needed Here theenrolments and SSECI are used as the illustration data sets forthe empirical analysis

There are some causes for the two time series to be thesubjects in our experiment There are two causes for thechoice of enrolment data at the University of Alabama Thefirst one is that most of the fuzzy time series studies havetaken this well-known time series as their experimentsThusthere are a lot of studies that can be used for our referenceThe other is that it is simple and easy to display the process ofthe proposedmodel Since time series models have been usedto make predictions in the areas of stock price forecasting formany years the daily SSECI covering the period from 1997 to2006 is adopted for further experiment

For the first data the determined length of the sevenintervals is 1000 All of the forecasting results from one orderto ten orders are compared with those of the conventionalfuzzy time series models based on fuzzy logical relationships

To discuss the effects of parameters 119872 and 119873 the ordernumber and hierarchies of fuzzy logical relationships on theforecasting results the ten-order (time-lag periods) fromone order to ten orders models are performed on thesecond data set with ten different lengths of intervals thatis 30 60 90 300 The forecasting results obtained by allof the high-order models are compared in terms of threeevaluation criteria reviewed in following subsection

42 Criteria of Evaluation In statistics the root meansquared error (RMSE) mean absolute error (MAE) andmean absolute percentage error (MAPE) are three typicalways to quantify the difference between values implied byan estimator and the actual values of the quantity beingestimated MSE is a risk function for measuring the averageof the squares of the difference For an unbiased estimatorthe MSE is the variance and the RMSE is the square rootof the variance known as the standard error FurthermoreRMSE is superior to MSE for the reason that its scale isthe same as the forecasts Thus we take RMSE as the firstrepresentative of the size of an average error As an averageof the absolute percent errors MAPE serves as a criterionfor the comparisons of forecasting results in the paper Somecomparisons of accuracy in the forecasted values of ourproposed models with other models are made on the basisof the three criteria

43 Performance Evaluation In Table 4 this study com-pares the RMSE MAE and MAPE forecasting value ofthe proposed method and the counterparts [28 29 31] onthe enrolment experiment Table 4 shows that the proposedmethod gets smaller RMSE MAE and MAPE than Hwangrsquosmodel [28] and also smaller than Chenrsquos model [31] of 2011 inmost cases Although there is a fly in the ointment that is theproposed model not always gets a higher average forecastingaccuracy rate than the model [29] of 2002 these results areimproving as the orders increase In summary the resultssuggest that the proposed model obtains better forecasts as

Mathematical Problems in Engineering 7

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1 Membership function 1

(a)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 2

(b)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 3

(c)

Figure 1 Three membership functions for forecasting SHI of 2003

Table 3 Forecasts of enrollment year actual data

Year Value 119872 = 1 119872 = 2 119872 = 3

119873 = 1 119873 = 2 119873 = 1 119873 = 2 119873 = 1 119873 = 2

1971 13055 mdash mdash mdash mdash mdash mdash1972 13563 13901 13902 mdash mdash mdash mdash1973 13867 13901 13902 13980 13968 mdash mdash1974 14696 13901 13902 13980 13968 14062 139901975 15460 15576 15576 15491 15510 15533 155111976 15311 15799 15576 15792 15590 15760 155901977 15603 15799 15576 15858 15604 15812 156081978 15861 15799 16246 15858 16221 15915 162281979 16807 15799 16246 15858 16342 15915 163471980 16919 16832 17251 16790 17267 16821 172721981 16388 16832 17251 16823 17240 16832 172451982 15433 16832 15576 16823 15698 16821 157191983 15497 15799 15576 15891 15590 15915 156141984 15145 15799 15576 15858 15604 15904 156081985 15163 15799 15576 15858 15604 15915 156201986 15984 15799 15576 15858 15604 15915 156201987 16859 15799 16246 15858 16221 15915 162281988 18150 16832 17251 16790 17267 16821 172601989 18970 19093 18591 18863 18473 18816 184631990 19328 19093 19596 19161 19614 18900 195811991 19337 19093 19094 19161 19151 19132 191631992 18876 19093 19094 19061 19070 19101 19070

MAE 473365 3720021 4713957 375308 5011709 3902802RMSE 6254334 4474897 6316806 4488908 6432859 4597683MAPE 00293 00227 00291 00227 00307 00236

the orders and hierarchies increase Unlike this point this isnot the trend in the three counterparts

Moreover we also apply the proposed method to handleforecasting the close price of Shanghai stock index of 2003with 119897 = 60 The comparison of the three criteria is listedin Table 5 From this table we can see that the proposed119866119879119878(119872119873) gets the best forecasts of the three counterpartswhen 119872 gt 5 and 119873 gt 2 as well as its better performancethan those of Hwangrsquos and Chenrsquos model [28 29] in all cases

and the shortcoming shown in Table 4 is gone On the wholethere is a ldquolawrdquo that forecasting errorswill be reducedwhen119872

or119873 is increased in the proposed model However this trendis not evident for the three counterparts by Tables 4 and 5 InTable 4 the three evaluation criteria of Hwangrsquos and Chenrsquosmodels [28 31] are decreasing while those of model [29] areincreasing In contrast it is obvious that the three evaluationcriteria of Hwangrsquos and Chenrsquos model are increasing whilethose of Chenrsquos model [29] are decreasing from Table 5 The

8 Mathematical Problems in Engineering

Table 4 Comparison of forecasting enrollment

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 625 632 643 641 658 575 562 289 203 191119866119879119878(119872 2) 447 449 460 436 448 456 437 413 391 284119866119879119878(119872 3) 447 449 460 436 448 456 406 382 365 225119866119879119878(119872 4) 447 449 460 434 445 453 423 415 397 253119866119879119878(119872 5) 447 449 460 434 445 453 423 415 397 253

[28] 588 588 578 608 602 620 632 575 554 498[29] 292 298 294 299 307 313 323 320 321 311[31] 543 543 522 420 456 443 452 463 480 498

MAE

119866119879119878(119872 1) 473 471 501 484 501 366 411 212 326 140119866119879119878(119872 2) 372 375 390 365 383 394 406 364 292 218119866119879119878(119872 3) 372 375 390 365 383 393 374 336 333 176119866119879119878(119872 4) 372 375 390 367 384 393 402 386 333 195119866119879119878(119872 5) 372 375 390 367 384 393 402 386 333 195

[28] 494 494 473 526 527 556 582 506 483 424[29] 252 262 256 259 272 277 289 284 282 271[31] 441 441 412 316 347 337 340 346 368 390

MAPE

119866119879119878(119872 1) 00293 00291 00307 00294 00305 00221 00246 00127 00103 00087119866119879119878(119872 2) 00227 00227 00236 00216 00227 00232 00239 00216 00195 00125119866119879119878(119872 3) 00227 00227 00236 00216 00227 00232 00220 00200 00176 00101119866119879119878(119872 4) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111119866119879119878(119872 5) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111

[28] 00299 00299 00283 00317 00313 00329 00345 00299 00285 00253[29] 00153 00159 00153 00154 00162 00164 00170 00166 00165 00158[31] 00268 00268 00248 00188 00203 00195 00196 00198 00211 00223

The bold data means the minimum error of the models with the same order

1300

1350

1400

1450

1500

1550

0 5 10 15 20 25 30 35 40 45 50Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(3 1) with L = 30

Figure 2 Actual values and forecasts of 2003 with the same L

use of different data sets seems to be the best reason toaccount for this contradiction From these discussions we geta conclusion that the proposed modelrsquos performance is morereasonable and robust than the three counterparts

Tables 4 and 5 depict that 119866119879119878(119872 3) 119866119879119878(119872 4) and119866119879119878(119872 5) have the same results although larger 119872 or 119873

is more easy to obtain the smaller forecasting errors In

0 5 10 15 20 25 30 35 40 45 501300

1350

1400

1450

1500

1550

Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(1 1) with L = 90

Figure 3 Actual values and forecasts of 2003 with different L

fact this trend is affected by the definitions of fuzzy setsand membership function As an in-depth analysis Figure 1shows three triangular membership functionsThe RMSEs offorecasts of 2003 by 119866119879119878(119872119873) with 119871 = 30 and the threemembership functions are listed in Table 6 The table tells usthat themore intervals that are concerned in themembership

Mathematical Problems in Engineering 9

Table 5 Comparison of forecasting SSECI

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 2274 2164 2121 2092 2086 2071 2066 2066 2025 2010119866119879119878(119872 2) 1832 1904 1790 1792 1769 1756 1754 1755 1695 1688119866119879119878(119872 3) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 4) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 5) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683

[28] 1841 1841 1942 1943 1951 1939 2005 1971 2000 2026[29] 2107 2102 2087 2073 2008 1988 1979 1969 1818 1821[31] 1756 1756 1781 1778 1838 1831 1867 1925 1947 1960

MAE

119866119879119878(119872 1) 1861 1710 1661 1612 1609 1608 1607 1611 1575 1550119866119879119878(119872 2) 1408 1389 1388 1391 1373 1353 1345 1346 1316 1317119866119879119878(119872 3) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 4) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 5) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306

[28] 1392 1392 1478 1502 1522 1518 1541 1538 1562 1590[29] 1731 1726 1715 1704 1677 1663 1656 1647 1604 1607[31] 1281 1281 1329 1306 1357 1338 1419 1437 1442 1551

MAPE

119866119879119878(119872 1) 00127 00117 00113 00110 00110 00110 00109 00110 00107 00105119866119879119878(119872 2) 00096 00095 00095 00095 00094 00092 00092 00092 00090 00090119866119879119878(119872 3) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 4) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 5) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089

[28] 00095 00095 00101 00102 00104 00103 00105 00105 00106 00108[29] 00118 00118 00117 00116 00114 00113 00113 00112 00109 00109[31] 00087 00087 00091 00089 00092 00091 00097 00098 00098 00099

The bold data means the minimum error of the models with the same order

Table 6 Effects of membership function

Model 119896 = 3 119896 = 6 119896 = 9

Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3119866119879119878(119896 1) 175133 173290 173755 171512 169790 170653 164968 162923 164014119866119879119878(119896 2) 158071 157476 159365 154880 154686 157182 147028 146918 147719119866119879119878(119896 3) 158071 156926 158736 154880 153137 154631 147028 145116 144828119866119879119878(119896 4) 158071 156902 158670 154880 153119 154813 147028 145218 144558119866119879119878(119896 5) 158071 156902 159157 154880 153119 155422 147028 145218 145082119866119879119878(119896 6) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 7) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 8) 158071 156902 159157 154880 153119 155424 147028 145218 145068The bold data means the minimum error of the models with the same order

function the more different forecasting accuracy obtainedIt is more obvious when 119872 is increasing There is stillanother conclusion that the forecasts of the thirdmembershipfunction are not always better than those of the first function

To further investigate the relation between the parameters119872 and the length of interval the proposed model has beenapplied to forecasting stock index close prices covering tenyears from 1997 to 2006 with 119872 = 1 2 50 and 119871 =

30 60 300 Since it has been affirmed by Tables 4 and5 that the characters of MAE and MAPE are similar to

those of RMSE MAE and MAPE we will only list theRMSE comparison of these experiments as follows Someexamples of actual values and forecasts of 2003 are depicted inFigures 2 and 3 Figure 2 shows us that 119866119879119878(3 1) has a betterperformance than 119866119879119878(1 1) in the same length of intervalsthat is 119871 = 30 Figure 3 illustrates that 119866119879119878(1 1) getsthe better forecasts when the lengths of intervals are smallFurthermore some properties will be further described inFigures 4 and 5 which depict the mean forecasting errorsof the ten years Figure 4 shows the relation between the

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

119877(22)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

1 1 6 1 0 0 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 1 1 0 1 1

0 0 0 0 0 1 0

0 0 0 0 0 0 1

]]]]]]]]]

]

(8)

119877(31)

=

[[[[[[[[[

[

0 1 2 0 0 0 0

0 0 1 0 0 0 0

0 0 3 4 0 2 0

0 0 3 2 0 0 1

0 0 0 0 0 0 0

0 0 0 0 0 1 1

0 0 0 0 0 0 0

]]]]]]]]]

]

119877(32)

=

[[[[[[[[[

[

0 0 0 0 0 0 0

0 1 5 2 0 1 0

0 0 2 0 0 0 0

0 0 0 2 0 1 0

0 0 2 0 0 0 2

0 0 0 0 0 0 0

0 0 0 0 0 1 0

]]]]]]]]]

]

(9)

Step 4 Calculate the forecasts To carry out the calculationsby the proposed principles we illustrate the forecastingprocess of enrolment of 1976 as follows From Table 1 thefuzzy set of enrolment of 1975 is (002 052 098 048 0 0 0)The two maximum membership attributes are the third andthe second Then 119896 = 1 the third row of 119877(11) is (0 0 7 20 0 0) and the second row of 119877(12) is (2 1 6 0 0 0 0) from(7) According to Definition 6 we have and

2(119860(11)

19751 119860(12)

19752) =

(0 0 1 0 0 0 0)

According to the second principle listed in Section 311198651(1976) then is119898

3 that is 15500 which is the middle point

of 1198603

When k = 2 the fuzzy set of 1974s is (0402 09020598 0098 0 0 0) shown in Table 1 The two maximummembership attributes are the second and the third Thesecond row of 119877(21) then is (0 0 1 0 0 0 0) and the thirdrow of 119877(22) is (0 0 2 0 0 0 0) from (8) With Definition 6we have and

2(119860(21)

19751 119860(22)

19752) = (0 0 1 0 0 0 0) By the first

principle listed in Section 31 1198652(1976) also is 15500In the similar way fromTable 1 andwith 119896 = 3 we can see

that the fuzzy set of 1973s is (08165 06835 01835 0 0 0 0)The two maximum membership attributes are the first andsecond ones The first row of 119877(31) is (0 1 2 0 0 0 0) andthe second row of 119877(32) is (0 1 5 2 0 0 0) from (9) WithDefinition 6 we have and

2(119860(31)

19751 119860(32)

19752) = (0 1 2 0 0 0 0)

With the second principle listed in Section 31 in mind1198653(1976) equals to (13) times 14500 + (23) times 15500 = 15167At last the forecasted value of 1976 that is 119865(1976) is

15590 which equals to 09130 times 15500 + 00790 times 15500 +

00142times1516 Also (1199081 1199082 1199083) = (09130 00790 00142)

is the regress result of the enrolment data Table 3 shows theactual and the forecasted enrolments of the data set

4 Empirical Analysis

41 Data Description To demonstrate the effectiveness ofthe proposed models amounts of data are needed Here theenrolments and SSECI are used as the illustration data sets forthe empirical analysis

There are some causes for the two time series to be thesubjects in our experiment There are two causes for thechoice of enrolment data at the University of Alabama Thefirst one is that most of the fuzzy time series studies havetaken this well-known time series as their experimentsThusthere are a lot of studies that can be used for our referenceThe other is that it is simple and easy to display the process ofthe proposedmodel Since time series models have been usedto make predictions in the areas of stock price forecasting formany years the daily SSECI covering the period from 1997 to2006 is adopted for further experiment

For the first data the determined length of the sevenintervals is 1000 All of the forecasting results from one orderto ten orders are compared with those of the conventionalfuzzy time series models based on fuzzy logical relationships

To discuss the effects of parameters 119872 and 119873 the ordernumber and hierarchies of fuzzy logical relationships on theforecasting results the ten-order (time-lag periods) fromone order to ten orders models are performed on thesecond data set with ten different lengths of intervals thatis 30 60 90 300 The forecasting results obtained by allof the high-order models are compared in terms of threeevaluation criteria reviewed in following subsection

42 Criteria of Evaluation In statistics the root meansquared error (RMSE) mean absolute error (MAE) andmean absolute percentage error (MAPE) are three typicalways to quantify the difference between values implied byan estimator and the actual values of the quantity beingestimated MSE is a risk function for measuring the averageof the squares of the difference For an unbiased estimatorthe MSE is the variance and the RMSE is the square rootof the variance known as the standard error FurthermoreRMSE is superior to MSE for the reason that its scale isthe same as the forecasts Thus we take RMSE as the firstrepresentative of the size of an average error As an averageof the absolute percent errors MAPE serves as a criterionfor the comparisons of forecasting results in the paper Somecomparisons of accuracy in the forecasted values of ourproposed models with other models are made on the basisof the three criteria

43 Performance Evaluation In Table 4 this study com-pares the RMSE MAE and MAPE forecasting value ofthe proposed method and the counterparts [28 29 31] onthe enrolment experiment Table 4 shows that the proposedmethod gets smaller RMSE MAE and MAPE than Hwangrsquosmodel [28] and also smaller than Chenrsquos model [31] of 2011 inmost cases Although there is a fly in the ointment that is theproposed model not always gets a higher average forecastingaccuracy rate than the model [29] of 2002 these results areimproving as the orders increase In summary the resultssuggest that the proposed model obtains better forecasts as

Mathematical Problems in Engineering 7

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1 Membership function 1

(a)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 2

(b)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 3

(c)

Figure 1 Three membership functions for forecasting SHI of 2003

Table 3 Forecasts of enrollment year actual data

Year Value 119872 = 1 119872 = 2 119872 = 3

119873 = 1 119873 = 2 119873 = 1 119873 = 2 119873 = 1 119873 = 2

1971 13055 mdash mdash mdash mdash mdash mdash1972 13563 13901 13902 mdash mdash mdash mdash1973 13867 13901 13902 13980 13968 mdash mdash1974 14696 13901 13902 13980 13968 14062 139901975 15460 15576 15576 15491 15510 15533 155111976 15311 15799 15576 15792 15590 15760 155901977 15603 15799 15576 15858 15604 15812 156081978 15861 15799 16246 15858 16221 15915 162281979 16807 15799 16246 15858 16342 15915 163471980 16919 16832 17251 16790 17267 16821 172721981 16388 16832 17251 16823 17240 16832 172451982 15433 16832 15576 16823 15698 16821 157191983 15497 15799 15576 15891 15590 15915 156141984 15145 15799 15576 15858 15604 15904 156081985 15163 15799 15576 15858 15604 15915 156201986 15984 15799 15576 15858 15604 15915 156201987 16859 15799 16246 15858 16221 15915 162281988 18150 16832 17251 16790 17267 16821 172601989 18970 19093 18591 18863 18473 18816 184631990 19328 19093 19596 19161 19614 18900 195811991 19337 19093 19094 19161 19151 19132 191631992 18876 19093 19094 19061 19070 19101 19070

MAE 473365 3720021 4713957 375308 5011709 3902802RMSE 6254334 4474897 6316806 4488908 6432859 4597683MAPE 00293 00227 00291 00227 00307 00236

the orders and hierarchies increase Unlike this point this isnot the trend in the three counterparts

Moreover we also apply the proposed method to handleforecasting the close price of Shanghai stock index of 2003with 119897 = 60 The comparison of the three criteria is listedin Table 5 From this table we can see that the proposed119866119879119878(119872119873) gets the best forecasts of the three counterpartswhen 119872 gt 5 and 119873 gt 2 as well as its better performancethan those of Hwangrsquos and Chenrsquos model [28 29] in all cases

and the shortcoming shown in Table 4 is gone On the wholethere is a ldquolawrdquo that forecasting errorswill be reducedwhen119872

or119873 is increased in the proposed model However this trendis not evident for the three counterparts by Tables 4 and 5 InTable 4 the three evaluation criteria of Hwangrsquos and Chenrsquosmodels [28 31] are decreasing while those of model [29] areincreasing In contrast it is obvious that the three evaluationcriteria of Hwangrsquos and Chenrsquos model are increasing whilethose of Chenrsquos model [29] are decreasing from Table 5 The

8 Mathematical Problems in Engineering

Table 4 Comparison of forecasting enrollment

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 625 632 643 641 658 575 562 289 203 191119866119879119878(119872 2) 447 449 460 436 448 456 437 413 391 284119866119879119878(119872 3) 447 449 460 436 448 456 406 382 365 225119866119879119878(119872 4) 447 449 460 434 445 453 423 415 397 253119866119879119878(119872 5) 447 449 460 434 445 453 423 415 397 253

[28] 588 588 578 608 602 620 632 575 554 498[29] 292 298 294 299 307 313 323 320 321 311[31] 543 543 522 420 456 443 452 463 480 498

MAE

119866119879119878(119872 1) 473 471 501 484 501 366 411 212 326 140119866119879119878(119872 2) 372 375 390 365 383 394 406 364 292 218119866119879119878(119872 3) 372 375 390 365 383 393 374 336 333 176119866119879119878(119872 4) 372 375 390 367 384 393 402 386 333 195119866119879119878(119872 5) 372 375 390 367 384 393 402 386 333 195

[28] 494 494 473 526 527 556 582 506 483 424[29] 252 262 256 259 272 277 289 284 282 271[31] 441 441 412 316 347 337 340 346 368 390

MAPE

119866119879119878(119872 1) 00293 00291 00307 00294 00305 00221 00246 00127 00103 00087119866119879119878(119872 2) 00227 00227 00236 00216 00227 00232 00239 00216 00195 00125119866119879119878(119872 3) 00227 00227 00236 00216 00227 00232 00220 00200 00176 00101119866119879119878(119872 4) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111119866119879119878(119872 5) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111

[28] 00299 00299 00283 00317 00313 00329 00345 00299 00285 00253[29] 00153 00159 00153 00154 00162 00164 00170 00166 00165 00158[31] 00268 00268 00248 00188 00203 00195 00196 00198 00211 00223

The bold data means the minimum error of the models with the same order

1300

1350

1400

1450

1500

1550

0 5 10 15 20 25 30 35 40 45 50Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(3 1) with L = 30

Figure 2 Actual values and forecasts of 2003 with the same L

use of different data sets seems to be the best reason toaccount for this contradiction From these discussions we geta conclusion that the proposed modelrsquos performance is morereasonable and robust than the three counterparts

Tables 4 and 5 depict that 119866119879119878(119872 3) 119866119879119878(119872 4) and119866119879119878(119872 5) have the same results although larger 119872 or 119873

is more easy to obtain the smaller forecasting errors In

0 5 10 15 20 25 30 35 40 45 501300

1350

1400

1450

1500

1550

Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(1 1) with L = 90

Figure 3 Actual values and forecasts of 2003 with different L

fact this trend is affected by the definitions of fuzzy setsand membership function As an in-depth analysis Figure 1shows three triangular membership functionsThe RMSEs offorecasts of 2003 by 119866119879119878(119872119873) with 119871 = 30 and the threemembership functions are listed in Table 6 The table tells usthat themore intervals that are concerned in themembership

Mathematical Problems in Engineering 9

Table 5 Comparison of forecasting SSECI

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 2274 2164 2121 2092 2086 2071 2066 2066 2025 2010119866119879119878(119872 2) 1832 1904 1790 1792 1769 1756 1754 1755 1695 1688119866119879119878(119872 3) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 4) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 5) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683

[28] 1841 1841 1942 1943 1951 1939 2005 1971 2000 2026[29] 2107 2102 2087 2073 2008 1988 1979 1969 1818 1821[31] 1756 1756 1781 1778 1838 1831 1867 1925 1947 1960

MAE

119866119879119878(119872 1) 1861 1710 1661 1612 1609 1608 1607 1611 1575 1550119866119879119878(119872 2) 1408 1389 1388 1391 1373 1353 1345 1346 1316 1317119866119879119878(119872 3) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 4) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 5) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306

[28] 1392 1392 1478 1502 1522 1518 1541 1538 1562 1590[29] 1731 1726 1715 1704 1677 1663 1656 1647 1604 1607[31] 1281 1281 1329 1306 1357 1338 1419 1437 1442 1551

MAPE

119866119879119878(119872 1) 00127 00117 00113 00110 00110 00110 00109 00110 00107 00105119866119879119878(119872 2) 00096 00095 00095 00095 00094 00092 00092 00092 00090 00090119866119879119878(119872 3) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 4) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 5) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089

[28] 00095 00095 00101 00102 00104 00103 00105 00105 00106 00108[29] 00118 00118 00117 00116 00114 00113 00113 00112 00109 00109[31] 00087 00087 00091 00089 00092 00091 00097 00098 00098 00099

The bold data means the minimum error of the models with the same order

Table 6 Effects of membership function

Model 119896 = 3 119896 = 6 119896 = 9

Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3119866119879119878(119896 1) 175133 173290 173755 171512 169790 170653 164968 162923 164014119866119879119878(119896 2) 158071 157476 159365 154880 154686 157182 147028 146918 147719119866119879119878(119896 3) 158071 156926 158736 154880 153137 154631 147028 145116 144828119866119879119878(119896 4) 158071 156902 158670 154880 153119 154813 147028 145218 144558119866119879119878(119896 5) 158071 156902 159157 154880 153119 155422 147028 145218 145082119866119879119878(119896 6) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 7) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 8) 158071 156902 159157 154880 153119 155424 147028 145218 145068The bold data means the minimum error of the models with the same order

function the more different forecasting accuracy obtainedIt is more obvious when 119872 is increasing There is stillanother conclusion that the forecasts of the thirdmembershipfunction are not always better than those of the first function

To further investigate the relation between the parameters119872 and the length of interval the proposed model has beenapplied to forecasting stock index close prices covering tenyears from 1997 to 2006 with 119872 = 1 2 50 and 119871 =

30 60 300 Since it has been affirmed by Tables 4 and5 that the characters of MAE and MAPE are similar to

those of RMSE MAE and MAPE we will only list theRMSE comparison of these experiments as follows Someexamples of actual values and forecasts of 2003 are depicted inFigures 2 and 3 Figure 2 shows us that 119866119879119878(3 1) has a betterperformance than 119866119879119878(1 1) in the same length of intervalsthat is 119871 = 30 Figure 3 illustrates that 119866119879119878(1 1) getsthe better forecasts when the lengths of intervals are smallFurthermore some properties will be further described inFigures 4 and 5 which depict the mean forecasting errorsof the ten years Figure 4 shows the relation between the

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1 Membership function 1

(a)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 2

(b)

Starting m1 m2 m3 m4 m5 m6 m7 Ending0

05

1Membership function 3

(c)

Figure 1 Three membership functions for forecasting SHI of 2003

Table 3 Forecasts of enrollment year actual data

Year Value 119872 = 1 119872 = 2 119872 = 3

119873 = 1 119873 = 2 119873 = 1 119873 = 2 119873 = 1 119873 = 2

1971 13055 mdash mdash mdash mdash mdash mdash1972 13563 13901 13902 mdash mdash mdash mdash1973 13867 13901 13902 13980 13968 mdash mdash1974 14696 13901 13902 13980 13968 14062 139901975 15460 15576 15576 15491 15510 15533 155111976 15311 15799 15576 15792 15590 15760 155901977 15603 15799 15576 15858 15604 15812 156081978 15861 15799 16246 15858 16221 15915 162281979 16807 15799 16246 15858 16342 15915 163471980 16919 16832 17251 16790 17267 16821 172721981 16388 16832 17251 16823 17240 16832 172451982 15433 16832 15576 16823 15698 16821 157191983 15497 15799 15576 15891 15590 15915 156141984 15145 15799 15576 15858 15604 15904 156081985 15163 15799 15576 15858 15604 15915 156201986 15984 15799 15576 15858 15604 15915 156201987 16859 15799 16246 15858 16221 15915 162281988 18150 16832 17251 16790 17267 16821 172601989 18970 19093 18591 18863 18473 18816 184631990 19328 19093 19596 19161 19614 18900 195811991 19337 19093 19094 19161 19151 19132 191631992 18876 19093 19094 19061 19070 19101 19070

MAE 473365 3720021 4713957 375308 5011709 3902802RMSE 6254334 4474897 6316806 4488908 6432859 4597683MAPE 00293 00227 00291 00227 00307 00236

the orders and hierarchies increase Unlike this point this isnot the trend in the three counterparts

Moreover we also apply the proposed method to handleforecasting the close price of Shanghai stock index of 2003with 119897 = 60 The comparison of the three criteria is listedin Table 5 From this table we can see that the proposed119866119879119878(119872119873) gets the best forecasts of the three counterpartswhen 119872 gt 5 and 119873 gt 2 as well as its better performancethan those of Hwangrsquos and Chenrsquos model [28 29] in all cases

and the shortcoming shown in Table 4 is gone On the wholethere is a ldquolawrdquo that forecasting errorswill be reducedwhen119872

or119873 is increased in the proposed model However this trendis not evident for the three counterparts by Tables 4 and 5 InTable 4 the three evaluation criteria of Hwangrsquos and Chenrsquosmodels [28 31] are decreasing while those of model [29] areincreasing In contrast it is obvious that the three evaluationcriteria of Hwangrsquos and Chenrsquos model are increasing whilethose of Chenrsquos model [29] are decreasing from Table 5 The

8 Mathematical Problems in Engineering

Table 4 Comparison of forecasting enrollment

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 625 632 643 641 658 575 562 289 203 191119866119879119878(119872 2) 447 449 460 436 448 456 437 413 391 284119866119879119878(119872 3) 447 449 460 436 448 456 406 382 365 225119866119879119878(119872 4) 447 449 460 434 445 453 423 415 397 253119866119879119878(119872 5) 447 449 460 434 445 453 423 415 397 253

[28] 588 588 578 608 602 620 632 575 554 498[29] 292 298 294 299 307 313 323 320 321 311[31] 543 543 522 420 456 443 452 463 480 498

MAE

119866119879119878(119872 1) 473 471 501 484 501 366 411 212 326 140119866119879119878(119872 2) 372 375 390 365 383 394 406 364 292 218119866119879119878(119872 3) 372 375 390 365 383 393 374 336 333 176119866119879119878(119872 4) 372 375 390 367 384 393 402 386 333 195119866119879119878(119872 5) 372 375 390 367 384 393 402 386 333 195

[28] 494 494 473 526 527 556 582 506 483 424[29] 252 262 256 259 272 277 289 284 282 271[31] 441 441 412 316 347 337 340 346 368 390

MAPE

119866119879119878(119872 1) 00293 00291 00307 00294 00305 00221 00246 00127 00103 00087119866119879119878(119872 2) 00227 00227 00236 00216 00227 00232 00239 00216 00195 00125119866119879119878(119872 3) 00227 00227 00236 00216 00227 00232 00220 00200 00176 00101119866119879119878(119872 4) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111119866119879119878(119872 5) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111

[28] 00299 00299 00283 00317 00313 00329 00345 00299 00285 00253[29] 00153 00159 00153 00154 00162 00164 00170 00166 00165 00158[31] 00268 00268 00248 00188 00203 00195 00196 00198 00211 00223

The bold data means the minimum error of the models with the same order

1300

1350

1400

1450

1500

1550

0 5 10 15 20 25 30 35 40 45 50Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(3 1) with L = 30

Figure 2 Actual values and forecasts of 2003 with the same L

use of different data sets seems to be the best reason toaccount for this contradiction From these discussions we geta conclusion that the proposed modelrsquos performance is morereasonable and robust than the three counterparts

Tables 4 and 5 depict that 119866119879119878(119872 3) 119866119879119878(119872 4) and119866119879119878(119872 5) have the same results although larger 119872 or 119873

is more easy to obtain the smaller forecasting errors In

0 5 10 15 20 25 30 35 40 45 501300

1350

1400

1450

1500

1550

Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(1 1) with L = 90

Figure 3 Actual values and forecasts of 2003 with different L

fact this trend is affected by the definitions of fuzzy setsand membership function As an in-depth analysis Figure 1shows three triangular membership functionsThe RMSEs offorecasts of 2003 by 119866119879119878(119872119873) with 119871 = 30 and the threemembership functions are listed in Table 6 The table tells usthat themore intervals that are concerned in themembership

Mathematical Problems in Engineering 9

Table 5 Comparison of forecasting SSECI

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 2274 2164 2121 2092 2086 2071 2066 2066 2025 2010119866119879119878(119872 2) 1832 1904 1790 1792 1769 1756 1754 1755 1695 1688119866119879119878(119872 3) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 4) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 5) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683

[28] 1841 1841 1942 1943 1951 1939 2005 1971 2000 2026[29] 2107 2102 2087 2073 2008 1988 1979 1969 1818 1821[31] 1756 1756 1781 1778 1838 1831 1867 1925 1947 1960

MAE

119866119879119878(119872 1) 1861 1710 1661 1612 1609 1608 1607 1611 1575 1550119866119879119878(119872 2) 1408 1389 1388 1391 1373 1353 1345 1346 1316 1317119866119879119878(119872 3) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 4) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 5) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306

[28] 1392 1392 1478 1502 1522 1518 1541 1538 1562 1590[29] 1731 1726 1715 1704 1677 1663 1656 1647 1604 1607[31] 1281 1281 1329 1306 1357 1338 1419 1437 1442 1551

MAPE

119866119879119878(119872 1) 00127 00117 00113 00110 00110 00110 00109 00110 00107 00105119866119879119878(119872 2) 00096 00095 00095 00095 00094 00092 00092 00092 00090 00090119866119879119878(119872 3) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 4) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 5) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089

[28] 00095 00095 00101 00102 00104 00103 00105 00105 00106 00108[29] 00118 00118 00117 00116 00114 00113 00113 00112 00109 00109[31] 00087 00087 00091 00089 00092 00091 00097 00098 00098 00099

The bold data means the minimum error of the models with the same order

Table 6 Effects of membership function

Model 119896 = 3 119896 = 6 119896 = 9

Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3119866119879119878(119896 1) 175133 173290 173755 171512 169790 170653 164968 162923 164014119866119879119878(119896 2) 158071 157476 159365 154880 154686 157182 147028 146918 147719119866119879119878(119896 3) 158071 156926 158736 154880 153137 154631 147028 145116 144828119866119879119878(119896 4) 158071 156902 158670 154880 153119 154813 147028 145218 144558119866119879119878(119896 5) 158071 156902 159157 154880 153119 155422 147028 145218 145082119866119879119878(119896 6) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 7) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 8) 158071 156902 159157 154880 153119 155424 147028 145218 145068The bold data means the minimum error of the models with the same order

function the more different forecasting accuracy obtainedIt is more obvious when 119872 is increasing There is stillanother conclusion that the forecasts of the thirdmembershipfunction are not always better than those of the first function

To further investigate the relation between the parameters119872 and the length of interval the proposed model has beenapplied to forecasting stock index close prices covering tenyears from 1997 to 2006 with 119872 = 1 2 50 and 119871 =

30 60 300 Since it has been affirmed by Tables 4 and5 that the characters of MAE and MAPE are similar to

those of RMSE MAE and MAPE we will only list theRMSE comparison of these experiments as follows Someexamples of actual values and forecasts of 2003 are depicted inFigures 2 and 3 Figure 2 shows us that 119866119879119878(3 1) has a betterperformance than 119866119879119878(1 1) in the same length of intervalsthat is 119871 = 30 Figure 3 illustrates that 119866119879119878(1 1) getsthe better forecasts when the lengths of intervals are smallFurthermore some properties will be further described inFigures 4 and 5 which depict the mean forecasting errorsof the ten years Figure 4 shows the relation between the

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 4 Comparison of forecasting enrollment

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 625 632 643 641 658 575 562 289 203 191119866119879119878(119872 2) 447 449 460 436 448 456 437 413 391 284119866119879119878(119872 3) 447 449 460 436 448 456 406 382 365 225119866119879119878(119872 4) 447 449 460 434 445 453 423 415 397 253119866119879119878(119872 5) 447 449 460 434 445 453 423 415 397 253

[28] 588 588 578 608 602 620 632 575 554 498[29] 292 298 294 299 307 313 323 320 321 311[31] 543 543 522 420 456 443 452 463 480 498

MAE

119866119879119878(119872 1) 473 471 501 484 501 366 411 212 326 140119866119879119878(119872 2) 372 375 390 365 383 394 406 364 292 218119866119879119878(119872 3) 372 375 390 365 383 393 374 336 333 176119866119879119878(119872 4) 372 375 390 367 384 393 402 386 333 195119866119879119878(119872 5) 372 375 390 367 384 393 402 386 333 195

[28] 494 494 473 526 527 556 582 506 483 424[29] 252 262 256 259 272 277 289 284 282 271[31] 441 441 412 316 347 337 340 346 368 390

MAPE

119866119879119878(119872 1) 00293 00291 00307 00294 00305 00221 00246 00127 00103 00087119866119879119878(119872 2) 00227 00227 00236 00216 00227 00232 00239 00216 00195 00125119866119879119878(119872 3) 00227 00227 00236 00216 00227 00232 00220 00200 00176 00101119866119879119878(119872 4) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111119866119879119878(119872 5) 00227 00227 00236 00217 00227 00231 00236 00228 00198 00111

[28] 00299 00299 00283 00317 00313 00329 00345 00299 00285 00253[29] 00153 00159 00153 00154 00162 00164 00170 00166 00165 00158[31] 00268 00268 00248 00188 00203 00195 00196 00198 00211 00223

The bold data means the minimum error of the models with the same order

1300

1350

1400

1450

1500

1550

0 5 10 15 20 25 30 35 40 45 50Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(3 1) with L = 30

Figure 2 Actual values and forecasts of 2003 with the same L

use of different data sets seems to be the best reason toaccount for this contradiction From these discussions we geta conclusion that the proposed modelrsquos performance is morereasonable and robust than the three counterparts

Tables 4 and 5 depict that 119866119879119878(119872 3) 119866119879119878(119872 4) and119866119879119878(119872 5) have the same results although larger 119872 or 119873

is more easy to obtain the smaller forecasting errors In

0 5 10 15 20 25 30 35 40 45 501300

1350

1400

1450

1500

1550

Time

Pric

e

Actual valuesForecasts of G(1 1) with L = 30

Forecasts of G(1 1) with L = 90

Figure 3 Actual values and forecasts of 2003 with different L

fact this trend is affected by the definitions of fuzzy setsand membership function As an in-depth analysis Figure 1shows three triangular membership functionsThe RMSEs offorecasts of 2003 by 119866119879119878(119872119873) with 119871 = 30 and the threemembership functions are listed in Table 6 The table tells usthat themore intervals that are concerned in themembership

Mathematical Problems in Engineering 9

Table 5 Comparison of forecasting SSECI

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 2274 2164 2121 2092 2086 2071 2066 2066 2025 2010119866119879119878(119872 2) 1832 1904 1790 1792 1769 1756 1754 1755 1695 1688119866119879119878(119872 3) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 4) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 5) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683

[28] 1841 1841 1942 1943 1951 1939 2005 1971 2000 2026[29] 2107 2102 2087 2073 2008 1988 1979 1969 1818 1821[31] 1756 1756 1781 1778 1838 1831 1867 1925 1947 1960

MAE

119866119879119878(119872 1) 1861 1710 1661 1612 1609 1608 1607 1611 1575 1550119866119879119878(119872 2) 1408 1389 1388 1391 1373 1353 1345 1346 1316 1317119866119879119878(119872 3) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 4) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 5) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306

[28] 1392 1392 1478 1502 1522 1518 1541 1538 1562 1590[29] 1731 1726 1715 1704 1677 1663 1656 1647 1604 1607[31] 1281 1281 1329 1306 1357 1338 1419 1437 1442 1551

MAPE

119866119879119878(119872 1) 00127 00117 00113 00110 00110 00110 00109 00110 00107 00105119866119879119878(119872 2) 00096 00095 00095 00095 00094 00092 00092 00092 00090 00090119866119879119878(119872 3) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 4) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 5) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089

[28] 00095 00095 00101 00102 00104 00103 00105 00105 00106 00108[29] 00118 00118 00117 00116 00114 00113 00113 00112 00109 00109[31] 00087 00087 00091 00089 00092 00091 00097 00098 00098 00099

The bold data means the minimum error of the models with the same order

Table 6 Effects of membership function

Model 119896 = 3 119896 = 6 119896 = 9

Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3119866119879119878(119896 1) 175133 173290 173755 171512 169790 170653 164968 162923 164014119866119879119878(119896 2) 158071 157476 159365 154880 154686 157182 147028 146918 147719119866119879119878(119896 3) 158071 156926 158736 154880 153137 154631 147028 145116 144828119866119879119878(119896 4) 158071 156902 158670 154880 153119 154813 147028 145218 144558119866119879119878(119896 5) 158071 156902 159157 154880 153119 155422 147028 145218 145082119866119879119878(119896 6) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 7) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 8) 158071 156902 159157 154880 153119 155424 147028 145218 145068The bold data means the minimum error of the models with the same order

function the more different forecasting accuracy obtainedIt is more obvious when 119872 is increasing There is stillanother conclusion that the forecasts of the thirdmembershipfunction are not always better than those of the first function

To further investigate the relation between the parameters119872 and the length of interval the proposed model has beenapplied to forecasting stock index close prices covering tenyears from 1997 to 2006 with 119872 = 1 2 50 and 119871 =

30 60 300 Since it has been affirmed by Tables 4 and5 that the characters of MAE and MAPE are similar to

those of RMSE MAE and MAPE we will only list theRMSE comparison of these experiments as follows Someexamples of actual values and forecasts of 2003 are depicted inFigures 2 and 3 Figure 2 shows us that 119866119879119878(3 1) has a betterperformance than 119866119879119878(1 1) in the same length of intervalsthat is 119871 = 30 Figure 3 illustrates that 119866119879119878(1 1) getsthe better forecasts when the lengths of intervals are smallFurthermore some properties will be further described inFigures 4 and 5 which depict the mean forecasting errorsof the ten years Figure 4 shows the relation between the

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 5 Comparison of forecasting SSECI

Model 119872 = 1 119872 = 2 119872 = 3 119872 = 4 119872 = 5 119872 = 6 119872 = 7 119872 = 8 119872 = 9 119872 = 10

RMSE

119866119879119878(119872 1) 2274 2164 2121 2092 2086 2071 2066 2066 2025 2010119866119879119878(119872 2) 1832 1904 1790 1792 1769 1756 1754 1755 1695 1688119866119879119878(119872 3) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 4) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683119866119879119878(119872 5) 1837 1806 1789 1793 1767 1754 1756 1756 1698 1683

[28] 1841 1841 1942 1943 1951 1939 2005 1971 2000 2026[29] 2107 2102 2087 2073 2008 1988 1979 1969 1818 1821[31] 1756 1756 1781 1778 1838 1831 1867 1925 1947 1960

MAE

119866119879119878(119872 1) 1861 1710 1661 1612 1609 1608 1607 1611 1575 1550119866119879119878(119872 2) 1408 1389 1388 1391 1373 1353 1345 1346 1316 1317119866119879119878(119872 3) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 4) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306119866119879119878(119872 5) 1417 1387 1381 1385 1367 1345 1346 1345 1313 1306

[28] 1392 1392 1478 1502 1522 1518 1541 1538 1562 1590[29] 1731 1726 1715 1704 1677 1663 1656 1647 1604 1607[31] 1281 1281 1329 1306 1357 1338 1419 1437 1442 1551

MAPE

119866119879119878(119872 1) 00127 00117 00113 00110 00110 00110 00109 00110 00107 00105119866119879119878(119872 2) 00096 00095 00095 00095 00094 00092 00092 00092 00090 00090119866119879119878(119872 3) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 4) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089119866119879119878(119872 5) 00097 00095 00094 00094 00093 00092 00092 00091 00089 00089

[28] 00095 00095 00101 00102 00104 00103 00105 00105 00106 00108[29] 00118 00118 00117 00116 00114 00113 00113 00112 00109 00109[31] 00087 00087 00091 00089 00092 00091 00097 00098 00098 00099

The bold data means the minimum error of the models with the same order

Table 6 Effects of membership function

Model 119896 = 3 119896 = 6 119896 = 9

Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3 Fun 1 Fun 2 Fun 3119866119879119878(119896 1) 175133 173290 173755 171512 169790 170653 164968 162923 164014119866119879119878(119896 2) 158071 157476 159365 154880 154686 157182 147028 146918 147719119866119879119878(119896 3) 158071 156926 158736 154880 153137 154631 147028 145116 144828119866119879119878(119896 4) 158071 156902 158670 154880 153119 154813 147028 145218 144558119866119879119878(119896 5) 158071 156902 159157 154880 153119 155422 147028 145218 145082119866119879119878(119896 6) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 7) 158071 156902 159157 154880 153119 155424 147028 145218 145068119866119879119878(119896 8) 158071 156902 159157 154880 153119 155424 147028 145218 145068The bold data means the minimum error of the models with the same order

function the more different forecasting accuracy obtainedIt is more obvious when 119872 is increasing There is stillanother conclusion that the forecasts of the thirdmembershipfunction are not always better than those of the first function

To further investigate the relation between the parameters119872 and the length of interval the proposed model has beenapplied to forecasting stock index close prices covering tenyears from 1997 to 2006 with 119872 = 1 2 50 and 119871 =

30 60 300 Since it has been affirmed by Tables 4 and5 that the characters of MAE and MAPE are similar to

those of RMSE MAE and MAPE we will only list theRMSE comparison of these experiments as follows Someexamples of actual values and forecasts of 2003 are depicted inFigures 2 and 3 Figure 2 shows us that 119866119879119878(3 1) has a betterperformance than 119866119879119878(1 1) in the same length of intervalsthat is 119871 = 30 Figure 3 illustrates that 119866119879119878(1 1) getsthe better forecasts when the lengths of intervals are smallFurthermore some properties will be further described inFigures 4 and 5 which depict the mean forecasting errorsof the ten years Figure 4 shows the relation between the

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 50 100 150 200 250 30010

20

30

40

50

60

70

80

Length

RMSE

K = 1

K = 3

K = 5

K = 7

K = 9

Figure 4 Comparison of RMSEs with different lengths of intervals

0 10 20 30 40 5010

20

30

40

50

60

70

Order

RMSE

L = 30

L = 90

L = 150

L = 210

L = 270

Figure 5 Comparison of RMSEs with different orders of themodel

RMSEs and lengths of intervals with different orders andFigure 5 shows the relation between the RMSEs and orderswith different lengths of intervals

From Figure 4 it is clear that the longer the interval thebigger the RMSE and the higher order models are betterthan the lowersThis conclusion also was testified by Figure 5which shows us another important message that the shorterlength of intervals can result in robuster forecasts Overallthese conclusions are important for the proposed mode to beapplied on other data set or area

5 Conclusion

After discussing the high-order fuzzy time series modelsand presenting the definition and operation for generalized

fuzzy logical relationship we have proposed a novel high-order fuzzy time series models based on the new relationshipThe work is driven by the three main reasons Firstly itis urged to generalize the fuzzy logical relationship by theadvanced fuzzy time series models Secondly it is to abstractthe relationships matrices among time series and find out thepatterns of time series fluctuations based on understandablefuzzy rulesThe last one is tomake the fuzzy time seriesmodelexplain more complex relationships

By using the enrolment of the University of Alabamaand close price of Shanghai Stock Exchange CompositeIndex as data sets for evaluating the models the experi-mental results give two conclusions (1) the performanceof 119866119879119878(119872119873) is more reasonable than the three conven-tional fuzzy time series models proposed earlier by Hwanget al [28] and Chen and Chen [29 31] (2) the numberof orders and principal fuzzy logical relationship affect theforecasting result slightly The higher the order the betterthe forecasting results the more hierarchy principal fuzzylogical relationships the less forecasts error but not infinitedecreasing

In the future research some suggestions are provided toimprove this paper The relation between the principal fuzzyrelationships and the conventional fuzzy relationships needsto be further discussed For example whatrsquos the effect broughtby exchange the definition of membership function and theoperations of principal fuzzy logical relationship How greatthis kind of effect Since the proposed model is on the basisof fuzzy logical relationship a generalized fuzzy relationshipstudy work is worth devoting into improvement of the modelhybridized with some advanced algorithms

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (nos 31260273 61261027)the Key Project of Chinese Ministry of Education (no210116) the Natural Science Foundation of Jiangxi ProvinceChina (nos 2010GQS0127 20114BAB211013 20122BAB21103320122BAB201044 20122BAB2010) and the JiangXi Provin-cial Foundation for Leaders of Disciplines in Science(20113BCB22008)

References

[1] Q Song and B S Chissom ldquoFuzzy time series and its modelsrdquoFuzzy Sets and Systems vol 54 no 3 pp 269ndash277 1993

[2] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part Irdquo Fuzzy Sets and Systems vol 54 no 1 pp 1ndash91993

[3] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime series Part IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp 1ndash81994

[4] S M Chen ldquoForecasting enrollments based on fuzzy timeseriesrdquo Fuzzy Sets and Systems vol 81 pp 311ndash319 1996

[5] K H Huarng ldquoEffective lengths of intervals to improve fore-casting in fuzzy time seriesrdquo Fuzzy Sets and Systems vol 123no 3 pp 387ndash394 2001

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

[6] K H Huarng and H K Yu ldquoA dynamic approach to adjustinglengths of intervals in fuzzy time series forecastingrdquo IntelligentData Analysis vol 8 no 1 pp 3ndash27 2004

[7] K H Huarng ldquoRatio-based lengths of intervals to improvefuzzy time series forecastingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 36 no 2 pp 328ndash340 2006

[8] Y H Leu C P Lee and Y Z Jou ldquoA distance-based fuzzy timeseries model for exchange rates forecastingrdquo Expert Systemswith Applications vol 36 no 4 pp 8107ndash8114 2009

[9] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoFinding an optimal interval length in high order fuzzytime seriesrdquo Expert Systems with Applications vol 37 pp 5052ndash5055 2010

[10] U Yolcu E Egrioglu V R Uslu M A Basaran and C HAladag ldquoAnew approach for determining the length of intervalsfor fuzzy time seriesrdquo Applied Soft Computing vol 9 no 2 pp647ndash651 2009

[11] H K Yu ldquoWeighted fuzzy time series model for TAIEXforecastingrdquo Physica A vol 349 no 3ndash4 pp 609ndash624 2005

[12] C H Cheng T L Chen and C H Chiang ldquoTrend-weightedfuzzy timeseries model for TAIEX forecastingrdquo Lecture Notes inComputer Science vol 4234 no 3 pp 469ndash477 2006

[13] M H Lee E Riswan and I Zuhaimy ldquoModified weighted forenrollment forecasting based on fuzzy time seriesrdquoMatematikavol 25 no 1 pp 67ndash78 2009

[14] W R Qiu X D Liu and H L Li ldquoA generalized method forforecasting based on fuzzy time seriesrdquo Expert Systems withApplications vol 38 no 8 pp 10446ndash10453 2011

[15] W R Qiu X D Liu and L D Wang ldquoForecasting Shanghaicomposite index based on fuzzy time series and improved C-fuzzy decision treesrdquo Expert Systems with Applications vol 39no 9 pp 7680ndash7689 2012

[16] S R Singh ldquoA robustmethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 188 no 1pp 472ndash484 2007

[17] S R Singh ldquoA simplemethod of forecasting based on fuzzy timeseriesrdquo Applied Mathematics and Computation vol 186 no 1pp 330ndash339 2007

[18] S R Singh ldquoA computational method of forecasting based onhigh-order fuzzy time seriesrdquo Expert Systems with Applicationsvol 36 pp 10551ndash10559 2009

[19] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on fuzzy logical relationshipsand genetic algorithmsrdquo Expert Systems with Applications vol33 pp 539ndash550 2007

[20] LW Lee L HWang and SM Chen ldquoTemperature predictionand TAIFEX forecasting based on high-order fuzzy logical rela-tionships and genetic simulated annealing techniquesrdquo ExpertSystems with Applications vol 34 pp 328ndash336 2008

[21] K H Huarng and H K Yu ldquoA type 2 fuzzy time series modelfor stock index forecastingrdquoPhysicaA vol 353 no 1ndash4 pp 445ndash462 2005

[22] L Youdthachai Y J Yang and R John ldquoHigh-order type-2fuzzy time seriesrdquo in Proceedings of the International Conferenceon Soft Computing and Pattern Recognition (SoCPaR rsquo10) pp363ndash368 Cergy Pontoise France 2010

[23] I H Kuo S J Horng T W Kao T L Lin C L Lee and YPan ldquoAn improved method for forecasting enrollments basedon fuzzy time series and particle swarm optimizationrdquo ExpertSystems with Applications vol 36 no 3 part 2 pp 6108ndash61172009

[24] I H Kuo S J Horng T W Kao et al ldquoAn efficient flow-shop scheduling algorithm based on a hybrid particle swarmoptimization modelrdquo Expert Systems with Applications vol 36no 3 part 2 pp 7027ndash7032 2009

[25] I H Kuo S J Horng and Y H Chen ldquoForecasting TAIFEXbased on fuzzy time series and particle swarm optimizationrdquoExpert Systems with Applications vol 37 pp 1494ndash1502 2010

[26] C H Aladag E Egrioglu M A Basaran U Yolcu and V RUslu ldquoForecasting in high order fuzzy times series by usingneural networks to define fuzzy relationsrdquo Expert Systems withApplications vol 36 pp 4228ndash4231 2009

[27] E Egrioglu C H Aladag U Yolcu V R Uslu and M ABasaran ldquoA new approach based on artificial neural networksfor high order multivariate fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 10589ndash10594 2009

[28] J R Hwang S M Chen and C H Lee ldquoHandling forecastingproblems using fuzzy time seriesrdquo Fuzzy Sets and Systems vol100 pp 217ndash228 1998

[29] S M Chen ldquoForecasting enrollements based on high-orderfuzzy time seriesrdquo Cybernetics and Systems vol 33 pp 1ndash162002

[30] S M Chen and N Y Chung ldquoForecasting enrollments usinghigh-order fuzzy time series and genetic algorithmsrdquo Interna-tional Journal of Intelligent Systems vol 21 no 5 pp 484ndash5012006

[31] S M Chen and C D Chen ldquoHandling forecasting problemsbased on high-order fuzzy logical relationshipsrdquo Expert Systemswith Applications vol 38 pp 3857ndash3864 2011

[32] C H Cheng H J Teoh and T L Chen ldquoHigh-order fuzzy timeseries based on rough set for forecasting TAIEXrdquo in Proceedingsof the 6th International Conference on Machine Learning andCybernetics vol 3 pp 1354ndash1358 Hong Kong China 2007

[33] L W Lee L H Wang S M Chen and Y H Leu ldquoHandlingforecasting problems based on two-factors high-order fuzzytime seriesrdquo IEEE Transactions on Fuzzy Systems vol 14 no 3pp 468ndash477 2006

[34] J I Park D J Lee C K Song and M G Chun ldquoTAIFEX andKOSPI 200 forecasting based on two-factors high-order fuzzytime series and particle swarm optimizationrdquo Expert Systemswith Applications vol 37 pp 959ndash967 2010

[35] H J Teoh T L Chen C H Cheng and H H Chu ldquoA hybridmulti-order fuzzy time series for forecasting stock marketsrdquoExpert Systems with Applications vol 36 pp 7888ndash7897 2009

[36] S T Li and Y C Cheng ldquoAn enhanced deterministic fuzzy timeseries forecasting modelrdquo Cybernetics and Systems vol 40 no3 pp 211ndash235 2009

[37] N Y Wang and S M Chen ldquoTemperature prediction andTAIFEX forecasting based on automatic clustering techniquesand two-factors high-order fuzzy time seriesrdquo Expert Systemswith Applications vol 36 pp 2143ndash2154 2009

[38] W R Qiu X D Liu and L DWang ldquoForecasting in time seriesbased on generalized fuzzy logical relationshiprdquo ICIC ExpressLetters A vol 4 no 5 pp 1431ndash1438 2010

[39] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[40] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 pp 199ndash249 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of