Intuitionistic Fuzzy Time Series Forecasting Model Based on Intuitionistic Fuzzy Reasoning

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.

Academic Editor:Reik Donner

Air and Missile Defense College, Air Force Engineering University, Xi'an 710051, China

Received 30 June 2015; Revised 10 October 2015; Accepted 20 October 2015; 13 January 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [figure omitted; refer to PDF] be a finite universal set. An intuitionistic fuzzy set [figure omitted; refer to PDF] in [figure omitted; refer to PDF] is an object having the form [figure omitted; refer to PDF] where the function [figure omitted; refer to PDF] defines the degree of membership and the function [figure omitted; refer to PDF] defines the degree of nonmembership of the element [figure omitted; refer to PDF] to set [figure omitted; refer to PDF] . For every [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is called the intuitionistic index of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . It is the hesitancy of [figure omitted; refer to PDF] to [figure omitted; refer to PDF] .

When [figure omitted; refer to PDF] is discrete, the intuitionistic fuzzy set [figure omitted; refer to PDF] can be noted as [figure omitted; refer to PDF]

Definition 2.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be two finite universal sets. A binary intuitionistic fuzzy relation [figure omitted; refer to PDF] from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] is an intuitionistic fuzzy set in the direct product space [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Definition 3.

Let [figure omitted; refer to PDF] , a subset of [figure omitted; refer to PDF] , be the universe of discourse on which intuitionistic fuzzy sets [figure omitted; refer to PDF] are defined. [figure omitted; refer to PDF] is a collection of [figure omitted; refer to PDF] and defines an intuitionistic fuzzy time series on [figure omitted; refer to PDF] .

Definition 4.

Let [figure omitted; refer to PDF] be an intuitionistic fuzzy relation from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] . Suppose that [figure omitted; refer to PDF] is caused only by [figure omitted; refer to PDF] , denoted as [figure omitted; refer to PDF] where " [figure omitted; refer to PDF] " is the intuitionistic fuzzy compositional operator. Then [figure omitted; refer to PDF] is called a first-order intuitionistic fuzzy logical relationship of [figure omitted; refer to PDF] .

Definition 5.

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

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

[figure omitted; refer to PDF] Standard deviation transformation is as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] will locate in [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] Range transformation is as follows: [figure omitted; refer to PDF] where, obviously, [figure omitted; refer to PDF] .

Step 2.

Establish the fuzzy similarity matrix [figure omitted; refer to PDF] .

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

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 [figure omitted; refer to PDF] . Secondly, draw the edges by the value of their weights [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] . The vertices of each connected branch make up a group.

Step 4.

Calculate the best [figure omitted; refer to PDF] .

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

The best classification can be noted as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] Therefore, we partition the universe [figure omitted; refer to PDF] into [figure omitted; refer to PDF] unequal intervals: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

3.2. Construction of Intuitionistic Fuzzy Sets

Corresponding to the above [figure omitted; refer to PDF] intervals, we define [figure omitted; refer to PDF] intuitionistic fuzzy sets representing [figure omitted; refer to PDF] linguistic values: [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] is located in the middle of an interval, namely, [figure omitted; refer to PDF] , we define that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

(2) When [figure omitted; refer to PDF] is located on the boundaries of an interval, namely, [figure omitted; refer to PDF] , we define that intuitionistic index has the maximum value and [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] ; then we can get [figure omitted; refer to PDF] .

Then, in view of above rules, the membership function is defined as a Gaussian function: [figure omitted; refer to PDF] The nonmembership function is a transformation of Gaussian function: [figure omitted; refer to PDF] Hence, the intuitionistic index function reads [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are important function parameters. The calculations of them are based on the above two rules: [figure omitted; refer to PDF]

Definition 6.

Let [figure omitted; refer to PDF] be an intuitionistic fuzzy set in a finite universe [figure omitted; refer to PDF] . If

(1) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ,

(2) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ,

(3) [figure omitted; refer to PDF] ,

then [figure omitted; refer to PDF] is a normal intuitionistic fuzzy set.

Therefore, we obtain the following theorem.

Theorem 7.

The membership function and nonmembership function of [figure omitted; refer to PDF] are standard; that is, [figure omitted; refer to PDF] is a normal intuitionistic fuzzy set.

Proof.

[figure omitted; refer to PDF] is a Gaussian function, so we have [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF]

[figure omitted; refer to PDF] Given [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] On the other hand, since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF]

[figure omitted; refer to PDF] According to the calculation of [figure omitted; refer to PDF] , it can be easily found that [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be intuitionistic fuzzy sets in universe [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be intuitionistic fuzzy sets in universe [figure omitted; refer to PDF] . The generalized multiple modus ponens based on intuitionistic fuzzy relation [27] is that a new proposition that " [figure omitted; refer to PDF] is [figure omitted; refer to PDF] " can be inferred from [figure omitted; refer to PDF] propositions: "if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is [figure omitted; refer to PDF] " and " [figure omitted; refer to PDF] is [figure omitted; refer to PDF] ." The reasoning model is as follows:

Rules:	IF [figure omitted; refer to PDF] is [figure omitted; refer to PDF]	THEN	[figure omitted; refer to PDF] is [figure omitted; refer to PDF]
	IF [figure omitted; refer to PDF] is [figure omitted; refer to PDF]	THEN	[figure omitted; refer to PDF] is [figure omitted; refer to PDF]
		[figure omitted; refer to PDF]
	IF [figure omitted; refer to PDF] is [figure omitted; refer to PDF]	THEN	[figure omitted; refer to PDF] is [figure omitted; refer to PDF]
Input:	IF [figure omitted; refer to PDF] is [figure omitted; refer to PDF]
Output:			[figure omitted; refer to PDF] is [figure omitted; refer to PDF]

Every rule has a corresponding input-output relation [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] , different operators result in different [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] is used here: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Then, according to the compositional operation of intuitionistic fuzzy rules, we get the total relation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The reasoning output is [figure omitted; refer to PDF] where " [figure omitted; refer to PDF] " is defined as the maximum and minimum operators: " [figure omitted; refer to PDF] " and " [figure omitted; refer to PDF] ": [figure omitted; refer to PDF]

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

Hence, we apply the intuitionistic fuzzy multiple modus ponens to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The reasoning model is as follows:

Rules:	IF x is [figure omitted; refer to PDF]	THEN	y is [figure omitted; refer to PDF]
	IF x is [figure omitted; refer to PDF]	THEN	y is [figure omitted; refer to PDF]
		[figure omitted; refer to PDF]
	IF x is [figure omitted; refer to PDF]	THEN	y is [figure omitted; refer to PDF]
		[figure omitted; refer to PDF]
	IF x is [figure omitted; refer to PDF]	THEN	y is [figure omitted; refer to PDF]
Input:	IF x is [figure omitted; refer to PDF]
Output:			y is [figure omitted; refer to PDF]

The reasoning output is as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] So, the membership and nonmembership of the output intuitionistic fuzzy set are [figure omitted; refer to PDF] That is to say, the membership and nonmembership of the forecasted result [figure omitted; refer to PDF] to every intuitionistic fuzzy set [figure omitted; refer to PDF] are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

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: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the output domain and [figure omitted; refer to PDF] is an intuitionistic fuzzy set in [figure omitted; refer to PDF] .

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, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , so the universe of discourse is set to [figure omitted; refer to PDF] .

Then [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] as shown in Table 1.

Table 1: The fuzzy similarity matrix [figure omitted; refer to PDF] .

	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 Section 3.1 to build a maximum spanning tree based on matrix [figure omitted; refer to PDF] . The tree is shown in Figure 1.

Figure 1: The maximum spanning tree of historical data.

[figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] 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 different [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	Number of classifications	Classification	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0.935	7	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	2.85	92.85
0.94	8	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	2.83	129.17
0.95	9	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	2.85	142.13
0.955	10	[figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	2.9	112.67

(4) For different classifications, calculate the values of [figure omitted; refer to PDF] according to (9). The results are also shown in Table 2. From Table 2, we can see that when [figure omitted; refer to PDF] , its corresponding [figure omitted; refer to PDF] is maximum and bigger than [figure omitted; refer to PDF] at the same time. So this classification is the best.

Therefore, the universe of discourse [figure omitted; refer to PDF] is partitioned into 9 unequal intervals according to the above classification. The boundaries of each interval are calculated according to (11). The intervals are [figure omitted; refer to PDF]

Step 2.

Construct intuitionistic fuzzy sets and intuitionistically fuzzify the historical data.

Corresponding to the 9 intervals, there should be 9 intuitionistic fuzzy sets [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] , 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.

Table 3: Function parameters of [figure omitted; refer to PDF] .

Intuitionistic fuzzy set	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	13154.5	99.5	13154.5	182.9
[figure omitted; refer to PDF]	13795.5	313.5	13795.5	576
[figure omitted; refer to PDF]	14601.5	205.9	14601.5	378.3
[figure omitted; refer to PDF]	15553.5	407.6	15553.5	748.9
[figure omitted; refer to PDF]	16392	132.8	16392	243.9
[figure omitted; refer to PDF]	17066.5	301.9	17066.5	554.7
[figure omitted; refer to PDF]	18047.5	330.3	18047.5	606.8
[figure omitted; refer to PDF]	18854.5	189.8	18854.5	348.7
[figure omitted; refer to PDF]	19574.5	274.2	19574.5	503.8

Figure 2: Membership functions.

[figure omitted; refer to PDF]

Figure 3: Nonmembership functions.

[figure omitted; refer to PDF]

Figure 4: Intuitionistic index functions.

[figure omitted; refer to PDF]

Step 3.

Establish forecast rules and forecast the enrollments.

The enrollments of year 1971 to 1991 can be denoted as [figure omitted; refer to PDF] , and the reasoning model based on Section 3.3.2 is as follows:

Rules:	IF x is [figure omitted; refer to PDF]	THEN	y is [figure omitted; refer to PDF]
	IF x is [figure omitted; refer to PDF]	THEN	y is [figure omitted; refer to PDF]
		[figure omitted; refer to PDF]
	IF x is [figure omitted; refer to PDF]	THEN	y is [figure omitted; refer to PDF]
Input:	IF x is [figure omitted; refer to PDF]
Output:			y is [figure omitted; refer to PDF]

Then we get [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where the membership of [figure omitted; refer to PDF] to [figure omitted; refer to PDF] is the biggest and the nonmembership is the smallest, so the intuitionistic forecasted result is [figure omitted; refer to PDF] .

Step 4.

Defuzzify and output the forecast result.

The defuzzification result based on Section 3.4 is [figure omitted; refer to PDF] 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: [figure omitted; refer to PDF]

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.

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	18855

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

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.

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	6094.66

Figure 5: Forecast results of TAIEX.

[figure omitted; refer to PDF]

The performance of all models is compared in Table 8.

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%

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.

Acknowledgment

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

Conflict of Interests

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

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