#### 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.

#### 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: 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 be a finite universal set. An intuitionistic fuzzy set in is an object having the formwhere the function defines the degree of membership and the function defines the degree of nonmembership of the element to set . For every , . is called the intuitionistic index of in . It is the hesitancy of to .

When is discrete, the intuitionistic fuzzy set can be noted as

*Definition 2. *Let and be two finite universal sets. A binary intuitionistic fuzzy relation from to is an intuitionistic fuzzy set in the direct product space :where , for , and .

*Definition 3. *Let , a subset of , be the universe of discourse on which intuitionistic fuzzy sets are defined. is a collection of and defines an intuitionistic fuzzy time series on .

*Definition 4. *Let be an intuitionistic fuzzy relation from to . Suppose that is caused only by , denoted aswhere “” is the intuitionistic fuzzy compositional operator. Then is called a first-order intuitionistic fuzzy logical relationship of .

*Definition 5. *If is independent of time ,then is called a time-invariant intuitionistic fuzzy time series. Otherwise, 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 should be defined, where and are the minimum and maximum historical data, respectively. and are two proper positive numbers. Usually, for simplicity, and are chosen to round down and round up to two proper integers.

Secondly, partition the universe 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].

Let be the universe of objects to be classified, where () has characteristics. Let be the similarity matrix of , where is the similarity between and (). A maximum spanning tree is a tree with all being the vertices and being the weights of every edge. Let be the clustering threshold. Cutting down the edges whose weights are smaller than , 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 , data in different dimensions should be transformed into the interval to meet the requirement of similarity matrix [26]. Generally, two kinds of transformation are required.

Standard deviation transformation is as follows:where and , , . 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 will locate in .

Range transformation is as follows:where, obviously, .

*Step 2. *Establish the fuzzy similarity matrix .

Let be the similarity between and ; then we will have a fuzzy similarity matrix . There are different ways to get . Since the Euclidean distance is widely used in similarity matrix establishment [26], we also choose it to calculate :

*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 . Secondly, draw the edges by the value of their weights 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 . The vertices of each connected branch make up a group.

*Step 4. *Calculate the best .

The value of varies from 0 to 1, and the best leads to the best classification. So how to get the best is an important step. In this paper, we also use a widely used -statistic to find the best :where is the number of groups for a given , is the number of objects in group (), is the average of the () characteristic of the objects in group , and is the characteristic of all objects. In (9), the numerator represents the distances between groups, and the denominator represents the distances within groups. So the bigger is, the better the classification we get. For a given confidence level , we can find several values of which are larger than . The which leads to the largest is the best , and the corresponding classification is the best as well.

The best classification can be noted aswhere , , , and .

LetTherefore, we partition the universe into unequal intervals: , , and .

##### 3.2. Construction of Intuitionistic Fuzzy Sets

Corresponding to the above intervals, we define intuitionistic fuzzy sets representing linguistic values: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)When is located in the middle of an interval, namely, , we define that and .(2)When is located on the boundaries of an interval, namely, , we define that intuitionistic index has the maximum value and . Let ; then we can get .

Then, in view of above rules, the membership function is defined as a Gaussian function:The nonmembership function is a transformation of Gaussian function:Hence, the intuitionistic index function readswhere and , , , and are important function parameters. The calculations of them are based on the above two rules:

*Definition 6. *Let be an intuitionistic fuzzy set in a finite universe . If(1), ,(2), ,(3),then is a normal intuitionistic fuzzy set.

Therefore, we obtain the following theorem.

Theorem 7. *The membership function and nonmembership function of are standard; that is, is a normal intuitionistic fuzzy set.*

*Proof. * is a Gaussian function, so we haveTherefore, Given and , we haveHence,On the other hand, since and , we haveTherefore,That is, According to the calculation of , it can be easily found that . 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 and be intuitionistic fuzzy sets in universe and let and be intuitionistic fuzzy sets in universe . The generalized multiple modus ponens based on intuitionistic fuzzy relation [27] is that a new proposition that “ is ” can be inferred from propositions: “if is , then is ” and “ is .” The reasoning model is as follows:

Rules: | IF is | THEN | is |

IF is | THEN | is | |

IF is | THEN | is | |

Input: | IF is | ||

Output: | is |

Every rule has a corresponding input-output relation . For , different operators result in different and , 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 is used here:where

Then, according to the compositional operation of intuitionistic fuzzy rules, we get the total relation:whereThe reasoning output iswhere “” is defined as the maximum and minimum operators: “” and “”:

###### 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 in the IFTS model; that is, let the historical data be intuitionistic fuzzy sets, noted as , and let be the elements in , and let and be the membership and nonmembership of to . Then is as follows:where

Hence, we apply the intuitionistic fuzzy multiple modus ponens to and . The reasoning model is as follows:

Rules: | IF x is | THEN | y is |

IF x is | THEN | y is | |

IF x is | THEN | y is | |

IF x is | THEN | y is | |

Input: | IF x is | ||

Output: | y is |

The reasoning output is as follows:whereSo, the membership and nonmembership of the output intuitionistic fuzzy set areThat is to say, the membership and nonmembership of the forecasted result to every intuitionistic fuzzy set are and .

##### 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:where is the output domain and is an intuitionistic fuzzy set in .

#### 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, and , so the universe of discourse is set to .

Then 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 as shown in Table 1.

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

Let 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 according to (9). The results are also shown in Table 2. From Table 2, we can see that when , its corresponding is maximum and bigger than at the same time. So this classification is the best.

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

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

Corresponding to the 9 intervals, there should be 9 intuitionistic fuzzy sets , 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 , 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 , and the reasoning model based on Section 3.3.2 is as follows:

Rules: | IF x is | THEN | y is |

IF x is | THEN | y is | |

IF x is | THEN | y is | |

Input: | IF x is | ||

Output: | y is |

Then we get :where the membership of to is the biggest and the nonmembership is the smallest, so the intuitionistic forecasted result is .

*Step 4. *Defuzzify and output the forecast result.

The defuzzification result based on Section 3.4 isThat 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: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.

#### 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).