# Intuitionistic Fuzzy Time Series Forecasting Model Based on Intuitionistic Fuzzy Reasoning

Authors: Ya’nan Wang; Yingjie Lei; Xiaoshi Fan; Yi Wang
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Fuzzy sets theory cannot describe the data comprehensively, which has greatly limited the objectivity of fuzzy time series in uncertain data forecasting. In this regard, an intuitionistic fuzzy time series forecasting model is built. In the new model, a fuzzy clustering algorithm is used to divide the universe of discourse into unequal intervals, and a more objective technique for ascertaining the membership function and nonmembership function of the intuitionistic fuzzy set is proposed. On these bases, forecast rules based on intuitionistic fuzzy approximate reasoning are established. At last, contrast experiments on the enrollments of the University of Alabama and the Taiwan Stock Exchange Capitalization Weighted Stock Index are carried out. The results show that the new model has a clear advantage of improving the forecast accuracy.


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## Body

## 1. Introduction

Time series forecasting theory plays an important role in the fields of economy, society, and nature. However, conventional forecasting methods are mainly based on statistical analysis, such as ARMA and ARIMA. These methods have two drawbacks: firstly, they need lots of historical data meeting certain conditions; secondly, they cannot handle linguistic values or imprecise data. Therefore, Song and Chissom [1–3] proposed the fuzzy time series (FTS) forecasting model which could effectively manage fuzzy information with the combination of fuzzy sets and fuzzy logic.The basic idea of FTS is that historical data are expressed as fuzzy sets and series variation trends are expressed as fuzzy relations. Data are forecasted by fuzzy reasoning while there are not enough historical data or just some imprecise data. The FTS theory has aroused wide concerns since its first appearance, and lots of excellent works have been done in the past twenty years. Song et al. [4] built a fuzzy stochastic fuzzy time series model focusing on a special kind of fuzzy historical data whose probabilities are also fuzzy sets. Hwang et al. [5] used the variations of historical data instead of the data themselves to build a time-variant FTS model. This model is quite different from Song and Chissom’s one, but it got a more accurate result. Cheng et al. [6] used the probabilities of fuzzy relations to construct a weighted 0-1 matrix for forecasting, which is simpler to calculate than previous models. Aladag and coworkers [7, 8] used an optimization algorithm and artificial neural networks to build a few high-order models, which were obviously superior to first-order models. Singh and Borah [9] developed the model of reference [6] by using the importance of fuzzy relations as their weights. According to this change, they also proposed a new defuzzification method. Huarng [10] discussed the effects of different lengths of intervals to forecast accuracy at the first time and put forward distribution-based length and average-based length to approach this issue. Lu et al. [11, 12] integrated the information granules and granular computing with Chen’s method to get better approaches for universe partition. S.-M. Chen and S.-W. Chen [13] classified the fuzzy relations into three groups: the “downtrend” group, “equal-trend” group, and “uptrend” group. The probabilities of three groups were used to build a two-factor second-order model. In FTS models, the Zadeh fuzzy set [14] is used to fuzzify historical data; namely, there is only one attribute—membership measuring the subjection degree. This is neither objective nor comprehensive and consequently limits the FTS models to deal with uncertain information and improve their forecast accuracy.The intuitionistic fuzzy set [15] has three indicators to describe data: the membership, the nonmembership, and the intuitionistic index, which make it more objective and careful in fuzzy information description. Therefore, Castillo et al. [16] combined the intuitionistic fuzzy set with time series analysis and put forward an intuitionistic fuzzy reasoning system for data forecasting. However, the main structure was just a weighted average of two subreasoning systems based on membership and nonmembership functions. Joshi and Kumar [17] built the first intuitionistic fuzzy time series (IFTS) forecasting model based on the FTS model, but there is a drawback in the construction of intuitionistic fuzzy set: the intuitionistic index is 0.2 all the time. Zheng et al. [18, 19] used the intuitionistic fuzzy c-means clustering algorithm to get unequal intervals of the universe of discourse, and they also used the trace-back mechanism and vector quantization to forecast. Their models effectively advanced the forecast results, but how to transform the historical data into a suitable form for the intuitionistic fuzzy c-means clustering algorithm is still an urgent problem. The introduction of intuitionistic fuzzy sets dramatically extends the ability for time series to handle with uncertain and imprecise data. It also sets a new research direction for FTS. However, the study on IFTS theory is just getting started. There are only a few academic achievements, and there is a lack of unification and theoretical depth; the forecast accuracy needs further improvement as well.In view of the above problems, we propose an IFTS model with modifications in three aspects: universe partition, intuitionistic fuzzy set construction, and forecast rules establishment. The paper is organized as follows: Section2 briefly reviews some concepts on intuitionistic fuzzy sets and intuitionistic fuzzy time series. Section 3 details how to establish the novel IFTS model in four steps. In Section 4, several existing models as well as the proposed model are used to perform profound experiments and validate the effectiveness of the proposed model. Finally, Section 5 gives some conclusions.


## 2. Basic Concepts

In this section, some basic definitions of intuitionistic fuzzy set and IFTS are presented.Definition 1.
LetX be a finite universal set. An intuitionistic fuzzy set A in X is an object having the form(1)A=x,μAx,γAx∣x∈X,where the function μA(x):X→[0,1] defines the degree of membership and the function γA(x):X→[0,1] defines the degree of nonmembership of the element x∈X to set A. For every x∈X, 0≤μA(x)+γA(x)≤1. πA(x)=1-μA(x)-γA(x) is called the intuitionistic index of x in A. It is the hesitancy of x to A.WhenX=x1,x2,…,xn is discrete, the intuitionistic fuzzy set A can be noted as(2)A=∑i=1nμAxi,γAxixi,xi∈X.Definition 2.
LetX and Y be two finite universal sets. A binary intuitionistic fuzzy relation R from X to Y is an intuitionistic fuzzy set in the direct product space X×Y:(3)R=x,y,μRx,y,γRx,y∣x,y∈X×Y,where 0≤μA(x,y)+γA(x,y)≤1, ∀(x,y)∈X×Y for μR(x,y):X×Y→[0,1], and γR(x,y):X×Y→[0,1].Definition 3.
LetX(t)(t=1,2,…), a subset of R, be the universe of discourse on which intuitionistic fuzzy sets fi(t)=μiX(t),γiX(t)(i=1,2,…) are defined. F(t)=f1(t),f2(t),… is a collection of fi(t) and defines an intuitionistic fuzzy time series on X(t).Definition 4.
LetRt,t-1 be an intuitionistic fuzzy relation from F(t-1) to F(t). Suppose that F(t) is caused only by F(t-1), denoted as(4)Ft=Ft-1∘Rt,t-1,where “∘” is the intuitionistic fuzzy compositional operator. Then R(t,t-1) is called a first-order intuitionistic fuzzy logical relationship of F(t).Definition 5.
IfRt,t-1 is independent of time t,(5)Rt,t-1=Rt-1,t-2∀t,then F(t) is called a time-invariant intuitionistic fuzzy time series. Otherwise, F(t) is called a time-variant intuitionistic fuzzy time series.The IFTS model studied in this paper is first order and time-invariant.


## 3. The Novel Intuitionistic Fuzzy Time Series Forecasting Model

The IFTS model can be summarized in four steps as the FTS model:(1)
Define and partition the universe of discourse.(2)
Construct intuitionistic fuzzy set and intuitionistically fuzzify the historical data.(3)
Establish forecast rules and get the forecasted value.(4)
Defuzzify and output the forecast result.The rest of this section will detail the proposed IFTS model following this procedure.


### 3.1. Unequal Universe Partition Based on Fuzzy Clustering

First of all, the universe of discourseU=[xmin-ε1,xmax+ε2] should be defined, where xmin and xmax are the minimum and maximum historical data, respectively. ε1 and ε2 are two proper positive numbers. Usually, for simplicity, ε1 and ε2 are chosen to round down xmin and round up xmax to two proper integers.Secondly, partition the universeU into several intervals. References [21–23] proved that unequal intervals do not only have actual meanings for regular understanding but also lead to a better outcome than equal ones. Some researchers [22, 24, 25] have already made achievements in this step by adopting methods such as genetic algorithms, particle swarm optimization, and fuzzy c-means clustering algorithm. But these kinds of methods usually need a huge amount of historical data to get a good performance, which deviates from the small database of historical information of IFTS. What is more, in practice, the IFTS model is generally used for problems which have not too many historical data such as in economic and environmental forecasting. So in this paper, we decide to use a more convenient and real-time method to solve this problem [26].LetX=x1,x2,…,xn be the universe of objects to be classified, where xj=(xj1,xj2,…,xjm) (j=1,2,…,n) has m characteristics. Let R=(rjk)n×n be the similarity matrix of X, where rjk is the similarity between xj and xk (j,k=1,2,…,n). A maximum spanning tree is a tree with all xj being the vertices and rjk being the weights of every edge. Let λ∈[0,1] be the clustering threshold. Cutting down the edges whose weights are smaller than λ, we can get a few subtrees. Hence, the vertices of different subtrees make up different groups. The main steps are as follows.Step 1.
Standardize historical data. Since the elements of fuzzy matrix should be in[0,1], data in different dimensions should be transformed into the interval [0,1] to meet the requirement of similarity matrix R [26]. Generally, two kinds of transformation are required.
(
1
) Standard deviation transformation is as follows:(6)xjh′=xjh-x-hsh,where x-h=(1/n)∑j=1nxjh and sh=(1/n)∑j=1n(xjh-x-h)2, j=1,2,…,n, h=1,2,…,m. With this transformation, the mean of every variable becomes 0, the standard deviation becomes 1, and the dimensional differences are eliminated. But it cannot ensure that xjh′ will locate in [0,1].
(
2
) Range transformation is as follows:(7)xjh′′=xjh′-min1≤j≤n⁡xjh′max1≤j≤n⁡xjh′-min1≤j≤n⁡xjh′,where, obviously, 0≤xjh′′≤1.Step 2.
Establish the fuzzy similarity matrixR.
Letrjk be the similarity between xj′′ and xk′′; then we will have a fuzzy similarity matrix R=(rjk)n×n. There are different ways to get rjk. Since the Euclidean distance is widely used in similarity matrix establishment [26], we also choose it to calculate rjk:(8)rjk=1-dxj′′,xk′′=1-∑h=1mxjh′′-xkh′′2.Step 3.
Build a maximum spanning tree and classify historical data.
In this step, the Kruskal algorithm [26] is used to build the maximum spanning tree. Firstly, draw every vertex xj. Secondly, draw the edges by the value of their weights rjk in descending order, until all of the vertices are connected but with no circles. At last, cut down the edges with smaller weights than the threshold λ. The vertices of each connected branch make up a group.Step 4.
Calculate the bestλ.
The value ofλ varies from 0 to 1, and the best λ leads to the best classification. So how to get the best λ is an important step. In this paper, we also use a widely used F-statistic to find the best λ:(9)F=∑i=1rni∑h=1mx-ih-x-h2/r-1∑i=1r∑j=1ni∑h=1mxjh-x-ih2/n-r~Fr-1,n-r,where r is the number of groups for a given λ,  ni is the number of objects in group i (i=1,2,…,r), x-ih=(1/n)∑j=1nixjh is the average of the hth (h=1,2,…,m) characteristic of the objects in group i, and x-h is the hth characteristic of all objects. In (9), the numerator represents the distances between groups, and the denominator represents the distances within groups. So the bigger F is, the better the classification we get. For a given confidence level α, we can find several values of F which are larger than Fα. The λ which leads to the largest F is the best λ, and the corresponding classification is the best as well.
The best classification can be noted as(10)x11,x12,…,x1n1,…,xi-11,xi-12,…,xi-1ni-1,xi1,xi2,…,xini,xi+11,xi+12,…,xi+1ni+1,…,xr1,xr2,…,xrnr,where i=1,2,…,r, xi-1ni-1≤xi1, xi1≤xi2≤⋯≤xini, and xini≤xi+11.
Let(11)di=xmin-ε1,i=0xini+xi+112,i=1,2,…,r-1xmax+ε2,i=r.Therefore, we partition the universe U into r unequal intervals: u1=[d0,d1],  u2=[d1,d2],…, and  ur=[dr-1,dr].


### 3.2. Construction of Intuitionistic Fuzzy Sets

Corresponding to the abover intervals, we define r intuitionistic fuzzy sets representing r linguistic values:(12)Ai=x,μAix,γAix∣x∈X,i=1,2,…,r.Constructing their membership functions and nonmembership functions is the key point in this section.Since the intuitionistic fuzzy set has a special characteristic, intuitionistic index, the design of membership function and nonmembership function has been quite comprehensive. However, existing methods based on fuzzy statistics, trichotomy, or binary comparison sequencing usually set the intuitionistic index to a fixed value, which does not take full advantage of the intuitionistic fuzzy set [27]. Therefore, according to the characteristics of IFTS intervals, a more objective method is proposed in this section.First of all, two rules based on objective analysis are as follows:(1)
Whenx is located in the middle of an interval, namely, x=(di-1+di)/2, we define that μAi(di-1+di)/2=1 and γAi(di-1+di)/2=0.(2)
Whenx is located on the boundaries of an interval, namely, x=di, we define that intuitionistic index has the maximum value and μAi(di)=γAi(di). Let πAidi=α(0≤α≤1); then we can get μAi(di)=γAi(di)=(1-α)/2.Then, in view of above rules, the membership function is defined as a Gaussian function:(13)μAi=exp⁡-x-cμi22σμi2.The nonmembership function is a transformation of Gaussian function:(14)γAix=1-exp⁡-x-cγi22σγi2.Hence, the intuitionistic index function reads(15)πAix=1-μAix-γAix,where i=1,2,…,r and cμi, σμi, cγi, and σγi are important function parameters. The calculations of them are based on the above two rules:(16)cμi=cγi=di-1+di2,σμi2=-di-1-cμi22ln⁡1-α/2,σγi2=-di-cγi22ln⁡1+1-α/2.Definition 6.
LetA be an intuitionistic fuzzy set in a finite universe X. If(1)
0≤μA(x)≤1, 0≤γA(x)≤1,(2)
0≤πA(x)≤1, 0≤μA(x)+γA(x)≤1,(3)
πA(x)+μA(x)+γA(x)=1,then A is a normal intuitionistic fuzzy set.Therefore, we obtain the following theorem.Theorem 7.
The membership function and nonmembership function ofAi(i=1,2,…,r) are standard; that is, Ai is a normal intuitionistic fuzzy set.Proof.
(
1
)
y
=
e
x
p
-
(
x
-
c
)
2
/
2
σ
2 is a Gaussian function, so we have(17)0≤exp-x-c22σ2≤1,0≤1-exp-x-c22σ2≤1.Therefore,(18)0≤μAix≤1,0≤γAix≤1.
(
2
) Given 0≤α≤1 and cμi=cγi=(di-1+di)/2, we have(19)σμi2≤σγi2.Hence,(20)exp⁡-x-cμi22σμi2≤exp⁡-x-cγi22σγi2,exp⁡-x-cμi22σμi2-exp⁡-x-cγi22σγi2≤0.On the other hand, since 0≤exp-(x-cμi)2/2σμi2≤1 and 0≤exp-(x-cγi)2/2σγi2≤1, we have(21)-1≤exp⁡-x-cμi22σμi2-exp⁡-x-cγi22σγi2≤1.Therefore,(22)-1≤exp⁡-x-cμi22σμi2-exp⁡-x-cγi22σγi2≤0,0≤exp⁡-x-cμi22σμi2+1-exp⁡-x-cγi22σγi2≤1.That is,(23)0≤μAix+γAix≤1,0≤πAix=1-μAix-γAix≤1.
(
3
) According to the calculation of πAi(x), it can be easily found that πAi(x)+μAi(x)+γAi(x)=1. This completes the proof.Theorem7 shows that the calculation of the membership function and nonmembership function of the intuitionistic fuzzy set is correct and appropriate.


### 3.3. Forecast Rules Based on Intuitionistic Fuzzy Reasoning

#### 3.3.1. Intuitionistic Fuzzy Multiple Modus Ponens

LetAi(i=1,2,…,n) and A∗ be intuitionistic fuzzy sets in universe U and let Bi(i=1,2,…,n) and B∗ be intuitionistic fuzzy sets in universe V. The generalized multiple modus ponens based on intuitionistic fuzzy relation [27] is that a new proposition that “y is B∗” can be inferred from n+1 propositions: “if x is Ai, then y is Bi” and “x is A∗.” The reasoning model is as follows:Rules:IFx is A1THENy is B1IFx is A2THENy is B2⋮IFx is AnTHENy is BnInput:IFx is A∗Output:y is B∗Every rule has a corresponding input-output relationRi. For Ri, different operators result in different μR and γR, but the reasoning outputs are all the same. Since it has a better performance and is easier to calculate than other operators [27], the Mamdani implication operator Rc is used here:(24)Ri=A×B=x,y,μRix,y,γRix,y∣x,y∈U×V,where(25)μRix,y=μAix∧μBix,γRix,y=γAix∨γBiy.Then, according to the compositional operation of intuitionistic fuzzy rules, we get the total relation:(26)R=⋃i=1nRi=x,y,μRx,y,γRx,y∣x,y∈U×V,where(27)μRx,y=⋁i=1nμRix,y=⋁i=1nμAix∧μBiy,γRx,y=⋀i=1nγRix,y=⋀i=1nγAix∨γBiy.The reasoning output is(28)B∗=A∗∘R,where “∘” is defined as the maximum and minimum operators: “∨” and “∧”:(29)μB∗y=⋁x∈UμA∗x∧μRx,y,γB∗y=⋀x∈UγA∗x∨γRx,y.


#### 3.3.2. Forecast Rules of IFTS Model

Inspired by the intuitionistic fuzzy multiple modus ponens, we exchange the positions of historical data and intuitionistic fuzzy setsAi(i=1,2,…,r) in the IFTS model; that is, let the historical data be intuitionistic fuzzy sets, noted as Fi(j=1,2,…,t), and let Ai be the elements in Fj, and let μAi(x) and γAi(x) be the membership and nonmembership of Ai to Fj. Then Fj is as follows:(30)Fj=∑i=1rμFjxi,γFjxixi,where(31)μFjxi=μAifit,γFjxi=γAifit.Hence, we apply the intuitionistic fuzzy multiple modus ponens toAi and Fj. The reasoning model is as follows:Rules:IFx is F1THENy is F2IFx is F2THENy is F3⋮IFx is FjTHENy is Fj+1⋮IFx is Ft-1THENy is FtInput:IFx is FtOutput:y is Ft+1∗The reasoning output is as follows:(32)Ft+1∗=Ft∘R,where(33)μRx,y=⋁j=1t-1μRjx,y=⋁j=1t-1μFjx∧μFj+1y,γRx,y=⋀j=1t-1γRjx,y=⋀j=1t-1γFjx∨γFj+1y.So, the membership and nonmembership of the output intuitionistic fuzzy set are(34)μFt+1∗y=⋁x∈UμFtx∧μRx,y,γFt+1∗y=⋀x∈UγFtx∨γRx,y.That is to say, the membership and nonmembership of the forecasted result ft+1(t) to every intuitionistic fuzzy set Ai are μFt+1∗(y) and γFt+1∗(y).


### 3.4. Defuzzification Algorithm

The widely used defuzzification algorithms include the maximum truth-value algorithm, gravity algorithm, and weighted average algorithm [27]. In this paper, we utilize the gravity algorithm, which has a more obvious and smoother output than others even when the input has tiny changes [27]. The calculation is as follows:(35)C=∫UcμFc+1/2πFcdc∫UμFc+1/2πFcdc=∫Uc1+μFc+γFcdc∫U1+μFc+γFcdc,where U is the output domain and F is an intuitionistic fuzzy set in U.


## 3.1. Unequal Universe Partition Based on Fuzzy Clustering

First of all, the universe of discourseU=[xmin-ε1,xmax+ε2] should be defined, where xmin and xmax are the minimum and maximum historical data, respectively. ε1 and ε2 are two proper positive numbers. Usually, for simplicity, ε1 and ε2 are chosen to round down xmin and round up xmax to two proper integers.Secondly, partition the universeU into several intervals. References [21–23] proved that unequal intervals do not only have actual meanings for regular understanding but also lead to a better outcome than equal ones. Some researchers [22, 24, 25] have already made achievements in this step by adopting methods such as genetic algorithms, particle swarm optimization, and fuzzy c-means clustering algorithm. But these kinds of methods usually need a huge amount of historical data to get a good performance, which deviates from the small database of historical information of IFTS. What is more, in practice, the IFTS model is generally used for problems which have not too many historical data such as in economic and environmental forecasting. So in this paper, we decide to use a more convenient and real-time method to solve this problem [26].LetX=x1,x2,…,xn be the universe of objects to be classified, where xj=(xj1,xj2,…,xjm) (j=1,2,…,n) has m characteristics. Let R=(rjk)n×n be the similarity matrix of X, where rjk is the similarity between xj and xk (j,k=1,2,…,n). A maximum spanning tree is a tree with all xj being the vertices and rjk being the weights of every edge. Let λ∈[0,1] be the clustering threshold. Cutting down the edges whose weights are smaller than λ, we can get a few subtrees. Hence, the vertices of different subtrees make up different groups. The main steps are as follows.Step 1.
Standardize historical data. Since the elements of fuzzy matrix should be in[0,1], data in different dimensions should be transformed into the interval [0,1] to meet the requirement of similarity matrix R [26]. Generally, two kinds of transformation are required.
(
1
) Standard deviation transformation is as follows:(6)xjh′=xjh-x-hsh,where x-h=(1/n)∑j=1nxjh and sh=(1/n)∑j=1n(xjh-x-h)2, j=1,2,…,n, h=1,2,…,m. With this transformation, the mean of every variable becomes 0, the standard deviation becomes 1, and the dimensional differences are eliminated. But it cannot ensure that xjh′ will locate in [0,1].
(
2
) Range transformation is as follows:(7)xjh′′=xjh′-min1≤j≤n⁡xjh′max1≤j≤n⁡xjh′-min1≤j≤n⁡xjh′,where, obviously, 0≤xjh′′≤1.Step 2.
Establish the fuzzy similarity matrixR.
Letrjk be the similarity between xj′′ and xk′′; then we will have a fuzzy similarity matrix R=(rjk)n×n. There are different ways to get rjk. Since the Euclidean distance is widely used in similarity matrix establishment [26], we also choose it to calculate rjk:(8)rjk=1-dxj′′,xk′′=1-∑h=1mxjh′′-xkh′′2.Step 3.
Build a maximum spanning tree and classify historical data.
In this step, the Kruskal algorithm [26] is used to build the maximum spanning tree. Firstly, draw every vertex xj. Secondly, draw the edges by the value of their weights rjk in descending order, until all of the vertices are connected but with no circles. At last, cut down the edges with smaller weights than the threshold λ. The vertices of each connected branch make up a group.Step 4.
Calculate the bestλ.
The value ofλ varies from 0 to 1, and the best λ leads to the best classification. So how to get the best λ is an important step. In this paper, we also use a widely used F-statistic to find the best λ:(9)F=∑i=1rni∑h=1mx-ih-x-h2/r-1∑i=1r∑j=1ni∑h=1mxjh-x-ih2/n-r~Fr-1,n-r,where r is the number of groups for a given λ,  ni is the number of objects in group i (i=1,2,…,r), x-ih=(1/n)∑j=1nixjh is the average of the hth (h=1,2,…,m) characteristic of the objects in group i, and x-h is the hth characteristic of all objects. In (9), the numerator represents the distances between groups, and the denominator represents the distances within groups. So the bigger F is, the better the classification we get. For a given confidence level α, we can find several values of F which are larger than Fα. The λ which leads to the largest F is the best λ, and the corresponding classification is the best as well.
The best classification can be noted as(10)x11,x12,…,x1n1,…,xi-11,xi-12,…,xi-1ni-1,xi1,xi2,…,xini,xi+11,xi+12,…,xi+1ni+1,…,xr1,xr2,…,xrnr,where i=1,2,…,r, xi-1ni-1≤xi1, xi1≤xi2≤⋯≤xini, and xini≤xi+11.
Let(11)di=xmin-ε1,i=0xini+xi+112,i=1,2,…,r-1xmax+ε2,i=r.Therefore, we partition the universe U into r unequal intervals: u1=[d0,d1],  u2=[d1,d2],…, and  ur=[dr-1,dr].


## 3.2. Construction of Intuitionistic Fuzzy Sets

Corresponding to the abover intervals, we define r intuitionistic fuzzy sets representing r linguistic values:(12)Ai=x,μAix,γAix∣x∈X,i=1,2,…,r.Constructing their membership functions and nonmembership functions is the key point in this section.Since the intuitionistic fuzzy set has a special characteristic, intuitionistic index, the design of membership function and nonmembership function has been quite comprehensive. However, existing methods based on fuzzy statistics, trichotomy, or binary comparison sequencing usually set the intuitionistic index to a fixed value, which does not take full advantage of the intuitionistic fuzzy set [27]. Therefore, according to the characteristics of IFTS intervals, a more objective method is proposed in this section.First of all, two rules based on objective analysis are as follows:(1)
Whenx is located in the middle of an interval, namely, x=(di-1+di)/2, we define that μAi(di-1+di)/2=1 and γAi(di-1+di)/2=0.(2)
Whenx is located on the boundaries of an interval, namely, x=di, we define that intuitionistic index has the maximum value and μAi(di)=γAi(di). Let πAidi=α(0≤α≤1); then we can get μAi(di)=γAi(di)=(1-α)/2.Then, in view of above rules, the membership function is defined as a Gaussian function:(13)μAi=exp⁡-x-cμi22σμi2.The nonmembership function is a transformation of Gaussian function:(14)γAix=1-exp⁡-x-cγi22σγi2.Hence, the intuitionistic index function reads(15)πAix=1-μAix-γAix,where i=1,2,…,r and cμi, σμi, cγi, and σγi are important function parameters. The calculations of them are based on the above two rules:(16)cμi=cγi=di-1+di2,σμi2=-di-1-cμi22ln⁡1-α/2,σγi2=-di-cγi22ln⁡1+1-α/2.Definition 6.
LetA be an intuitionistic fuzzy set in a finite universe X. If(1)
0≤μA(x)≤1, 0≤γA(x)≤1,(2)
0≤πA(x)≤1, 0≤μA(x)+γA(x)≤1,(3)
πA(x)+μA(x)+γA(x)=1,then A is a normal intuitionistic fuzzy set.Therefore, we obtain the following theorem.Theorem 7.
The membership function and nonmembership function ofAi(i=1,2,…,r) are standard; that is, Ai is a normal intuitionistic fuzzy set.Proof.
(
1
)
y
=
e
x
p
-
(
x
-
c
)
2
/
2
σ
2 is a Gaussian function, so we have(17)0≤exp-x-c22σ2≤1,0≤1-exp-x-c22σ2≤1.Therefore,(18)0≤μAix≤1,0≤γAix≤1.
(
2
) Given 0≤α≤1 and cμi=cγi=(di-1+di)/2, we have(19)σμi2≤σγi2.Hence,(20)exp⁡-x-cμi22σμi2≤exp⁡-x-cγi22σγi2,exp⁡-x-cμi22σμi2-exp⁡-x-cγi22σγi2≤0.On the other hand, since 0≤exp-(x-cμi)2/2σμi2≤1 and 0≤exp-(x-cγi)2/2σγi2≤1, we have(21)-1≤exp⁡-x-cμi22σμi2-exp⁡-x-cγi22σγi2≤1.Therefore,(22)-1≤exp⁡-x-cμi22σμi2-exp⁡-x-cγi22σγi2≤0,0≤exp⁡-x-cμi22σμi2+1-exp⁡-x-cγi22σγi2≤1.That is,(23)0≤μAix+γAix≤1,0≤πAix=1-μAix-γAix≤1.
(
3
) According to the calculation of πAi(x), it can be easily found that πAi(x)+μAi(x)+γAi(x)=1. This completes the proof.Theorem7 shows that the calculation of the membership function and nonmembership function of the intuitionistic fuzzy set is correct and appropriate.


## 3.3. Forecast Rules Based on Intuitionistic Fuzzy Reasoning

### 3.3.1. Intuitionistic Fuzzy Multiple Modus Ponens

LetAi(i=1,2,…,n) and A∗ be intuitionistic fuzzy sets in universe U and let Bi(i=1,2,…,n) and B∗ be intuitionistic fuzzy sets in universe V. The generalized multiple modus ponens based on intuitionistic fuzzy relation [27] is that a new proposition that “y is B∗” can be inferred from n+1 propositions: “if x is Ai, then y is Bi” and “x is A∗.” The reasoning model is as follows:Rules:IFx is A1THENy is B1IFx is A2THENy is B2⋮IFx is AnTHENy is BnInput:IFx is A∗Output:y is B∗Every rule has a corresponding input-output relationRi. For Ri, different operators result in different μR and γR, but the reasoning outputs are all the same. Since it has a better performance and is easier to calculate than other operators [27], the Mamdani implication operator Rc is used here:(24)Ri=A×B=x,y,μRix,y,γRix,y∣x,y∈U×V,where(25)μRix,y=μAix∧μBix,γRix,y=γAix∨γBiy.Then, according to the compositional operation of intuitionistic fuzzy rules, we get the total relation:(26)R=⋃i=1nRi=x,y,μRx,y,γRx,y∣x,y∈U×V,where(27)μRx,y=⋁i=1nμRix,y=⋁i=1nμAix∧μBiy,γRx,y=⋀i=1nγRix,y=⋀i=1nγAix∨γBiy.The reasoning output is(28)B∗=A∗∘R,where “∘” is defined as the maximum and minimum operators: “∨” and “∧”:(29)μB∗y=⋁x∈UμA∗x∧μRx,y,γB∗y=⋀x∈UγA∗x∨γRx,y.


### 3.3.2. Forecast Rules of IFTS Model

Inspired by the intuitionistic fuzzy multiple modus ponens, we exchange the positions of historical data and intuitionistic fuzzy setsAi(i=1,2,…,r) in the IFTS model; that is, let the historical data be intuitionistic fuzzy sets, noted as Fi(j=1,2,…,t), and let Ai be the elements in Fj, and let μAi(x) and γAi(x) be the membership and nonmembership of Ai to Fj. Then Fj is as follows:(30)Fj=∑i=1rμFjxi,γFjxixi,where(31)μFjxi=μAifit,γFjxi=γAifit.Hence, we apply the intuitionistic fuzzy multiple modus ponens toAi and Fj. The reasoning model is as follows:Rules:IFx is F1THENy is F2IFx is F2THENy is F3⋮IFx is FjTHENy is Fj+1⋮IFx is Ft-1THENy is FtInput:IFx is FtOutput:y is Ft+1∗The reasoning output is as follows:(32)Ft+1∗=Ft∘R,where(33)μRx,y=⋁j=1t-1μRjx,y=⋁j=1t-1μFjx∧μFj+1y,γRx,y=⋀j=1t-1γRjx,y=⋀j=1t-1γFjx∨γFj+1y.So, the membership and nonmembership of the output intuitionistic fuzzy set are(34)μFt+1∗y=⋁x∈UμFtx∧μRx,y,γFt+1∗y=⋀x∈UγFtx∨γRx,y.That is to say, the membership and nonmembership of the forecasted result ft+1(t) to every intuitionistic fuzzy set Ai are μFt+1∗(y) and γFt+1∗(y).


## 3.3.1. Intuitionistic Fuzzy Multiple Modus Ponens

LetAi(i=1,2,…,n) and A∗ be intuitionistic fuzzy sets in universe U and let Bi(i=1,2,…,n) and B∗ be intuitionistic fuzzy sets in universe V. The generalized multiple modus ponens based on intuitionistic fuzzy relation [27] is that a new proposition that “y is B∗” can be inferred from n+1 propositions: “if x is Ai, then y is Bi” and “x is A∗.” The reasoning model is as follows:Rules:IFx is A1THENy is B1IFx is A2THENy is B2⋮IFx is AnTHENy is BnInput:IFx is A∗Output:y is B∗Every rule has a corresponding input-output relationRi. For Ri, different operators result in different μR and γR, but the reasoning outputs are all the same. Since it has a better performance and is easier to calculate than other operators [27], the Mamdani implication operator Rc is used here:(24)Ri=A×B=x,y,μRix,y,γRix,y∣x,y∈U×V,where(25)μRix,y=μAix∧μBix,γRix,y=γAix∨γBiy.Then, according to the compositional operation of intuitionistic fuzzy rules, we get the total relation:(26)R=⋃i=1nRi=x,y,μRx,y,γRx,y∣x,y∈U×V,where(27)μRx,y=⋁i=1nμRix,y=⋁i=1nμAix∧μBiy,γRx,y=⋀i=1nγRix,y=⋀i=1nγAix∨γBiy.The reasoning output is(28)B∗=A∗∘R,where “∘” is defined as the maximum and minimum operators: “∨” and “∧”:(29)μB∗y=⋁x∈UμA∗x∧μRx,y,γB∗y=⋀x∈UγA∗x∨γRx,y.


## 3.3.2. Forecast Rules of IFTS Model

Inspired by the intuitionistic fuzzy multiple modus ponens, we exchange the positions of historical data and intuitionistic fuzzy setsAi(i=1,2,…,r) in the IFTS model; that is, let the historical data be intuitionistic fuzzy sets, noted as Fi(j=1,2,…,t), and let Ai be the elements in Fj, and let μAi(x) and γAi(x) be the membership and nonmembership of Ai to Fj. Then Fj is as follows:(30)Fj=∑i=1rμFjxi,γFjxixi,where(31)μFjxi=μAifit,γFjxi=γAifit.Hence, we apply the intuitionistic fuzzy multiple modus ponens toAi and Fj. The reasoning model is as follows:Rules:IFx is F1THENy is F2IFx is F2THENy is F3⋮IFx is FjTHENy is Fj+1⋮IFx is Ft-1THENy is FtInput:IFx is FtOutput:y is Ft+1∗The reasoning output is as follows:(32)Ft+1∗=Ft∘R,where(33)μRx,y=⋁j=1t-1μRjx,y=⋁j=1t-1μFjx∧μFj+1y,γRx,y=⋀j=1t-1γRjx,y=⋀j=1t-1γFjx∨γFj+1y.So, the membership and nonmembership of the output intuitionistic fuzzy set are(34)μFt+1∗y=⋁x∈UμFtx∧μRx,y,γFt+1∗y=⋀x∈UγFtx∨γRx,y.That is to say, the membership and nonmembership of the forecasted result ft+1(t) to every intuitionistic fuzzy set Ai are μFt+1∗(y) and γFt+1∗(y).


## 3.4. Defuzzification Algorithm

The widely used defuzzification algorithms include the maximum truth-value algorithm, gravity algorithm, and weighted average algorithm [27]. In this paper, we utilize the gravity algorithm, which has a more obvious and smoother output than others even when the input has tiny changes [27]. The calculation is as follows:(35)C=∫UcμFc+1/2πFcdc∫UμFc+1/2πFcdc=∫Uc1+μFc+γFcdc∫U1+μFc+γFcdc,where U is the output domain and F is an intuitionistic fuzzy set in U.


## 4. Applications

In this section, we focus on two numerical experiments to demonstrate the performance of the proposed IFTS model. In each experiment, several existing FTS and IFTS models are also applied on the same data set to make comparisons. The experimental results and associating analyses are shown, respectively.


### 4.1. Enrollments of the University of Alabama

The enrollment of the University of Alabama has been firstly used in Song’s paper on FTS model [2]. Since then, this data set has been 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.
Define and partition the universe of discourse.
The enrollments from year 1971 to 1991 are chosen as historical data to forecast the enrollment of year 1992. In the historical data,xmin=13055 and xmax=19337, so the universe of discourse is set to U=[13000,20000].
ThenU is partitioned into unequal intervals based on the fuzzy clustering algorithm designed in Section 3.1. The step-by-step details are as follows:(1) Standardize historical data according to (6) and (7).(2) Establish the fuzzy similarity matrixR as shown in Table 1.Table 1
The fuzzy similarity matrixR.
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1971
1.000
0.919
0.871
0.739
0.617
0.641
0.594
0.553
0.403
0.385
0.469
0.622
0.611
0.667
0.664
0.534
0.395
0.189
0.058
0.001
0.000
1972
0.919
1.000
0.952
0.820
0.698
0.722
0.675
0.634
0.484
0.466
0.550
0.702
0.692
0.748
0.745
0.615
0.475
0.270
0.139
0.082
0.081
1973
0.871
0.952
1.000
0.868
0.746
0.770
0.724
0.683
0.532
0.514
0.599
0.751
0.741
0.797
0.794
0.663
0.524
0.318
0.188
0.131
0.129
1974
0.739
0.820
0.868
1.000
0.878
0.902
0.856
0.815
0.664
0.646
0.731
0.883
0.873
0.929
0.926
0.795
0.656
0.450
0.320
0.263
0.261
1975
0.617
0.698
0.746
0.878
1.000
0.976
0.977
0.936
0.786
0.768
0.852
0.996
0.994
0.950
0.953
0.917
0.777
0.572
0.441
0.384
0.383
1976
0.641
0.722
0.770
0.902
0.976
1.000
0.954
0.912
0.762
0.744
0.829
0.981
0.970
0.974
0.976
0.893
0.754
0.548
0.418
0.361
0.359
1977
0.594
0.675
0.724
0.856
0.977
0.953
1.000
0.959
0.808
0.791
0.875
0.973
0.983
0.927
0.930
0.939
0.800
0.595
0.464
0.407
0.406
1978
0.553
0.634
0.683
0.815
0.936
0.912
0.959
1.000
0.849
0.832
0.916
0.932
0.942
0.886
0.889
0.980
0.841
0.636
0.505
0.448
0.447
1979
0.403
0.484
0.532
0.664
0.786
0.762
0.808
0.849
1.000
0.982
0.933
0.781
0.791
0.735
0.738
0.869
0.992
0.786
0.656
0.599
0.597
1980
0.385
0.466
0.514
0.646
0.768
0.744
0.791
0.832
0.982
1.000
0.916
0.763
0.774
0.718
0.721
0.851
0.990
0.804
0.673
0.616
0.615
1981
0.469
0.550
0.599
0.731
0.852
0.829
0.875
0.916
0.933
0.915
1.000
0.848
0.858
0.802
0.805
0.936
0.925
0.720
0.589
0.532
0.531
1982
0.621
0.702
0.751
0.883
0.996
0.981
0.973
0.932
0.781
0.763
0.848
1.000
0.990
0.954
0.957
0.912
0.773
0.568
0.437
0.380
0.379
1983
0.611
0.692
0.741
0.872
0.994
0.970
0.983
0.942
0.791
0.774
0.858
0.990
1.000
0.944
0.947
0.922
0.783
0.578
0.447
0.390
0.389
1984
0.667
0.748
0.797
0.929
0.950
0.974
0.927
0.886
0.735
0.718
0.802
0.954
0.944
1.000
0.997
0.866
0.727
0.522
0.391
0.334
0.333
1985
0.664
0.745
0.794
0.926
0.953
0.976
0.930
0.889
0.738
0.720
0.805
0.957
0.947
0.997
1.000
0.869
0.730
0.525
0.394
0.337
0.336
1986
0.534
0.615
0.663
0.795
0.917
0.893
0.939
0.980
0.869
0.851
0.936
0.912
0.922
0.866
0.869
1.000
0.861
0.655
0.525
0.468
0.466
1987
0.394
0.475
0.524
0.656
0.777
0.754
0.800
0.841
0.992
0.990
0.925
0.773
0.783
0.727
0.730
0.861
1.000
0.795
0.664
0.607
0.606
1988
0.189
0.270
0.318
0.450
0.572
0.548
0.595
0.636
0.786
0.804
0.720
0.567
0.578
0.522
0.525
0.655
0.794
1.000
0.869
0.812
0.811
1989
0.058
0.139
0.188
0.320
0.441
0.418
0.464
0.505
0.656
0.674
0.589
0.437
0.447
0.391
0.394
0.525
0.664
0.869
1.000
0.943
0.942
1990
0.001
0.082
0.131
0.263
0.384
0.361
0.407
0.448
0.599
0.617
0.532
0.380
0.390
0.334
0.337
0.468
0.607
0.812
0.943
1.000
0.999
1991
0.000
0.081
0.129
0.261
0.383
0.359
0.406
0.447
0.597
0.615
0.531
0.379
0.389
0.333
0.336
0.466
0.606
0.811
0.942
0.999
1.000(3) Use the Kruskal algorithm mentioned in Section3.1 to build a maximum spanning tree based on matrix R. The tree is shown in Figure 1.Figure 1
The maximum spanning tree of historical data.Letλ be 0.93, 0.94, 0.95, and 0.96, respectively. We can get different classifications of historical data as shown in Table 2.Table 2
Classifications of differentλ.
λ
Number of classifications
Classification
F
0.05
F
0.935
7
1971, 1972,1973, 1974, 1979,1980,1987, 1988, 1989,1990,1991, 1975,1976,1977,1978,1981,1982,1983,1984,1985,1986
2.85
92.85
0.94
8
1971, 1972,1973, 1974, 1981, 1979,1980,1987, 1988, 1989,1990,1991, 1975,1976,1977,1978,1982,1983,1984,1985,1986
2.83
129.17
0.95
9
1971, 1972,1973, 1974, 1981, 1979,1980,1987, 1988, 1989, 1990,1991, 1975,1976,1977,1978,1982,1983,1984,1985,1986
2.85
142.13
0.955
10
1971, 1972, 1973, 1974, 1981, 1979,1980,1987, 1988, 1989, 1990,1991, 1975,1976,1977,1978,1982,1983,1984,1985,1986
2.9
112.67(4) For different classifications, calculate the values ofF according to (9). The results are also shown in Table 2. From Table 2, we can see that when λ=0.95, its corresponding F is maximum and bigger than Fα(α=0.05) at the same time. So this classification is the best.Therefore, the universe of discourseU is partitioned into 9 unequal intervals according to the above classification. The boundaries of each interval are calculated according to (11). The intervals are(36)u1=13000,13309,u2=13309,14282.u3=14282,14921,u4=14921,16186,u5=16186,16598,u6=16598,17535,u7=17535,18560,u8=18560,19149,u9=19149,20000.Step 2.
Construct intuitionistic fuzzy sets and intuitionistically fuzzify the historical data.
Corresponding to the 9 intervals, there should be 9 intuitionistic fuzzy setsA1,A2,…,A9, and their realistic significance is as follows: “very very very few”, “very very few”, “very few”, “few”, “normal”, “many”, “very many”, “very very many”, “very very very many”. Then calculate the parameters of the membership and nonmembership functions based on Section 3.2. For α=0.4, the parameters are shown in Table 3.
The membership function, nonmembership function, and intuitionistic index function of every intuitionistic fuzzy set are shown in Figures2, 3, and 4, respectively.
Then we can calculate the membership, nonmembership, and intuitionistic index of every historical value to every intuitionistic fuzzy set.Table 3
Function parameters ofAi.
Intuitionistic fuzzy set
c
μ
i
σ
μ
i
c
γ
i
σ
γ
i
A
1
13154.5
99.5
13154.5
182.9
A
2
13795.5
313.5
13795.5
576
A
3
14601.5
205.9
14601.5
378.3
A
4
15553.5
407.6
15553.5
748.9
A
5
16392
132.8
16392
243.9
A
6
17066.5
301.9
17066.5
554.7
A
7
18047.5
330.3
18047.5
606.8
A
8
18854.5
189.8
18854.5
348.7
A
9
19574.5
274.2
19574.5
503.8Figure 2
Membership functions.Figure 3
Nonmembership functions.Figure 4
Intuitionistic index functions.Step 3.
Establish forecast rules and forecast the enrollments.
The enrollments of year 1971 to 1991 can be denoted asF1,F2,…,F21, and the reasoning model based on Section 3.3.2 is as follows:Rules:IFx is F1THENy is F2IFx is F2THENy is F3⋮IFx is F20THENy is F21Input:IFx is F21Output:y is F22∗
Then we getF22∗:(37)F22∗=0,1A1+0,1A2+0,1A3+0,1A4+0,1A5+0,1A6+0,0.895A7+0.687,0.207A8+0.039,0.616A9,where the membership of F22∗ to A8 is the biggest and the nonmembership is the smallest, so the intuitionistic forecasted result is A8.Step 4.
Defuzzify and output the forecast result.
The defuzzification result based on Section3.4 is(38)C=∫Uc2+exp⁡-c-18854.52/2×189.82-exp⁡-c-18854.52/2×347.82dc∫U2+exp⁡-c-18854.52/2×189.82-exp⁡-c-18854.52/2×347.82dc=18854.5≈18855.That is to say, the enrollment of year 1992 is 18855.
To test the performance of our model, we use the models of inference [2], [12], and [17] as well as ours to forecast every year’s enrollment, respectively. The results are shown in Table 4. The models of inference [2, 12] are FTS models, and the model of inference [17] is an IFTS model. In inference [12], there are three kinds of universe partition: 7, 17, and 22 intervals. Since there are only 22 historical data, the 17-interval partition and 22-interval partition are not applicable, so we choose the 7-interval partition.
The root mean square error (RMSE) and average forecast error (AFE) are exploited to evaluate the performance of every model:(39)RMSE=1n∑i=1nactualvaluei-forecastvaluei2,AFE=1n∑i=1nactualvaluei-forecastvalueiactualvaluei×100%.
The results are shown in Table5.
The results in Tables4 and 5 indicate that our model can not only reach the forecast goal but also achieve a better result than the other tested models. That is to say, the proposed model is feasible and efficient.Table 4
Forecast results of the enrollments.
Year
Actual enrollment
Forecasted enrollment
Model [2]
Model [12]
Model [17]
Our model
1972
13563
14000
14279
14250
13500
1973
13867
14000
14279
14246
14155
1974
14696
14000
14279
14246
14155
1975
15460
15500
15392
15491
15539
1976
15311
16000
15392
15491
15539
1977
15603
16000
15392
15491
15502
1978
15861
16000
16467
16345
15502
1979
16807
16000
16467
16345
16667
1980
16919
16813
17161
15850
16667
1981
16388
16813
17161
15850
15669
1982
15433
16789
14916
15850
15564
1983
15497
16000
15392
15450
15564
1984
15145
16000
15392
15450
15564
1985
15163
16000
15392
15491
15523
1986
15984
16000
15470
15491
15523
1987
16859
16000
16467
16345
16799
1988
18150
16813
17161
17950
18268
1989
18970
19000
19257
18961
18268
1990
19328
19000
19257
18961
18780
1991
19337
19000
19257
18961
19575
1992
18876
19000
19257
18961
18855Table 5
Forecast performance of enrollments.
Criterion
Model [2]
Model [12]
Model [17]
Our model
RMSE
677.1
445.5
418.9
350.9
AFE
3.35%
2.3%
2.07%
1.72%


### 4.2. Experiments on TAIEX

The Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) is a typical economic data set widely used in fuzzy time series forecasting [13, 20, 23, 25, 28, 29]. In this experiment, TAIEX values from 11/1/2004 to 12/31/2004 are used as historical data, which are shown in Table 6. The intuitionistic fuzzified value of historical data when forecasted by our model are also shown in Table 6.Table 6
Historical data of TAIEX.
Date
TAIEX
Intuitionistic fuzzified value
11/1/2004
5656.17
A
1
11/2/2004
5759.61
A
2
11/3/2004
5862.85
A
5
11/4/2004
5860.73
A
5
11/5/2004
5931.31
A
7
11/8/2004
5937.46
A
7
11/9/2004
5945.2
A
8
11/10/2004
5948.49
A
8
11/11/2004
5874.52
A
5
11/12/2004
5917.16
A
7
11/15/2004
5906.69
A
7
11/16/2004
5910.85
A
7
11/17/2004
6028.68
A
11
11/18/2004
6049.49
A
12
11/19/2004
6026.55
A
11
11/22/2004
5838.42
A
5
11/23/2004
5851.1
A
5
11/24/2004
5911.31
A
7
11/25/2004
5855.24
A
5
11/26/2004
5778.26
A
3
11/29/2004
5785.26
A
3
11/30/2004
5844.76
A
5
12/1/2004
5798.62
A
4
12/2/2004
5867.95
A
5
12/3/2004
5893.27
A
6
12/6/2004
5919.17
A
7
12/7/2004
5925.28
A
7
12/8/2004
5892.51
A
6
12/9/2004
5913.97
A
7
12/10/2004
5911.63
A
7
12/13/2004
5878.89
A
5
12/14/2004
5909.65
A
7
12/15/2004
6002.58
A
10
12/16/2004
6019.23
A
11
12/17/2004
6009.32
A
10
12/20/2004
5985.94
A
9
12/21/2004
5987.85
A
9
12/22/2004
6001.52
A
10
12/23/2004
5997.67
A
10
12/24/2004
6019.42
A
11
12/27/2004
5985.94
A
9
12/28/2004
6000.57
A
10
12/29/2004
6088.49
A
13
12/30/2004
6100.86
A
14
12/31/2004
6139.69
A
15For comparison, we also applied the models of reference [2, 13, 17, 20] to forecast TAIEX at the same time. The forecast results of every model are shown in Table 7 and Figure 5.Table 7
Forecast results of TAIEX.
Actual TAIEX
Forecasted TAIEX
Model [2]
Model [20]
Model [13]
Model [17]
Our model
5656.17
—
—
—
—
—
5759.61
5675
5658.97
—
5680.5
5678.95
5862.85
5825
5754.24
5756.8
5836.8
5738.41
5860.73
5875
5863.09
5865.05
5866.5
5852.11
5931.31
5875
5868.04
5868.44
5890.23
5852.11
5937.46
5900
5927.67
5935.51
5911.52
5886.39
5945.2
5900
5942.9
5932.96
5968
5937.47
5948.49
5925
5942.85
5940.7
5969.6
5954.27
5874.52
5925
5952.69
5945.72
5969.6
5852.11
5917.16
5900
5882.56
5873.1
5923
5909.01
5906.69
5925
5931.63
5919.36
5887
5920.66
5910.85
5925
5908.03
5905.27
5887
5920.66
6028.68
5925
5912.13
5906.35
5995
5954.27
6049.49
5975
6037.08
6030.88
5942
6026.68
6026.55
5975
6042.75
6051.69
5933
6040.36
5838.42
6025
6010.72
6025.63
5877
5909.01
5851.1
5925
5862.23
5837
5896
5852.11
5911.31
5908.33
5855.49
5853.3
5911.52
5886.39
5855.24
5900
5911.03
5913.51
5933
5852.11
5778.26
5900
5858.01
5862.16
5833
5886.39
5785.26
5858.33
5773.75
5785.57
5790
5816.28
5844.76
5775
5783.93
5780.76
5782
5852.11
5798.62
5850
5844.68
5841.95
5790
5780.44
5867.95
5825
5817.26
5797.2
5778
5828.67
5893.27
5850
5861.34
5865.14
5757
5828.67
5919.17
5825
5897.53
5895.47
5824
5888.54
5925.28
5900
5916.83
5916.36
5911
5920.66
5892.51
5925
5910.24
5920.78
5942
5920.66
5913.97
5900
5895.93
5891.09
5900
5920.66
5911.63
5925
5918.24
5946.17
5900
5906.75
5878.89
5925
5915.62
5910.71
5852
5886.39
5909.65
5900
5882.3
5877.47
5852
5886.39
6002.58
5900
5911.86
5911.86
5990
5920.66
6019.23
5941.67
6002.5
6004.78
6000
6003.52
6009.32
6025
6027.15
6026.94
6025
6015.1
5985.94
6025
6012.86
6004.82
6009
6003.39
5987.85
6000
5991.1
5984.52
5946
5991.75
6001.52
5975
6005.92
5995.56
5965
5979.99
5997.67
6000
6005.86
6009.23
6009
5991.75
6019.42
6000
5997.68
6005.38
6030
6026.68
5985.94
6025
6026.35
6016.65
6030
6026.68
6000.57
6025
5987.87
5985.02
6015
6003.39
6088.49
6025
6018.68
6008.28
6033
6015.1
6100.86
6025
6086.83
6085.68
6098
6081.83
6139.69
6050
6099.45
6098.05
6125
6094.66Figure 5
Forecast results of TAIEX.The performance of all models is compared in Table8.Table 8
Forecast performance of TAIEX.
Criterion
Model [2]
Model [20]
Model [13]
Model [17]
Our model
RMSE
61.17
52.63
53.63
50.27
43.23
AFE
0.83%
0.51%
0.65%
0.65%
0.51%Table8 indicates that the RMSE and MSE of proposed model are both smaller than the other models. Therefore, our two experiments both indicate that the IFTS model proposed in this paper could effectively increase forecast accuracy.


## 4.1. Enrollments of the University of Alabama

The enrollment of the University of Alabama has been firstly used in Song’s paper on FTS model [2]. Since then, this data set has been 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.
Define and partition the universe of discourse.
The enrollments from year 1971 to 1991 are chosen as historical data to forecast the enrollment of year 1992. In the historical data,xmin=13055 and xmax=19337, so the universe of discourse is set to U=[13000,20000].
ThenU is partitioned into unequal intervals based on the fuzzy clustering algorithm designed in Section 3.1. The step-by-step details are as follows:(1) Standardize historical data according to (6) and (7).(2) Establish the fuzzy similarity matrixR as shown in Table 1.Table 1
The fuzzy similarity matrixR.
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1971
1.000
0.919
0.871
0.739
0.617
0.641
0.594
0.553
0.403
0.385
0.469
0.622
0.611
0.667
0.664
0.534
0.395
0.189
0.058
0.001
0.000
1972
0.919
1.000
0.952
0.820
0.698
0.722
0.675
0.634
0.484
0.466
0.550
0.702
0.692
0.748
0.745
0.615
0.475
0.270
0.139
0.082
0.081
1973
0.871
0.952
1.000
0.868
0.746
0.770
0.724
0.683
0.532
0.514
0.599
0.751
0.741
0.797
0.794
0.663
0.524
0.318
0.188
0.131
0.129
1974
0.739
0.820
0.868
1.000
0.878
0.902
0.856
0.815
0.664
0.646
0.731
0.883
0.873
0.929
0.926
0.795
0.656
0.450
0.320
0.263
0.261
1975
0.617
0.698
0.746
0.878
1.000
0.976
0.977
0.936
0.786
0.768
0.852
0.996
0.994
0.950
0.953
0.917
0.777
0.572
0.441
0.384
0.383
1976
0.641
0.722
0.770
0.902
0.976
1.000
0.954
0.912
0.762
0.744
0.829
0.981
0.970
0.974
0.976
0.893
0.754
0.548
0.418
0.361
0.359
1977
0.594
0.675
0.724
0.856
0.977
0.953
1.000
0.959
0.808
0.791
0.875
0.973
0.983
0.927
0.930
0.939
0.800
0.595
0.464
0.407
0.406
1978
0.553
0.634
0.683
0.815
0.936
0.912
0.959
1.000
0.849
0.832
0.916
0.932
0.942
0.886
0.889
0.980
0.841
0.636
0.505
0.448
0.447
1979
0.403
0.484
0.532
0.664
0.786
0.762
0.808
0.849
1.000
0.982
0.933
0.781
0.791
0.735
0.738
0.869
0.992
0.786
0.656
0.599
0.597
1980
0.385
0.466
0.514
0.646
0.768
0.744
0.791
0.832
0.982
1.000
0.916
0.763
0.774
0.718
0.721
0.851
0.990
0.804
0.673
0.616
0.615
1981
0.469
0.550
0.599
0.731
0.852
0.829
0.875
0.916
0.933
0.915
1.000
0.848
0.858
0.802
0.805
0.936
0.925
0.720
0.589
0.532
0.531
1982
0.621
0.702
0.751
0.883
0.996
0.981
0.973
0.932
0.781
0.763
0.848
1.000
0.990
0.954
0.957
0.912
0.773
0.568
0.437
0.380
0.379
1983
0.611
0.692
0.741
0.872
0.994
0.970
0.983
0.942
0.791
0.774
0.858
0.990
1.000
0.944
0.947
0.922
0.783
0.578
0.447
0.390
0.389
1984
0.667
0.748
0.797
0.929
0.950
0.974
0.927
0.886
0.735
0.718
0.802
0.954
0.944
1.000
0.997
0.866
0.727
0.522
0.391
0.334
0.333
1985
0.664
0.745
0.794
0.926
0.953
0.976
0.930
0.889
0.738
0.720
0.805
0.957
0.947
0.997
1.000
0.869
0.730
0.525
0.394
0.337
0.336
1986
0.534
0.615
0.663
0.795
0.917
0.893
0.939
0.980
0.869
0.851
0.936
0.912
0.922
0.866
0.869
1.000
0.861
0.655
0.525
0.468
0.466
1987
0.394
0.475
0.524
0.656
0.777
0.754
0.800
0.841
0.992
0.990
0.925
0.773
0.783
0.727
0.730
0.861
1.000
0.795
0.664
0.607
0.606
1988
0.189
0.270
0.318
0.450
0.572
0.548
0.595
0.636
0.786
0.804
0.720
0.567
0.578
0.522
0.525
0.655
0.794
1.000
0.869
0.812
0.811
1989
0.058
0.139
0.188
0.320
0.441
0.418
0.464
0.505
0.656
0.674
0.589
0.437
0.447
0.391
0.394
0.525
0.664
0.869
1.000
0.943
0.942
1990
0.001
0.082
0.131
0.263
0.384
0.361
0.407
0.448
0.599
0.617
0.532
0.380
0.390
0.334
0.337
0.468
0.607
0.812
0.943
1.000
0.999
1991
0.000
0.081
0.129
0.261
0.383
0.359
0.406
0.447
0.597
0.615
0.531
0.379
0.389
0.333
0.336
0.466
0.606
0.811
0.942
0.999
1.000(3) Use the Kruskal algorithm mentioned in Section3.1 to build a maximum spanning tree based on matrix R. The tree is shown in Figure 1.Figure 1
The maximum spanning tree of historical data.Letλ be 0.93, 0.94, 0.95, and 0.96, respectively. We can get different classifications of historical data as shown in Table 2.Table 2
Classifications of differentλ.
λ
Number of classifications
Classification
F
0.05
F
0.935
7
1971, 1972,1973, 1974, 1979,1980,1987, 1988, 1989,1990,1991, 1975,1976,1977,1978,1981,1982,1983,1984,1985,1986
2.85
92.85
0.94
8
1971, 1972,1973, 1974, 1981, 1979,1980,1987, 1988, 1989,1990,1991, 1975,1976,1977,1978,1982,1983,1984,1985,1986
2.83
129.17
0.95
9
1971, 1972,1973, 1974, 1981, 1979,1980,1987, 1988, 1989, 1990,1991, 1975,1976,1977,1978,1982,1983,1984,1985,1986
2.85
142.13
0.955
10
1971, 1972, 1973, 1974, 1981, 1979,1980,1987, 1988, 1989, 1990,1991, 1975,1976,1977,1978,1982,1983,1984,1985,1986
2.9
112.67(4) For different classifications, calculate the values ofF according to (9). The results are also shown in Table 2. From Table 2, we can see that when λ=0.95, its corresponding F is maximum and bigger than Fα(α=0.05) at the same time. So this classification is the best.Therefore, the universe of discourseU is partitioned into 9 unequal intervals according to the above classification. The boundaries of each interval are calculated according to (11). The intervals are(36)u1=13000,13309,u2=13309,14282.u3=14282,14921,u4=14921,16186,u5=16186,16598,u6=16598,17535,u7=17535,18560,u8=18560,19149,u9=19149,20000.Step 2.
Construct intuitionistic fuzzy sets and intuitionistically fuzzify the historical data.
Corresponding to the 9 intervals, there should be 9 intuitionistic fuzzy setsA1,A2,…,A9, and their realistic significance is as follows: “very very very few”, “very very few”, “very few”, “few”, “normal”, “many”, “very many”, “very very many”, “very very very many”. Then calculate the parameters of the membership and nonmembership functions based on Section 3.2. For α=0.4, the parameters are shown in Table 3.
The membership function, nonmembership function, and intuitionistic index function of every intuitionistic fuzzy set are shown in Figures2, 3, and 4, respectively.
Then we can calculate the membership, nonmembership, and intuitionistic index of every historical value to every intuitionistic fuzzy set.Table 3
Function parameters ofAi.
Intuitionistic fuzzy set
c
μ
i
σ
μ
i
c
γ
i
σ
γ
i
A
1
13154.5
99.5
13154.5
182.9
A
2
13795.5
313.5
13795.5
576
A
3
14601.5
205.9
14601.5
378.3
A
4
15553.5
407.6
15553.5
748.9
A
5
16392
132.8
16392
243.9
A
6
17066.5
301.9
17066.5
554.7
A
7
18047.5
330.3
18047.5
606.8
A
8
18854.5
189.8
18854.5
348.7
A
9
19574.5
274.2
19574.5
503.8Figure 2
Membership functions.Figure 3
Nonmembership functions.Figure 4
Intuitionistic index functions.Step 3.
Establish forecast rules and forecast the enrollments.
The enrollments of year 1971 to 1991 can be denoted asF1,F2,…,F21, and the reasoning model based on Section 3.3.2 is as follows:Rules:IFx is F1THENy is F2IFx is F2THENy is F3⋮IFx is F20THENy is F21Input:IFx is F21Output:y is F22∗
Then we getF22∗:(37)F22∗=0,1A1+0,1A2+0,1A3+0,1A4+0,1A5+0,1A6+0,0.895A7+0.687,0.207A8+0.039,0.616A9,where the membership of F22∗ to A8 is the biggest and the nonmembership is the smallest, so the intuitionistic forecasted result is A8.Step 4.
Defuzzify and output the forecast result.
The defuzzification result based on Section3.4 is(38)C=∫Uc2+exp⁡-c-18854.52/2×189.82-exp⁡-c-18854.52/2×347.82dc∫U2+exp⁡-c-18854.52/2×189.82-exp⁡-c-18854.52/2×347.82dc=18854.5≈18855.That is to say, the enrollment of year 1992 is 18855.
To test the performance of our model, we use the models of inference [2], [12], and [17] as well as ours to forecast every year’s enrollment, respectively. The results are shown in Table 4. The models of inference [2, 12] are FTS models, and the model of inference [17] is an IFTS model. In inference [12], there are three kinds of universe partition: 7, 17, and 22 intervals. Since there are only 22 historical data, the 17-interval partition and 22-interval partition are not applicable, so we choose the 7-interval partition.
The root mean square error (RMSE) and average forecast error (AFE) are exploited to evaluate the performance of every model:(39)RMSE=1n∑i=1nactualvaluei-forecastvaluei2,AFE=1n∑i=1nactualvaluei-forecastvalueiactualvaluei×100%.
The results are shown in Table5.
The results in Tables4 and 5 indicate that our model can not only reach the forecast goal but also achieve a better result than the other tested models. That is to say, the proposed model is feasible and efficient.Table 4
Forecast results of the enrollments.
Year
Actual enrollment
Forecasted enrollment
Model [2]
Model [12]
Model [17]
Our model
1972
13563
14000
14279
14250
13500
1973
13867
14000
14279
14246
14155
1974
14696
14000
14279
14246
14155
1975
15460
15500
15392
15491
15539
1976
15311
16000
15392
15491
15539
1977
15603
16000
15392
15491
15502
1978
15861
16000
16467
16345
15502
1979
16807
16000
16467
16345
16667
1980
16919
16813
17161
15850
16667
1981
16388
16813
17161
15850
15669
1982
15433
16789
14916
15850
15564
1983
15497
16000
15392
15450
15564
1984
15145
16000
15392
15450
15564
1985
15163
16000
15392
15491
15523
1986
15984
16000
15470
15491
15523
1987
16859
16000
16467
16345
16799
1988
18150
16813
17161
17950
18268
1989
18970
19000
19257
18961
18268
1990
19328
19000
19257
18961
18780
1991
19337
19000
19257
18961
19575
1992
18876
19000
19257
18961
18855Table 5
Forecast performance of enrollments.
Criterion
Model [2]
Model [12]
Model [17]
Our model
RMSE
677.1
445.5
418.9
350.9
AFE
3.35%
2.3%
2.07%
1.72%


## 4.2. Experiments on TAIEX

The Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) is a typical economic data set widely used in fuzzy time series forecasting [13, 20, 23, 25, 28, 29]. In this experiment, TAIEX values from 11/1/2004 to 12/31/2004 are used as historical data, which are shown in Table 6. The intuitionistic fuzzified value of historical data when forecasted by our model are also shown in Table 6.Table 6
Historical data of TAIEX.
Date
TAIEX
Intuitionistic fuzzified value
11/1/2004
5656.17
A
1
11/2/2004
5759.61
A
2
11/3/2004
5862.85
A
5
11/4/2004
5860.73
A
5
11/5/2004
5931.31
A
7
11/8/2004
5937.46
A
7
11/9/2004
5945.2
A
8
11/10/2004
5948.49
A
8
11/11/2004
5874.52
A
5
11/12/2004
5917.16
A
7
11/15/2004
5906.69
A
7
11/16/2004
5910.85
A
7
11/17/2004
6028.68
A
11
11/18/2004
6049.49
A
12
11/19/2004
6026.55
A
11
11/22/2004
5838.42
A
5
11/23/2004
5851.1
A
5
11/24/2004
5911.31
A
7
11/25/2004
5855.24
A
5
11/26/2004
5778.26
A
3
11/29/2004
5785.26
A
3
11/30/2004
5844.76
A
5
12/1/2004
5798.62
A
4
12/2/2004
5867.95
A
5
12/3/2004
5893.27
A
6
12/6/2004
5919.17
A
7
12/7/2004
5925.28
A
7
12/8/2004
5892.51
A
6
12/9/2004
5913.97
A
7
12/10/2004
5911.63
A
7
12/13/2004
5878.89
A
5
12/14/2004
5909.65
A
7
12/15/2004
6002.58
A
10
12/16/2004
6019.23
A
11
12/17/2004
6009.32
A
10
12/20/2004
5985.94
A
9
12/21/2004
5987.85
A
9
12/22/2004
6001.52
A
10
12/23/2004
5997.67
A
10
12/24/2004
6019.42
A
11
12/27/2004
5985.94
A
9
12/28/2004
6000.57
A
10
12/29/2004
6088.49
A
13
12/30/2004
6100.86
A
14
12/31/2004
6139.69
A
15For comparison, we also applied the models of reference [2, 13, 17, 20] to forecast TAIEX at the same time. The forecast results of every model are shown in Table 7 and Figure 5.Table 7
Forecast results of TAIEX.
Actual TAIEX
Forecasted TAIEX
Model [2]
Model [20]
Model [13]
Model [17]
Our model
5656.17
—
—
—
—
—
5759.61
5675
5658.97
—
5680.5
5678.95
5862.85
5825
5754.24
5756.8
5836.8
5738.41
5860.73
5875
5863.09
5865.05
5866.5
5852.11
5931.31
5875
5868.04
5868.44
5890.23
5852.11
5937.46
5900
5927.67
5935.51
5911.52
5886.39
5945.2
5900
5942.9
5932.96
5968
5937.47
5948.49
5925
5942.85
5940.7
5969.6
5954.27
5874.52
5925
5952.69
5945.72
5969.6
5852.11
5917.16
5900
5882.56
5873.1
5923
5909.01
5906.69
5925
5931.63
5919.36
5887
5920.66
5910.85
5925
5908.03
5905.27
5887
5920.66
6028.68
5925
5912.13
5906.35
5995
5954.27
6049.49
5975
6037.08
6030.88
5942
6026.68
6026.55
5975
6042.75
6051.69
5933
6040.36
5838.42
6025
6010.72
6025.63
5877
5909.01
5851.1
5925
5862.23
5837
5896
5852.11
5911.31
5908.33
5855.49
5853.3
5911.52
5886.39
5855.24
5900
5911.03
5913.51
5933
5852.11
5778.26
5900
5858.01
5862.16
5833
5886.39
5785.26
5858.33
5773.75
5785.57
5790
5816.28
5844.76
5775
5783.93
5780.76
5782
5852.11
5798.62
5850
5844.68
5841.95
5790
5780.44
5867.95
5825
5817.26
5797.2
5778
5828.67
5893.27
5850
5861.34
5865.14
5757
5828.67
5919.17
5825
5897.53
5895.47
5824
5888.54
5925.28
5900
5916.83
5916.36
5911
5920.66
5892.51
5925
5910.24
5920.78
5942
5920.66
5913.97
5900
5895.93
5891.09
5900
5920.66
5911.63
5925
5918.24
5946.17
5900
5906.75
5878.89
5925
5915.62
5910.71
5852
5886.39
5909.65
5900
5882.3
5877.47
5852
5886.39
6002.58
5900
5911.86
5911.86
5990
5920.66
6019.23
5941.67
6002.5
6004.78
6000
6003.52
6009.32
6025
6027.15
6026.94
6025
6015.1
5985.94
6025
6012.86
6004.82
6009
6003.39
5987.85
6000
5991.1
5984.52
5946
5991.75
6001.52
5975
6005.92
5995.56
5965
5979.99
5997.67
6000
6005.86
6009.23
6009
5991.75
6019.42
6000
5997.68
6005.38
6030
6026.68
5985.94
6025
6026.35
6016.65
6030
6026.68
6000.57
6025
5987.87
5985.02
6015
6003.39
6088.49
6025
6018.68
6008.28
6033
6015.1
6100.86
6025
6086.83
6085.68
6098
6081.83
6139.69
6050
6099.45
6098.05
6125
6094.66Figure 5
Forecast results of TAIEX.The performance of all models is compared in Table8.Table 8
Forecast performance of TAIEX.
Criterion
Model [2]
Model [20]
Model [13]
Model [17]
Our model
RMSE
61.17
52.63
53.63
50.27
43.23
AFE
0.83%
0.51%
0.65%
0.65%
0.51%Table8 indicates that the RMSE and MSE of proposed model are both smaller than the other models. Therefore, our two experiments both indicate that the IFTS model proposed in this paper could effectively increase forecast accuracy.


## 5. Conclusions

In this paper, a novel IFTS model is proposed for improving the performance of FTS model. In order to be succinct, we use the maximum spanning tree based fuzzy clustering algorithm to partition the universe of discourse into unequal intervals. According to the characteristics of partitioned data, a more objective method is proposed to ascertain membership function and nonmembership function of the intuitionistic fuzzy set. Besides, intuitionistic fuzzy reasoning is utilized to establish forecast rules, which make the model more sensitive to the fuzzy variation of uncertain data. Finally, based on experiments with two data sets, the feasibility and advantage of the new model are verified.



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*Source: 5035160-2016-01-13.xml*