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

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.  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.  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.  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  developed the model of reference  by using the importance of fuzzy relations as their weights. According to this change, they also proposed a new defuzzification method. Huarng  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  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  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  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.  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  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: Section 2 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. Let X be a finite universal set. An intuitionistic fuzzy set A in X is an object having the form

A = [(x, [[mu].sub.A](x), [[gamma].sub.A](x)> | x [member of] X}, (1)

where the function [[mu].sub.A](x) : X [right arrow] [0,1] defines the degree of membership and the function [[gamma].sub.A](x) : X [right arrow] [0,1] defines the degree of nonmembership of the element x [member of] X to set A. For every x [member of] X, 0 [less than or equal to] [[mu].sub.A](x) + [[gamma].sub.A](x)[less than or equal to] 1. [[pi].sub.A](x) = 1 - [[mu].sub.A](x) - [[gamma].sub.A](x) is called the intuitionistic index of x in A. It is the hesitancy of x to A.

When X = {[x.sub.1],[x.sub.2], ..., [x.sub.n]} is discrete, the intuitionistic fuzzy set A can be noted as

A = [n.summation over (i=1)]<[[mu].sub.A]([x.sub.i]), [[gamma].sub.A]([x.sub.i])>/[x.sub.i], [x.sub.i][member of] X. (2)

Definition 2. Let X 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 x Y:

R = {<((x, y), [[mu].sub.R](x,y), [[gamma].sub.R](x, y)> | (x, y) [member of] X x Y}, (3)

where 0 [less than or equal to] [[mu].sup.A](x, y) + [[gamma].sup.A](x, y) [less than or equal to] 1, [for all](x, y) [member of] X x Y for [[mu].sub.R](x, y) : X x Y [right arrow] [0,1], and [[gamma].sub.R](x, y) : X x Y [right arrow] [0,1].

Definition 3. Let X(t) (t = 1,2, ...), a subset of R, be the universe of discourse on which intuitionistic fuzzy sets [f.sub.i](t) = <[[mu].sub.i](X(t)), [[gamma].sub.i](X(t))> (i = 1,2, ...) are defined. F(t) = {[f.sub.1](t), [f.sub.2](t), ...} is a collection of [f.sub.i](t) and defines an intuitionistic fuzzy time series on X(t).

Definition 4. Let R(t, 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

F(t) = F(t - 1) x R(t, t - 1), (4)

where "x" 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. If R(t, t - 1) is independent of time t,

R(t, t - 1) = R(t - 1, t - 2) [for all]t, (5)

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 discourse U = [[x.sub.min] - [[epsilon].sub.1], [x.sub.max] + [[epsilon].sub.2]] should be defined, where [x.sub.min] and [x.sub.max] are the minimum and maximum historical data, respectively. [[epsilon].sub.1] and [[epsilon].sub.2] are two proper positive numbers. Usually, for simplicity, [[epsilon].sub.1] and [[epsilon].sub.2] are chosen to round down [x.sub.min] and round up [x.sub.max] to two proper integers.

Secondly, partition the universe U 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 .

Let X = {[x.sub.1],[x.sub.2], ..., [x.sub.n]} be the universe of objects to be classified, where [x.sub.j] = ([x.sub.j1],[x.sub.j2], ..., [x.sub.jm]) (j = 1,2, ..., n) has m characteristics. Let R = [([r.sub.jk]).sub.nxn] be the similarity matrix of X, where [r.sub.jk] is the similarity between [x.sub.j] and [x.sub.k] (j, k = 1,2, ..., n). A maximum spanning tree is a tree with all [x.sub.j] being the vertices and [r.sub.jk] being the weights of every edge. Let [lambda][member of] [0,1] be the clustering threshold. Cutting down the edges whose weights are smaller than [lambda], 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 . Generally, two kinds of transformation are required.

(1) Standard deviation transformation is as follows:

[x'.sub.jh] = [x.sub.jh] - [[bar.x].sub.h]/[s.sub.h], (6)

where [[bar.x].sub.h] = (1/n) [[summation].sup.n.sub.j=1] [x.sub.jh] and [s.sub.h] = [square root of ((1/n)[[summation].sup.n.sub.j=1][([x.sub.jh] - [[bar.x].sub.h]).sup.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 [x'.sub.jh] will locate in [0,1].

(2) Range transformation is as follows:

[mathematical expression not reproducible], (7)

where, obviously, 0 [less than or equal to] [x".sub.jh] [less than or equal to] 1.

Step 2. Establish the fuzzy similarity matrix R.

Let [r.sub.jk] be the similarity between [x".sub.j] and [x".sub.k]; then we will have a fuzzy similarity matrix R = [([r.sub.jk]).sub.nxn]. There are different ways to get [r.sub.jk]. Since the Euclidean distance is widely used in similarity matrix establishment , we also choose it to calculate [r.sub.jk]:

[mathematical expression not reproducible]. (8)

Step 3. Build a maximum spanning tree and classify historical data.

In this step, the Kruskal algorithm  is used to build the maximum spanning tree. Firstly, draw every vertex [x.sub.j]. Secondly, draw the edges by the value of their weights [r.sub.jk] 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 [lambda]. The vertices of each connected branch make up a group.

Step 4. Calculate the best [lambda].

The value of [lambda] varies from 0 to 1, and the best [lambda] leads to the best classification. So how to get the best [lambda] is an important step. In this paper, we also use a widely used F-statistic to find the best [lambda]:

[mathematical expression not reproducible], (9)

where r is the number of groups for a given [lambda], [n.sub.i] is the number of objects in group i [mathematical expression not reproducible] is the average of the hth (h = 1,2, ..., m) characteristic of the objects in group i, and [[bar.x].sub.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 [alpha], we can find several values of F which are larger than [F.sub.[alpha]]. The [lambda] which leads to the largest F is the best [lambda], and the corresponding classification is the best as well.

The best classification can be noted as

[mathematical expression not reproducible], (10)

where [mathematical expression not reproducible], and [mathematical expression not reproducible].

Let

[mathematical expression not reproducible]. (11)

Therefore, we partition the universe U into r unequal intervals: [u.sub.1] = [[d.sub.0],[d.sub.1]], [u.sub.2] = [[d.sub.1],[d.sub.2]], ..., and [u.sub.r] = [[d.sub.r-1],[d.sub.r]].

3.2. Construction of Intuitionistic Fuzzy Sets. Corresponding to the above r intervals, we define r intuitionistic fuzzy sets representing r linguistic values:

[mathematical expression not reproducible]. (12)

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 . 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) When x is located in the middle of an interval, namely, x = ([d.sub.i-l] + [d.sub.i])/2, we define that [mathematical expression not reproducible] and [mathematical expression not reproducible].

(2) When x is located on the boundaries of an interval, namely, x = [d.sub.i], we define that intuitionistic index has the maximum value and [mathematical expression not reproducible]. Let [mathematical expression not reproducible]; then we can get [mathematical expression not reproducible].

Then, in view of above rules, the membership function is defined as a Gaussian function:

[mathematical expression not reproducible]. (13)

The nonmembership function is a transformation of Gaussian function:

[mathematical expression not reproducible]. (14)

Hence, the intuitionistic index function reads

[mathematical expression not reproducible], (15)

where i = 1,2, ..., r and [c.sub.[mu]i], [[sigma].sub.[mu]i], [c.sub.[gamma]i], and [[sigma].sub.[gamma]i] are important function parameters. The calculations of them are based on the above two rules:

[mathematical expression not reproducible]. (16)

Definition 6. Let A be an intuitionistic fuzzy set in a finite universe X. If

(1) 0 [less than or equal to] [[mu].sub.A](x) [less than or equal to] 1, 0 [less than or equal to] [[gamma].sub.A](x) [less than or equal to] 1,

(2) 0 [less than or equal to] [[pi].sub.A](x) [less than or equal to] 1, 0 [less than or equal to] [[mu].sub.A](x) + [[gamma].sub.A](x) [less than or equal to] 1,

(3) [[pi].sub.A](x) + [[mu].sub.A](x) + [[gamma].sub.A](x) = 1,

then A is a normal intuitionistic fuzzy set.

Therefore, we obtain the following theorem.

Theorem 7. The membership function and nonmembership function of [A.sub.i] (i = 1, 2, ..., r) are standard; that is, [A.sub.i] is a normal intuitionistic fuzzy set.

Proof. (1) y = exp[(-(x - c).sup.2]/2[[sigma].sup.2]) is a Gaussian function, so we have

[mathematical expression not reproducible]. (17)

Therefore,

[mathematical expression not reproducible]. (18)

(2) Given 0 [less than or equal to] [alpha] [less than or equal to] 1 and [c.sub.[mu]i] = [c.sub.[gamma]i] = ([d.sub.i-1] + [d.sub.i])/2, we have

[[sigma].sup.2.sub.[mu]i] [less than or equal to][[sigma].sup.2.sub.[gamma]i]. (19)

Hence,

[mathematical expression not reproducible]. (20)

On the other hand, since 0 [less than or equal to] exp[(-(x - [c.sub.[mu]i]).sup.2]/2[[sigma].sup.2.sub.[mu]i]) [less than or equal to] 1 and 0 [less than or equal to] exp[(-(x - [c.sub.[gamma]i]).sup.2]/2[[sigma].sup.2.sub.[gamma]i]) [less than or equal to] 1, we have

[mathematical expression not reproducible]. (21)

Therefore,

[mathematical expression not reproducible]. (22)

That is,

[mathematical expression not reproducible]. (23)

(3) According to the calculation of [mathematical expression not reproducible], it can be easily found that [mathematical expression not reproducible]. This completes the proof.

Theorem 7 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. Let [A.sub.i] (i = 1, 2, ..., n) and [A.sup.*] be intuitionistic fuzzy sets in universe U and let [B.sub.i] (i = 1, 2, ..., n) and [B.sup.*] be intuitionistic fuzzy sets in universe V. The generalized multiple modus ponens based on intuitionistic fuzzy relation  is that a new proposition that "y is [B.sup.*]" can be inferred from n + 1 propositions: "if x is [A.sup.i], then y is [B.sub.i]" and "x is [A.sup.*]." The reasoning model is as follows:

[mathematical expression not reproducible]

Every rule has a corresponding input-output relation [R.sub.i]. For [R.sub.i], different operators result in different [[mu].sub.R] and [[gamma].sub.R], but the reasoning outputs are all the same. Since it has a better performance and is easier to calculate than other operators , the Mamdani implication operator [R.sub.c] is used here:

[mathematical expression not reproducible], (24)

where

[mathematical expression not reproducible]. (25)

Then, according to the compositional operation of intuitionistic fuzzy rules, we get the total relation:

[mathematical expression not reproducible], (26)

where

[mathematical expression not reproducible]. (27)

The reasoning output is

[B.sup.*] = [A.sup.*] x R, (28)

where "x" is defined as the maximum and minimum operators: "[disjunction]" and "[conjunction]":

[mathematical expression not reproducible]. (29)

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 sets [A.sub.i] (i = 1, 2, ..., r) in the IFTS model; that is, let the historical data be intuitionistic fuzzy sets, noted as [F.sub.i] (j = 1, 2, ..., t), and let [A.sub.i] be the elements in [F.sub.j], and let [mathematical expression not reproducible] and [mathematical expression not reproducible] be the membership and nonmembership of [A.sub.i] to [F.sub.j]. Then [F.sub.j] is as follows:

[mathematical expression not reproducible], (30)

where

[mathematical expression not reproducible]. (31)

Hence, we apply the intuitionistic fuzzy multiple modus ponens to [A.sub.i] and [F.sub.j]. The reasoning model is as follows:

[mathematical expression not reproducible]

The reasoning output is as follows:

[F.sup.*.sub.t+1] = [F.sub.t] x R, (32)

where

[mathematical expression not reproducible]. (33)

So, the membership and nonmembership of the output intuitionistic fuzzy set are

[mathematical expression not reproducible]. (34)

That is to say, the membership and nonmembership of the forecasted result [f.sub.t+1](t) to every intuitionistic fuzzy set [A.sub.i] are [mathematical expression not reproducible] and [mathematical expression not reproducible].

3.4. Defuzzification Algorithm. The widely used defuzzification algorithms include the maximum truth-value algorithm, gravity algorithm, and weighted average algorithm . 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 . The calculation is as follows:

[mathematical expression not reproducible], (25)

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 . 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, [x.sub.min] = 13055 and [x.sub.max] = 19337, so the universe of discourse is set to U = [13000, 20000].

Then U 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 matrix R as shown in Table 1.

(3) Use the Kruskal algorithm mentioned in Section 3.1 to build a maximum spanning tree based on matrix R. The tree is shown in Figure 1.

Let [lambda] be 0.93,0.94,0.95, and 0.96, respectively. We can get different classifications of historical data as shown in Table 2.

(4) For different classifications, calculate the values of F according to (9). The results are also shown in Table 2. From Table 2, we can see that when [lambda] = 0.95, its corresponding F is maximum and bigger than [F.sub.[alpha]] ([alpha] = 0.05) at the same time. So this classification is the best.

Therefore, the universe of discourse U is partitioned into 9 unequal intervals according to the above classification. The boundaries of each interval are calculated according to (11). The intervals are

[mathematical expression not reproducible]. (36)

Step 2. Construct intuitionistic fuzzy sets and intuitionistically fuzzify the historical data.

Corresponding to the 9 intervals, there should be 9 intuitionistic fuzzy sets [A.sub.1],[A.sub.2],...,[A.sub.9], 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 [alpha] = 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 Figures 2, 3, and 4, respectively.

Then we can calculate the membership, nonmembership, and intuitionistic index of every historical value to every intuitionistic fuzzy set.

Step 3. Establish forecast rules and forecast the enrollments.

The enrollments of year 1971 to 1991 can be denoted as [F.sub.1], [F.sub.2], ..., [F.sub.21], and the reasoning model based on Section 3.3.2 is as follows:

[mathematical expression not reproducible]

Then we get [F.sup.*.sub.22]:

[mathematical expression not reproducible], (37)

where the membership of [F.sup.*.sub.22] to [A.sub.8] is the biggest and the nonmembership is the smallest, so the intuitionistic forecasted result is [A.sub.8].

Step 4. Defuzzify and output the forecast result.

The defuzzification result based on Section 3.4 is

[mathematical expression not reproducible]. (38)

That is to say, the enrollment of year 1992 is 18855.

To test the performance of our model, we use the models of inference , , and  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  is an IFTS model. In inference , 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:

[mathematical expression not reproducible]. (39)

The results are shown in Table 5.

The results in Tables 4 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.

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.

For 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.

The performance of all models is compared in Table 8.

Table 8 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.

http://dx.doi.org/10.1155/2016/5035160

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research is sponsored by Natural Science Foundation of China (Grant no. 61402517).

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Ya'nan Wang, Yingjie Lei, Xiaoshi Fan, and Yi Wang

Air and Missile Defense College, Air Force Engineering University, Xian 710051, China

Correspondence should be addressed to Ya'nan Wang; wyn1988814@163.com

Received 30 June 2015; Revised 10 October 2015; Accepted 20 October 2015

Academic Editor: Reik Donner

Caption: Figure 1: The maximum spanning tree of historical data.

Caption: Figure 2: Membership functions.

Caption: Figure 3: Nonmembership functions.

Caption: Figure 4: Intuitionistic index functions.

Caption: Figure 5: Forecast results of TAIEX.
```Table 1: The fuzzy similarity matrix R.

1971    1972    1973    1974    1975    1976    1977    1978

1971   1.000   0.919   0.871   0.739   0.617   0.641   0.594   0.553
1972   0.919   1.000   0.952   0.820   0.698   0.722   0.675   0.634
1973   0.871   0.952   1.000   0.868   0.746   0.770   0.724   0.683
1974   0.739   0.820   0.868   1.000   0.878   0.902   0.856   0.815
1975   0.617   0.698   0.746   0.878   1.000   0.976   0.977   0.936
1976   0.641   0.722   0.770   0.902   0.976   1.000   0.954   0.912
1977   0.594   0.675   0.724   0.856   0.977   0.953   1.000   0.959
1978   0.553   0.634   0.683   0.815   0.936   0.912   0.959   1.000
1979   0.403   0.484   0.532   0.664   0.786   0.762   0.808   0.849
1980   0.385   0.466   0.514   0.646   0.768   0.744   0.791   0.832
1981   0.469   0.550   0.599   0.731   0.852   0.829   0.875   0.916
1982   0.621   0.702   0.751   0.883   0.996   0.981   0.973   0.932
1983   0.611   0.692   0.741   0.872   0.994   0.970   0.983   0.942
1984   0.667   0.748   0.797   0.929   0.950   0.974   0.927   0.886
1985   0.664   0.745   0.794   0.926   0.953   0.976   0.930   0.889
1986   0.534   0.615   0.663   0.795   0.917   0.893   0.939   0.980
1987   0.394   0.475   0.524   0.656   0.777   0.754   0.800   0.841
1988   0.189   0.270   0.318   0.450   0.572   0.548   0.595   0.636
1989   0.058   0.139   0.188   0.320   0.441   0.418   0.464   0.505
1990   0.001   0.082   0.131   0.263   0.384   0.361   0.407   0.448
1991   0.000   0.081   0.129   0.261   0.383   0.359   0.406   0.447

1979    1980    1981    1982    1983    1984    1985    1986

1971   0.403   0.385   0.469   0.622   0.611   0.667   0.664   0.534
1972   0.484   0.466   0.550   0.702   0.692   0.748   0.745   0.615
1973   0.532   0.514   0.599   0.751   0.741   0.797   0.794   0.663
1974   0.664   0.646   0.731   0.883   0.873   0.929   0.926   0.795
1975   0.786   0.768   0.852   0.996   0.994   0.950   0.953   0.917
1976   0.762   0.744   0.829   0.981   0.970   0.974   0.976   0.893
1977   0.808   0.791   0.875   0.973   0.983   0.927   0.930   0.939
1978   0.849   0.832   0.916   0.932   0.942   0.886   0.889   0.980
1979   1.000   0.982   0.933   0.781   0.791   0.735   0.738   0.869
1980   0.982   1.000   0.916   0.763   0.774   0.718   0.721   0.851
1981   0.933   0.915   1.000   0.848   0.858   0.802   0.805   0.936
1982   0.781   0.763   0.848   1.000   0.990   0.954   0.957   0.912
1983   0.791   0.774   0.858   0.990   1.000   0.944   0.947   0.922
1984   0.735   0.718   0.802   0.954   0.944   1.000   0.997   0.866
1985   0.738   0.720   0.805   0.957   0.947   0.997   1.000   0.869
1986   0.869   0.851   0.936   0.912   0.922   0.866   0.869   1.000
1987   0.992   0.990   0.925   0.773   0.783   0.727   0.730   0.861
1988   0.786   0.804   0.720   0.567   0.578   0.522   0.525   0.655
1989   0.656   0.674   0.589   0.437   0.447   0.391   0.394   0.525
1990   0.599   0.617   0.532   0.380   0.390   0.334   0.337   0.468
1991   0.597   0.615   0.531   0.379   0.389   0.333   0.336   0.466

1987    1988      1989      1990      1991

1971   0.395   0.189     0.058     0.001     0.000
1972   0.475   0.270     0.139     0.082     0.081
1973   0.524   0.318     0.188     0.131     0.129
1974   0.656   0.450     0.320     0.263     0.261
1975   0.777   0.572     0.441     0.384     0.383
1976   0.754   0.548     0.418     0.361     0.359
1977   0.800   0.595     0.464     0.407     0.406
1978   0.841   0.636     0.505     0.448     0.447
1979   0.992   0.786     0.656     0.599     0.597
1980   0.990   0.804     0.673     0.616     0.615
1981   0.925   0.720     0.589     0.532     0.531
1982   0.773   0.568     0.437     0.380     0.379
1983   0.783   0.578     0.447     0.390     0.389
1984   0.727   0.522     0.391     0.334     0.333
1985   0.730   0.525     0.394     0.337     0.336
1986   0.861   0.655     0.525     0.468     0.466
1987   1.000   0.795     0.664     0.607     0.606
1988   0.794   1.000     0.869     0.812     0.811
1989   0.664   0.869     1.000     0.943     0.942
1990   0.607   0.812     0.943     1.000     0.999
1991   0.606   0.811     0.942     0.999     1.000

Table 2: Classifications of different [lambda].

Number of
[lambda]   classifications   Classification

0.935             7          {1971}, {1972, 1973}, {1974}, {1979, 1980,
1987}, {1988}, {1989, 1990, 1991}, {1975,
1976, 1977, 1978, 1981, 1982, 1983, 1984,
1985, 1986}
0.94                         {1971}, {1972, 1973}, {1974}, {1981},
{1979, 1980, 1987}, {1988}, {1989, 1990,
1991}, {1975, 1976, 1977, 1978, 1982,
1983, 1984, 1985, 1986}
0.95              9          {1971}, {1972, 1973}, {1974}, {1981},
{1979, 1980, 1987}, {1988}, {1989},
{1990, 1991}, {1975, 1976, 1977, 1978,
1982, 1983, 1984, 1985, 1986}
0.955                        {1971}, {1972}, {1973}, {1974}, {1981},
{1979, 1980, 1987}, {1988}, {1989}, {1990,
1991}, {1975, 1976, 1977, 1978, 1982,
1983, 1984, 1985, 1986}

[lambda]   [F.sub.0.05]     F

0.935          2.85       92.85
0.94           2.83       129.17
0.95           2.85       142.13
0.955          2.9        112.67

Table 3: Function parameters of [A.sub.i].

Intuitionistic
fuzzy set        [C.sub.[mu]i]    [[sigma].sub.[mu]i]

[A.sub.1]           13154.5              99.5
[A.sub.2]           13795.5              313.5
[A.sub.3]           14601.5              205.9
[A.sub.4]           15553.5              407.6
[A.sub.5]            16392               132.8
[A.sub.6]           17066.5              301.9
[A.sub.7]           18047.5              330.3
[A.sub.8]           18854.5              189.8
[A.sub.9]           19574.5              274.2

Intuitionistic
fuzzy set        [c.sub.[gamma]i]    [[sigma].sub.[gamma]i]

[A.sub.1]             13154.5                 182.9
[A.sub.2]             13795.5                  576
[A.sub.3]             14601.5                 378.3
[A.sub.4]             15553.5                 748.9
[A.sub.5]              16392                  243.9
[A.sub.6]             17066.5                 554.7
[A.sub.7]             18047.5                 606.8
[A.sub.8]             18854.5                 348.7
[A.sub.9]             19574.5                 503.8

Table 4: Forecast results of the enrollments.

Forecasted enrollment

Actual
Year   enrollment   Model    Model    Model    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        18855

Table 5: Forecast performance of enrollments.

Criterion   Model    Model    Model    Our model

RMSE          677.1        445.5        418.9       350.9
AFE           3.35%        2.3%         2.07%       1.72%

Table 6: Historical data of TAIEX.

Date            TAIEX     Intuitionistic
fuzzified value

11/1/2004      5656.17       [A.sub.1]
11/2/2004      5759.61       [A.sub.2]
11/3/2004      5862.85       [A.sub.5]
11/4/2004      5860.73       [A.sub.5]
11/5/2004      5931.31       [A.sub.7]
11/8/2004      5937.46       [A.sub.7]
11/9/2004      5945.2        [A.sub.8]
11/10/2004     5948.49       [A.sub.8]
11/11/2004     5874.52       [A.sub.5]
11/12/2004     5917.16       [A.sub.7]
11/15/2004     5906.69       [A.sub.7]
11/16/2004     5910.85       [A.sub.7]
11/17/2004     6028.68      [A.sub.11]
11/18/2004     6049.49      [A.sub.12]
11/19/2004     6026.55      [A.sub.11]
11/22/2004     5838.42       [A.sub.5]
11/23/2004     5851.1        [A.sub.5]
11/24/2004     5911.31       [A.sub.7]
11/25/2004     5855.24       [A.sub.5]
11/26/2004     5778.26       [A.sub.3]
11/29/2004     5785.26       [A.sub.3]
11/30/2004     5844.76       [A.sub.5]
12/1/2004      5798.62       [A.sub.4]
12/2/2004      5867.95       [A.sub.5]
12/3/2004      5893.27       [A.sub.6]
12/6/2004      5919.17       [A.sub.7]
12/7/2004      5925.28       [A.sub.7]
12/8/2004      5892.51       [A.sub.6]
12/9/2004      5913.97       [A.sub.7]
12/10/2004     5911.63       [A.sub.7]
12/13/2004     5878.89       [A.sub.5]
12/14/2004     5909.65       [A.sub.7]
12/15/2004     6002.58      [A.sub.10]
12/16/2004     6019.23      [A.sub.11]
12/17/2004     6009.32      [A.sub.10]
12/20/2004     5985.94       [A.sub.9]
12/21/2004     5987.85       [A.sub.9]
12/22/2004     6001.52      [A.sub.10]
12/23/2004     5997.67      [A.sub.10]
12/24/2004     6019.42      [A.sub.11]
12/27/2004     5985.94       [A.sub.9]
12/28/2004     6000.57      [A.sub.10]
12/29/2004     6088.49      [A.sub.13]
12/30/2004     6100.86      [A.sub.14]
12/31/2004     6139.69      [A.sub.15]

Table 7: Forecast results of TAIEX.

Forecasted TAIEX
Actual
TAIEX     Model    Model    Model    Model    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       6094.66

Table 8: Forecast performance of TAIEX.

Our
Criterion   Model    Model    Model    Model    model

RMSE          61.17       52.63        53.63        50.27      43.23
AFE           0.83%       0.51%        0.65%        0.65%      0.51%
```
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