Research Article Intuitionistic Fuzzy Time Series Forecasting Model Based on Intuitionistic Fuzzy Reasoning

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1 Mathematical Problems i Egieerig Volume 2016, Article ID , 12 pages Research Article Ituitioistic Fuzzy Time Series Forecastig Model Based o Ituitioistic Fuzzy Reasoig Ya a Wag, Yigjie Lei, Xiaoshi Fa, ad Yi Wag Air ad Missile Defese College, Air Force Egieerig Uiversity, Xi a , Chia Correspodece should be addressed to Ya a Wag; wy @163.com Received 30 Jue 2015; Revised 10 October 2015; Accepted 20 October 2015 Academic Editor: Reik Doer Copyright 2016 Ya a Wag et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Fuzzy sets theory caot describe the data comprehesively, which has greatly limited the objectivity of fuzzy time series i ucertai data forecastig. I this regard, a ituitioistic fuzzy time series forecastig model is built. I the ew model, a fuzzy clusterig algorithm is used to divide the uiverse of discourse ito uequal itervals, ad a more objective techique for ascertaiig the membership fuctio ad omembership fuctio of the ituitioistic fuzzy set is proposed. O these bases, forecast rules based o ituitioistic fuzzy approximate reasoig are established. At last, cotrast experimets o the erollmets of the Uiversity of Alabama ad the Taiwa Stock Exchage Capitalizatio Weighted Stock Idex are carried out. The results show that the ew model has a clear advatage of improvig the forecast accuracy. 1. Itroductio Time series forecastig theory plays a importat role i the fields of ecoomy, society, ad ature. However, covetioal forecastigmethodsar ybasedostatisticalaalysis, such as ARMA ad ARIMA. These methods have two drawbacks: firstly, they eed lots of historical data meetig certai coditios; secodly, they caot hadle liguistic values or imprecise data. Therefore, Sog ad Chissom [1 3] proposed the fuzzy time series (FTS) forecastig model which could effectively maage fuzzy iformatio with the combiatio of fuzzy sets ad fuzzy logic. The basic idea of FTS is that historical data are expressed as fuzzy sets ad series variatio treds are expressed as fuzzy relatios. Data are forecasted by fuzzy reasoig while there are ot eough historical data or just some imprecise data. The FTS theory has aroused wide cocers sice its first appearace, ad lots of excellet works have bee doe i the past twety years. Sog et al. [4] built a fuzzy stochastic fuzzy time series model focusig o a special kid of fuzzy historicaldatawhoseprobabilitiesarealsofuzzysets.hwagetal. [5] used the variatios of historical data istead of the data themselves to build a time-variat FTS model. This model is quite differet from Sog ad Chissom s oe, but it got a more accurate result. Cheg et al. [6] used the probabilities of fuzzy relatios to costruct a weighted 0-1 matrix for forecastig, which is simpler to calculate tha previous models. Aladag adcoworkers[7,8]usedaoptimizatioalgorithmad artificial eural etworks to build a few high-order models, which were obviously superior to first-order models. Sigh ad Borah [9] developed the model of referece [6] by usig the importace of fuzzy relatios as their weights. Accordig to this chage, they also proposed a ew defuzzificatio method. Huarg [10] discussed the effects of differet legths of itervals to forecast accuracy at the first time ad put forward distributio-based legth ad average-based legth to approach this issue. Lu et al. [11, 12] itegrated the iformatio graules ad graular computig with Che s method to get better approaches for uiverse partitio. S.-M. Che ad S.-W. Che [13] classified the fuzzy relatios ito three groups: the dowtred group, equal-tred group, ad uptred group. The probabilities of three groups were used to build a two-factor secod-order model. I FTS models, thezadehfuzzyset[14]isusedtofuzzifyhistoricaldata; amely, there is oly oe attribute membership measurig the subjectio degree. This is either objective or comprehesive ad cosequetly limits the FTS models to deal with ucertai iformatio ad improve their forecast accuracy.

2 2 Mathematical Problems i Egieerig The ituitioistic fuzzy set [15] has three idicators to describe data: the membership, the omembership, ad the ituitioistic idex, which make it more objective ad careful i fuzzy iformatio descriptio. Therefore, Castillo et al. [16] combied the ituitioistic fuzzy set with time series aalysis ad put forward a ituitioistic fuzzy reasoig system for data forecastig. However, the mai structure was just a weighted average of two subreasoig systems based o membership ad omembership fuctios. Joshi ad Kumar [17] built the first ituitioistic fuzzy time series (IFTS) forecastig model based o the FTS model, but there is a drawback i the costructio of ituitioistic fuzzy set: the ituitioistic idex is 0.2 all the time. Zheg et al. [18, 19] used the ituitioistic fuzzy c-meas clusterig algorithm to get uequal itervals of the uiverse of discourse, ad they also used the trace-back mechaism ad vector quatizatio to forecast. Their models effectively advaced the forecast results, but how to trasform the historical data ito a suitable form for the ituitioistic fuzzy c-meas clusterig algorithm is still a urget problem. The itroductio of ituitioistic fuzzy sets dramatically exteds the ability for time series to hadle with ucertai ad imprecise data. It also sets a ew research directio for FTS. However, the study o IFTS theory is just gettig started. There are oly a few academic achievemets, ad there is a lack of uificatio ad theoretical depth; the forecast accuracy eeds further improvemet as well. I view of the above problems, we propose a IFTS model with modificatios i three aspects: uiverse partitio, ituitioistic fuzzy set costructio, ad forecast rules establishmet.thepaperisorgaizedasfollows:sectio2briefly reviews some cocepts o ituitioistic fuzzy sets ad ituitioistic fuzzy time series. Sectio 3 details how to establish the ovel IFTS model i four steps. I Sectio 4, several existigmodelsaswellastheproposedmodelareusedtoperform profoud experimets ad validate the effectiveess of the proposed model. Fially, Sectio 5 gives some coclusios. 2. Basic Cocepts I this sectio, some basic defiitios of ituitioistic fuzzy set ad IFTS are preseted. Defiitio 1. Let X be a fiite uiversal set. A ituitioistic fuzzy set A i X is a object havig the form A = { x, μ A (x),γ A (x) x X}, (1) where the fuctio μ A (x) : X [0, 1] defies the degree of membership ad the fuctio γ A (x) : X [0, 1] defies the degree of omembership of the elemet x Xto set A.For every x X, 0 μ A (x)+γ A (x) 1. π A (x) = 1 μ A (x) γ A (x) is called the ituitioistic idex of x i A. It is the hesitacy of x to A. Whe X={x 1,x 2,...,x } is discrete, the ituitioistic fuzzy set A ca be oted as A= μ A (x i ),γ A (x i ) x i, x i X. (2) Defiitio 2. Let X ad Y be two fiite uiversal sets. A biary ituitioistic fuzzy relatio R from X to Y is a ituitioistic fuzzy set i the direct product space X Y: R = { (x, y), μ R (x, y), γ R (x, y) (x, y) X Y}, (3) where 0 μ A (x, y) + γ A (x, y) 1, (x, y) X Y for μ R (x,y):x Y [0,1],adγ R (x,y):x Y [0,1]. Defiitio 3. Let X(t) (t = 1,2,...),asubsetofR, be the uiverse of discourse o which ituitioistic fuzzy sets f i (t) = μ i (X(t)), γ i (X(t)) (i = 1, 2,...) are defied. F(t) = {f 1 (t), f 2 (t),...} is a collectio of f i (t) ad defies a ituitioistic fuzzy time series o X(t). Defiitio 4. Let R(t, t 1) be a ituitioistic fuzzy relatio from F(t 1) to F(t).SupposethatF(t) is caused oly by F(t 1), deoted as F (t) =F(t 1) R(t, t 1), (4) where is the ituitioistic fuzzy compositioal operator. The R(t, t 1) is called a first-order ituitioistic fuzzy logical relatioship of F(t). Defiitio 5. If R(t, t 1) is idepedet of time t, R (t, t 1) =R(t 1,t 2) t, (5) the F(t) is called a time-ivariat ituitioistic fuzzy time series. Otherwise, F(t) is called a time-variat ituitioistic fuzzy time series. The IFTS model studied i this paper is first order ad time-ivariat. 3. The Novel Ituitioistic Fuzzy Time Series Forecastig Model The IFTS model ca be summarized i four steps as the FTS model: (1)Defieadpartitiotheuiverseofdiscourse. (2) Costruct ituitioistic fuzzy set ad ituitioistically fuzzify the historical data. (3) Establish forecast rules ad get the forecasted value. (4) Defuzzify ad output the forecast result. The rest of this sectio will detail the proposed IFTS model followig this procedure Uequal Uiverse Partitio Based o Fuzzy Clusterig. First of all, the uiverse of discourse U=[x mi ε 1,x max +ε 2 ] should be defied, where x mi ad x max are the miimum ad maximum historical data, respectively. ε 1 ad ε 2 are two proper positive umbers. Usually, for simplicity, ε 1 ad ε 2 are chose to roud dow x mi ad roud up x max to two proper itegers. Secodly, partitio the uiverse U ito several itervals. Refereces [21 23] proved that uequal itervals do ot oly

3 Mathematical Problems i Egieerig 3 have actual meaigs for regular uderstadig but also lead to a better outcome tha equal oes. Some researchers [22, 24, 25] have already made achievemets i this step byadoptigmethodssuchasgeeticalgorithms,particle swarm optimizatio, ad fuzzy c-meas clusterig algorithm. Butthesekidsofmethodsusuallyeedahugeamoutof historical data to get a good performace, which deviates from the small database of historical iformatio of IFTS. What is more, i practice, the IFTS model is geerally used for problems which have ot too may historical data such as i ecoomic ad evirometal forecastig. So i this paper, we decide to use a more coveiet ad real-time method to solve this problem [26]. Let X={x 1,x 2,...,x } be the uiverse of objects to be classified, where x j =(x j1,x j2,...,x jm ) (j = 1,2,...,)has m characteristics. Let R =(r jk ) be the similarity matrix of X, wherer jk is the similarity betwee x j ad x k (j, k = 1,2,...,). A maximum spaig tree is a tree with all x j beig the vertices ad r jk beig the weights of every edge. Let λ [0,1]be the clusterig threshold. Cuttig dow the edges whoseweightsaresmallerthaλ, wecagetafewsubtrees. Hece, the vertices of differet subtrees make up differet groups. The mai steps are as follows. Step 1. Stadardize historical data. Sice the elemets of fuzzy matrix should be i [0, 1], data i differet dimesios should be trasformed ito the iterval [0, 1] to meet the requiremet of similarity matrix R [26]. Geerally, two kids of trasformatio are required. (1) Stadard deviatio trasformatio is as follows: x jh = x jh x h s h, (6) where x h =(1/) j=1 x jh ad s h = (1/) j=1 (x jh x h ) 2, j = 1,2,...,, h = 1,2,...,m. With this trasformatio, themeaofeveryvariablebecomes0,thestadarddeviatio becomes 1, ad the dimesioal differeces are elimiated. But it caot esure that x jh will locate i [0, 1]. (2) Rage trasformatio is as follows: x jh = x jh mi 1 j {x jh } max 1 j {x jh } mi 1 j {x jh }, (7) where, obviously, 0 x jh 1. Step 2. Establish the fuzzy similarity matrix R. Let r jk be the similarity betwee x j ad x k ;thewewill have a fuzzy similarity matrix R =(r jk ). There are differet ways to get r jk. Sice the Euclidea distace is widely used i similarity matrix establishmet [26], we also choose it to calculate r jk : r jk =1 d(x m j,x k )=1 h=1 (x jh x kh )2. (8) Step 3. Buildamaximumspaigtreeadclassifyhistorical data. I this step, the Kruskal algorithm [26] is used to build the maximum spaig tree. Firstly, draw every vertex x j. Secodly, draw the edges by the value of their weights r jk i descedig order, util all of the vertices are coected but with o circles. At last, cut dow the edges with smaller weights tha the threshold λ.the vertices of each coected brach make up a group. Step 4. Calculate the best λ. The value of λ varies from 0 to 1, ad the best λ leads to thebestclassificatio.sohowtogetthebestλ is a importat step. I this paper, we also use a widely used F-statistic to fid the best λ: F= r i m h=1 (x ih x h ) 2 / (r 1) r i j=1 m h=1 (x jh x ih ) 2 / ( r) F(r 1, r), where r istheumberofgroupsforagiveλ, i is the umber of objects i group i (i = 1,2,...,r), x ih = (1/) i j=1 x jh is the average of the hth (h = 1,2,...,m) characteristic of the objects i group i,adx h is the hth characteristic of all objects. I (9), the umerator represets the distaces betwee groups, ad the deomiator represets the distaces withi groups. So the bigger F is, the better the classificatio we get. For a give cofidece level α, weca fid several values of F which are larger tha F α.theλwhich leads to the largest F is the best λ, ad the correspodig classificatio is the best as well. The best classificatio ca be oted as {x 1 1,x2 1,...,x 1 1 },...,{x1 i 1,x2 i 1,...,x i 1 i 1 }, {x 1 i,x2 i,...,x i i },{x 1 i+1,x2 i+1,...,x i+1 i+1 },..., {x 1 r,x2 r,...,x r r }, (9) (10) where i = 1,2,...,r, x i 1 i 1 x1 i, x1 i x 2 i x i i,ad x i i x 1 i+1. Let x mi ε 1, i = 0 { d i = x i i +x 1 i+1, { 2,2,...,r 1 { x max +ε 2, i = r. (11) Therefore, we partitio the uiverse U ito r uequal itervals: u 1 =[d 0,d 1 ], u 2 =[d 1,d 2 ],...,adu r =[d r 1,d r ] Costructio of Ituitioistic Fuzzy Sets. Correspodig to the above r itervals, we defie r ituitioistic fuzzy sets represetig r liguistic values: A i ={ x,μ Ai (x),γ Ai (x) x X},,2,...,r. (12)

4 4 Mathematical Problems i Egieerig Costructig their membership fuctios ad omembership fuctios is the key poit i this sectio. Sice the ituitioistic fuzzy set has a special characteristic, ituitioistic idex, the desig of membership fuctio ad omembership fuctio has bee quite comprehesive. However, existig methods based o fuzzy statistics, trichotomy, or biary compariso sequecig usually set the ituitioistic idex to a fixed value, which does ot take full advatage of the ituitioistic fuzzy set [27]. Therefore, accordig to the characteristics of IFTS itervals, a more objective method is proposed i this sectio. First of all, two rules based o objective aalysis are as follows: (1) Whe x is located i the middle of a iterval, amely, x=(d i 1 +d i )/2, we defie that μ Ai ((d i 1 +d i )/2) = 1 ad γ Ai ((d i 1 +d i )/2) = 0. (2) Whe x is located o the boudaries of a iterval, amely, x = d i, we defie that ituitioistic idex has the maximum value ad μ Ai (d i ) = γ Ai (d i ).Let π Ai (d i )=α(0 α 1);thewecagetμ Ai (d i )= γ Ai (d i ) = (1 α)/2. The, i view of above rules, the membership fuctio is defied as a Gaussia fuctio: μ Ai = exp ( (x c μi) 2 ). (13) 2σ 2 μi The omembership fuctio is a trasformatio of Gaussia fuctio: γ Ai (x) =1 exp ( (x c γi) 2 ). (14) 2σ 2 γi Hece, the ituitioistic idex fuctio reads π Ai (x) =1 μ Ai (x) γ Ai (x), (15) where i = 1,2,...,r ad c μi, σ μi, c γi,adσ γi are importat fuctio parameters. The calculatios of them are based o the above two rules: c μi =c γi = d i 1 +d i, 2 σ 2 μi = (d i 1 c μi ) 2 2 l ((1 α) /2), σ 2 γi = (d i c γi ) 2 2 l (1+(1 α) /2). (16) Defiitio 6. Let A be a ituitioistic fuzzy set i a fiite uiverse X.If (1) 0 μ A (x) 1, 0 γ A (x) 1, (2) 0 π A (x) 1, 0 μ A (x) + γ A (x) 1, (3) π A (x) + μ A (x) + γ A (x) = 1, the A is a ormal ituitioistic fuzzy set. Therefore, we obtai the followig theorem. Theorem 7. The membership fuctio ad omembership fuctio of A i (i = 1,2,...,r) are stadard; that is, A i is a ormal ituitioistic fuzzy set. Proof. (1) y = exp( (x c) 2 /2σ 2 ) is a Gaussia fuctio, so we have 0 exp ( (x c)2 2σ 2 ) 1, (17) 0 1 exp ( (x c)2 2σ 2 ) 1. Therefore, 0 μ Ai (x) 1, (18) 0 γ Ai (x) 1. (2) Give 0 α 1ad c μi =c γi =(d i 1 +d i )/2,wehave Hece, exp ( (x c μi) 2 2σ 2 μi exp ( (x c μi) 2 2σ 2 μi σ 2 μi σ2 γi. (19) ) exp ( (x c γi) 2 2σ 2 γi ) exp ( (x c γi) 2 2σ 2 γi ), ) 0. (20) O the other had, sice 0 exp( (x c μi ) 2 /2σ 2 μi ) 1ad 0 exp( (x c γi ) 2 /2σ 2 γi ) 1,wehave 1 exp ( (x c μi) 2 Therefore, 2σ 2 μi 1 exp ( (x c μi) 2 That is, 0, 2σ 2 μi 0 exp ( (x c μi) σ 2 μi 0 μ Ai (x) +γ Ai (x) 1, ) exp ( (x c γi) 2 2σ 2 γi ) exp ( (x c γi) 2 2σ 2 γi ) 1. (21) ) )+1 exp ( (x c γi) 2 2σ 2 γi 0 π Ai (x) =1 μ Ai (x) γ Ai (x) 1. ) (22) (23) (3) Accordig to the calculatio of π Ai (x),itcabeeasily foud that π Ai (x) + μ Ai (x) + γ Ai (x) = 1. Thiscompletesthe proof.

5 Mathematical Problems i Egieerig 5 Theorem 7 shows that the calculatio of the membership fuctio ad omembership fuctio of the ituitioistic fuzzy set is correct ad appropriate Forecast Rules Based o Ituitioistic Fuzzy Reasoig Ituitioistic Fuzzy Multiple Modus Poes. Let A i (i = 1,2,...,) ad A be ituitioistic fuzzy sets i uiverse U ad let B i (,2,...,)ad B be ituitioistic fuzzy sets i uiverse V. The geeralized multiple modus poes based o ituitioistic fuzzy relatio [27] is that a ew propositio that y is B cabeiferredfrom+1 propositios: if x is A i,they is B i ad x is A. The reasoig model is as follows: Rules: IF x is A 1 THEN y is B 1 IF x is A 2 THEN y is B 2. IF x is A THEN y is B Iput: IF x is A Output: y is B Every rule has a correspodig iput-output relatio R i. For R i, differet operators result i differet μ R ad γ R,but the reasoig outputs are all the same. Sice it has a better performace ad is easier to calculate tha other operators [27], the Mamdai implicatio operator R c is used here: R i = (A B) ={ (x,y),μ Ri (x, y), γ Ri (x, y) (x, y) U V}, where μ Ri (x, y) = μ Ai (x) μ Bi (x), γ Ri (x, y) = γ Ai (x) γ Bi (y). (24) (25) The, accordig to the compositioal operatio of ituitioistic fuzzy rules, we get the total relatio: R= where R i ={ (x,y),μ R (x, y), γ R (x,y) (x,y) U V}, μ R (x, y) = γ R (x, y) = The reasoig output is μ Ri (x, y) = γ Ri (x, y) = (μ Ai (x) μ Bi (y)), (γ Ai (x) γ Bi (y)). (26) (27) B =A R, (28) where is defied as the maximum ad miimum operators: ad : μ B (y) = (μ A (x) μ R (x, y)), x U γ B (y) = (γ A (x) γ R (x, y)). x U (29) Forecast Rules of IFTS Model. Ispired by the ituitioistic fuzzy multiple modus poes, we exchage the positios of historical data ad ituitioistic fuzzy sets A i (i = 1,2,...,r) i the IFTS model; that is, let the historical data be ituitioistic fuzzy sets, oted as F i (j = 1,2,...,t),ad let A i be the elemets i F j, ad let μ Ai (x) ad γ Ai (x) be the membership ad omembership of A i to F j.thef j is as follows: F j = r μ Fj (x i ),γ Fj (x i ) x i, (30) where μ Fj (x i )=μ Ai (f i (t)), (31) γ Fj (x i )=γ Ai (f i (t)). Hece, we apply the ituitioistic fuzzy multiple modus poes to A i ad F j.thereasoigmodelisasfollows: Rules: IF x is F 1 THEN y is F 2 IF x is F 2 THEN y is F 3. IF x is F j THEN y is F j+1. IF x is F t 1 THEN y is F t Iput: IF x is F t Output: y is F t+1 Thereasoigoutputisasfollows: F t+1 =F t R, (32) where μ R (x, y) = γ R (x, y) = t 1 j=1 t 1 j=1 μ Rj (x, y) = γ Rj (x, y) = t 1 j=1 t 1 j=1 (μ Fj (x) μ Fj+1 (y)), (γ Fj (x) γ Fj+1 (y)). (33) So, the membership ad omembership of the output ituitioistic fuzzy set are μ F t+1 (y) = (μ Ft (x) μ R (x, y)), x U (34) γ F t+1 (y) = (γ Ft (x) γ R (x, y)). x U That is to say, the membership ad omembership of the forecasted result f t+1 (t) to every ituitioistic fuzzy set A i are μ F t+1 (y) ad γ F t+1 (y).

6 6 Mathematical Problems i Egieerig Figure 1: The maximum spaig tree of historical data Defuzzificatio Algorithm. The widely used defuzzificatio algorithms iclude the maximum truth-value algorithm, gravity algorithm, ad weighted average algorithm [27]. I this paper, we utilize the gravity algorithm, which has a more obvious ad smoother output tha others eve whe the iput has tiy chages [27]. The calculatio is as follows: C= U c(μ F (c) + (1/2) π F (c)) dc U (μ F (c) + (1/2) π F (c)) dc = U c(1+μ F (c) +γ F (c)) dc U (1 + μ F (c) +γ F (c)) dc, (35) where U is the output domai ad F is a ituitioistic fuzzy set i U. 4. Applicatios I this sectio, we focus o two umerical experimets to demostrate the performace of the proposed IFTS model. I each experimet, several existig FTS ad IFTS models are also applied o the same data set to make comparisos. The experimetal results ad associatig aalyses are show, respectively Erollmets of the Uiversity of Alabama. The erollmet of the Uiversity of Alabama has bee firstly used i Sog s paper o FTS model [2]. Sice the, this data set has bee used by most of the scholars to test their FTS or IFTS models. The detailed test process of our model is as follows. Step 1. Defie ad partitio the uiverse of discourse. The erollmets from year 1971 to 1991 are chose as historical data to forecast the erollmet of year I the historical data, x mi = ad x max = 19337, sothe uiverse of discourse is set to U = [13000, 20000]. The U is partitioed ito uequal itervals based o the fuzzy clusterig algorithm desiged i Sectio 3.1. The stepby-step details are as follows: (1) Stadardize historical data accordig to (6) ad (7). (2) Establish the fuzzy similarity matrix R as show i Table 1. (3) Use the Kruskal algorithm metioed i Sectio 3.1 to buildamaximumspaigtreebasedomatrixr.thetree is show i Figure 1. Let λ be 0.93, 0.94, 0.95, ad 0.96, respectively. We ca get differet classificatios of historical data as show i Table 2. (4) For differet classificatios, calculate the values of F accordig to (9). The results are also show i Table 2. From Table 2, we ca see that whe λ = 0.95, its correspodig F is maximum ad bigger tha F α (α = 0.05) at the same time. So this classificatio is the best. Therefore, the uiverse of discourse U is partitioed ito 9 uequal itervals accordig to the above classificatio. The boudaries of each iterval are calculated accordig to (11). The itervals are u 1 = [13000, 13309], u 2 = [13309, 14282]. u 3 = [14282, 14921], u 4 = [14921, 16186], u 5 = [16186, 16598], u 6 = [16598, 17535], u 7 = [17535, 18560], u 8 = [18560, 19149], u 9 = [19149, 20000]. (36) Step 2. Costruct ituitioistic fuzzy sets ad ituitioistically fuzzify the historical data. Correspodig to the 9 itervals, there should be 9 ituitioistic fuzzy sets A 1,A 2,...,A 9,adtheirrealistic sigificace is as follows: very very very few, very very few, very few, few, ormal, may, very may, very very may, very very very may. The calculate the parameters of the membership ad omembership fuctios based o Sectio 3.2. For α=0.4, the parameters are show i Table 3. The membership fuctio, omembership fuctio, ad ituitioistic idex fuctio of every ituitioistic fuzzy set are show i Figures 2, 3, ad 4, respectively.

7 Mathematical Problems i Egieerig 7 Table 1: The fuzzy similarity matrix R

8 8 Mathematical Problems i Egieerig Figure 2: Membership fuctios. λ Table 2: Classificatios of differet λ. Number of classificatios Classificatio F 0.05 F {1971}, {1972, 1973}, {1974}, {1979, 1980, 1987}, {1988}, {1989, 1990, 1991}, {1975, 1976, 1977, 1978, 1981, 1982, 1983, 1984, 1985, 1986} {1971}, {1972, 1973}, {1974}, {1981}, {1979, 1980, 1987}, {1988}, {1989, 1990, 1991}, {1975, 1976, 1977, 1978, 1982, 1983, 1984, 1985, 1986} {1971}, {1972, 1973}, {1974}, {1981}, {1979, 1980, 1987}, {1988}, {1989}, {1990, 1991}, {1975, 1976, 1977, 1978, 1982, 1983, 1984, 1985, 1986} {1971}, {1972}, {1973}, {1974}, {1981}, {1979, 1980, 1987}, {1988}, {1989}, {1990, 1991}, {1975, 1976, 1977, 1978, 1982, 1983, 1984, 1985, 1986} Figure 3: Nomembership fuctios Figure 4: Ituitioistic idex fuctios.

9 Mathematical Problems i Egieerig 9 TAIEX Sectio is as follows: Rules: IF x is F 1 THEN y is F 2 IF x is F 2 THEN y is F 3. IF x is F 20 THEN y is F 21 Iput: IF x is F 21 Output: y is F 22 ThewegetF 22 : Date F 22 0, 1 0, 1 0, 1 0, 1 0, 1 = A 1 A 2 A 3 A 4 A 5 Actual values Model [17] Model [13] Model [20] Model [2] Our model Figure 5: Forecast results of TAIEX , 1 0, , A 6 A 7 A , A 9, (37) The we ca calculate the membership, omembership, ad ituitioistic idex of every historical value to every ituitioistic fuzzy set. Step 3. Establish forecast rules ad forecast the erollmets. The erollmets of year 1971 to 1991 ca be deoted as F 1,F 2,...,F 21, ad the reasoig model based o where the membership of F 22 to A 8 is the biggest ad the omembership is the smallest, so the ituitioistic forecasted result is A 8. Step 4. Defuzzify ad output the forecast result. The defuzzificatio result based o Sectio 3.4 is C= U c(2+exp ( (c )2 / ( )) exp ( (c ) 2 / ( ))) dc U (2 + exp ( (c ) 2 / ( )) exp ( (c ) 2 / ( ))) dc = (38) That is to say, the erollmet of year 1992 is To test the performace of our model, we use the models ofiferece[2],[12],ad[17]aswellasourstoforecast every year s erollmet, respectively. The results are show i Table 4. The models of iferece [2, 12] are FTS models, ad the model of iferece [17] is a IFTS model. I iferece [12],therearethreekidsofuiversepartitio:7,17,ad 22 itervals. Sice there are oly 22 historical data, the 17- iterval partitio ad 22-iterval partitio are ot applicable, so we choose the 7-iterval partitio. The root mea square error (RMSE) ad average forecast error (AFE) are exploited to evaluate the performace of every model: RMSE = 1 AFE = 1 (actual value i forecast value i ) 2, actual value i forecast value i actual value i 100%. (39) The results are show i Table 5. TheresultsiTables4ad5idicatethatourmodelca ot oly reach the forecast goal but also achieve a better result tha the other tested models. That is to say, the proposed model is feasible ad efficiet Experimets o TAIEX. The Taiwa Stock Exchage Capitalizatio Weighted Stock Idex (TAIEX) is a typical ecoomic data set widely used i fuzzy time series forecastig [13, 20, 23, 25, 28, 29]. I this experimet, TAIEX values from 11/1/2004 to 12/31/2004 are used as historical data, which are show i Table 6. The ituitioistic fuzzified value of historical data whe forecasted by our model are also show i Table 6. For compariso, we also applied the models of referece [2,13,17,20]toforecastTAIEXatthesametime.Theforecast results of every model are show i Table 7 ad Figure 5. The performace of all models is compared i Table 8. Table 8 idicates that the RMSE ad MSE of proposed model are both smaller tha the other models. Therefore, our two experimets both idicate that the IFTS model proposed i this paper could effectively icrease forecast accuracy. 5. Coclusios I this paper, a ovel IFTS model is proposed for improvig the performace of FTS model. I order to be succict, we use

10 10 Mathematical Problems i Egieerig Ituitioistic fuzzy set Table 3: Fuctio parameters of A i. c μi σ μi c γi σ γi A A A A A A A A A Table 4: Forecast results of the erollmets. Year Actual Forecasted erollmet erollmet Model [2] Model [12] Model [17] Our model Table 5: Forecast performace of erollmets. Criterio Model [2] Model [12] Model [17] Our model RMSE AFE 3.35% 2.3% 2.07% 1.72% Date Table 6: Historical data of TAIEX. TAIEX Ituitioistic fuzzified value 11/1/ A 1 11/2/ A 2 11/3/ A 5 11/4/ A 5 11/5/ A 7 11/8/ A 7 11/9/ A 8 11/10/ A 8 11/11/ A 5 11/12/ A 7 11/15/ A 7 11/16/ A 7 11/17/ A 11 11/18/ A 12 11/19/ A 11 11/22/ A 5 11/23/ A 5 11/24/ A 7 11/25/ A 5 11/26/ A 3 11/29/ A 3 11/30/ A 5 12/1/ A 4 12/2/ A 5 12/3/ A 6 12/6/ A 7 12/7/ A 7 12/8/ A 6 12/9/ A 7 12/10/ A 7 12/13/ A 5 12/14/ A 7 12/15/ A 10 12/16/ A 11 12/17/ A 10 12/20/ A 9 12/21/ A 9 12/22/ A 10 12/23/ A 10 themaximumspaigtreebasedfuzzyclusterigalgorithm to partitio the uiverse of discourse ito uequal itervals. Accordig to the characteristics of partitioed data, a more objective method is proposed to ascertai membership fuctio ad omembership fuctio of the ituitioistic fuzzy set. Besides, ituitioistic fuzzy reasoig is utilized to establish forecast rules, which make the model more sesitive 12/24/ A 11 12/27/ A 9 12/28/ A 10 12/29/ A 13 12/30/ A 14 12/31/ A 15

11 Mathematical Problems i Egieerig 11 Table 7: Forecast results of TAIEX. Actual TAIEX Forecasted TAIEX Model [2] Model [20] Model [13] Model [17] Our model Table 8: Forecast performace of TAIEX. Criterio Model [2] Model [20] Model [13] Model [17] Our model RMSE AFE 0.83% 0.51% 0.65% 0.65% 0.51% to the fuzzy variatio of ucertai data. Fially, based o experimets with two data sets, the feasibility ad advatage of the ew model are verified. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmet The research is sposored by Natural Sciece Foudatio of Chia (Grat o ). Refereces [1] Q. Sog ad B. S. Chissom, Fuzzy time series ad its models, Fuzzy Sets ad Systems,vol.54,o.3,pp ,1993. [2] Q. Sog ad B. S. Chissom, Forecastig erollmets with fuzzy time series part I, Fuzzy Sets ad Systems,vol.54,o.1,pp.1 9, [3] Q. Sog ad B. S. Chissom, Forecastig erollmets with fuzzy time series part II, Fuzzy Sets ad Systems, vol.62,o.1,pp. 1 8, [4] Q. Sog, R. P. Lelad, ad B. S. Chissom, Fuzzy stochastic fuzzy time series ad its models, Fuzzy Sets ad Systems, vol. 88, o. 3, pp , [5] J.-R. Hwag, S.-M. Che, ad C.-H. Lee, Hadlig forecastig problems usig fuzzy time series, Fuzzy Sets ad Systems, vol. 100, o. 1 3, pp , [6] C.-H. Cheg, Y.-S. Che, ad Y.-L. Wu, Forecastig iovatio diffusio of products usig tred-weighted fuzzy time-series model, Expert Systems with Applicatios, vol. 36, o. 2, pp , [7] C.H.Aladag,M.A.Basara,E.Egrioglu,U.Yolcu,adV.R. Uslu, Forecastig i high order fuzzy times series by usig eural etworks to defie fuzzy relatios, Expert Systems with Applicatios,vol.36,o.3,pp ,2009. [8]E.Egrioglu,C.H.Aladag,U.Yolcu,V.R.Uslu,adM.A. Basara, Fidig a optimal iterval legth i high order fuzzy time series, Expert Systems with Applicatios,vol.37,o.7,pp , [9] P. Sigh ad B. Borah, A efficiet time series forecastig model based o fuzzy time series, Egieerig Applicatios of Artificial Itelligece, vol. 26, o. 10, pp , [10] K. Huarg, Effective legths of itervals to improve forecastig i fuzzy time series, Fuzzy Sets ad Systems, vol. 123, o. 3, pp , [11] W. Lu, W. Pedrycz, X. Liu, J. Yag, ad P. Li, The modelig of time series based o fuzzy iformatio graules, Expert Systems with Applicatios,vol.41,o.8,pp ,2014. [12] W.Lu,X.Y.Che,W.Pedrycz,X.D.Liu,adJ.Yag, Usig iterval iformatio graules to improve forecastig i fuzzy

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