Fuzzy Forecast Based on Fuzzy Time Series

1. Introduction

In 1965, Zadeh [1] proposed the concept of fuzzy sets as a tool to test the unknown degree of membership. Many fuzzy studies then attempted to use this method as a theoretical framework, which is widely used in the research fields of natural sciences and social sciences, obtaining good study achievements [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. The fuzzy time series is also an analysis method derived from the concept of fuzzy sets. In 1993, Song and Chissom [18, 19, 20, 21] successfully combined the concept of fuzzy sets with the time series model and began studies on fuzzy time series. Chen [3] proposed the simplified and easy-to-calculate method for Song and Chissom’s model [18, 19, 20, 21], so that the computation complexity of fuzzy time series is dramatically reduced. Lee and Chou [14] also proposed that rational settings of the lower and upper boundary in intervals of the universal set for fuzzy time series have improved their accuracy and reliability. Liaw [17] proposed a simple test method for whether a fuzzy time series has a fuzzy trend, in which the method is used to determine whether the data for analysis is in a steady state. Chou [12] added to Chen and Hsieh’s defuzzification method [2] in the fuzzy time series, so that the long-term level for the series can be obtained, and the model originally used for single-point prediction can be applied to long-term prediction and interval prediction. This article mainly uses the algorithmic method of Chou’s [12] research process for illustrating the fuzzy time series, taking the Taiwan Shipping and Transportation Index (STI) [23] as an example.

The remainder of this chapter is organized as follows. Section 2 presents the definition of fuzzy time series and Section 3 defines the long-term predictive significance level process. A numerical example of STI is shown in Section 4, and concluding remarks are mentioned in conclusion.

2. Definition of fuzzy time series

Fuzzy sets, presented by Zadeh [1], have numerous presentations, such as in fuzzy sets, fuzzy decision analysis, and fuzzy time series. The concept is also widely applied in social science article and applications [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Fuzzy time series is developed rapidly since their introduction by Song and Chissom [18, 19, 20, 21]. Current fuzzy time series methods have benefited from both theoretical developments as well as relevant applications in research [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], which has led to more diverse uses. This trend indicates that the development of fuzzy time series has markedly improved. The definitions of the fuzzy time series used in this article are described as follows.

Definition 1 [18, 19, 20, 21]. A fuzzy number on the real line ℜis a fuzzy subset of ℜthat is normal and convex.

Definition 2 [18, 19, 20, 21]. Let Ytt=…012…, a subset of ℜ, be the universe of discourse on which the fuzzy sets fitt=12…are defined, and let Ftbe the collection of fitt=12…. Then, Ftis called fuzzy time series on Ytt=…012….

Definition 3 [18, 19, 20, 21]. Let Iand Jbe the index sets for Ft−1and Ft, respectively. If for any fjt∈Ft, where j∈J, there then exists fit−1∈Ft−1, where i∈I, such that there exists a fuzzy relation Rijtt−1and fjt=fit−1∘Rijtt−1, where ‘∘’ is the max–min composition. Then, Ftis said to be caused by only Ft−1. Denote this as fit−1→fjt, or equivalently, Ft−1→Ft.

Definition 4 [18, 19, 20, 21]. If, for any fjt−1∈Ft, where j∈J, there exists fit−1∈Ft−1, where i∈I, and a fuzzy relation Rijtt−1, such that fjt=fit−1∘Rijtt−1. Let Rtt−1=∪ijRijtt−1, where ∪is the union operator. Then, Rtt−1is called the fuzzy relation between Ftand Ft−1. Thus, we define this as the following fuzzy relational equation:

Definition 5 [18, 19, 20, 21]. Suppose that R1tt−1=∪ijR1ijtt−1and R2tt−1=∪ijR2ijtt−1are two fuzzy relations between Ftand Ft−1. If, for any fjt∈Ft, where j∈J, there exists fit−1∈Ft−1, where i∈I, and fuzzy relations R1ijtt−1and R2ijtt−1such that fit=fit−1∘Rij1tt−1and fit=fit−1∘Rij2tt−1, then define R1tt−1=R2tt−1.

Definition 6 [18, 19, 20, 21]. Suppose that Ftis only caused by Ft−1, Ft−2, …, or Ft−mm>0. This relation can be expressed as the following fuzzy relational equation: Ft=Ft−1∘R0tt−m, which is called the first-order model of Ft.

Definition 7 [18, 19, 20, 21]. Suppose that Ftis simultaneously caused by Ft−1, Ft−2, …, and Ft−mm>0. This relation can be expressed as the following fuzzy relational equation: Ft=Ft−1×Ft−2×…×Ft−m∘Ratt−m, which is called the m^th-order model of Ft.

Definition 8 [3]. Ftis fuzzy time series if Ftis a fuzzy set. The transition is denoted as Ft−1→Ft.

Definition 9 [7]. Let dtbe a set of real numbers: dt⊆R. We define an exponential function where

1. y=expdt⇔lny=dtand

2. explndt=dt,lnexpx=dt.

Definition 10 [14]. The universe of discourse U=DLDUis defined such that DL=Dmin−stαn/nand DU=Dmax+stαn/nwhen n≤30or DL=Dmin−σZα/nand DU=Dmax+σZα/nwhen n>30, where tαnis the 1001−αpercentile of the tdistribution with ndegrees of freedom. zαis the 1001−αpercentile of the standard normal distribution. Briefly, if Zis an N(0, 1) distribution, then PZ≥zα=α.

Definition 11 [14]. Assuming that there are mlinguistic values under consideration, let Aibe the fuzzy number that represents the i^th linguistic value of the linguistic variable, where 1≤i≤m. The support of Aiis defined as follows:

Definition 12 [17]. For a test H0:nonfuzzy trendagainst H1:fuzzy trend, where the critical region C∗=CC2k+C2n−k>Cλ=C2n×1−λ, the initial value of the significance level αis 0.2.

Definition 13 [8]. Let dtbe a set of real numbers dt⊆R. An upper interval for dtis a number b such that x≤bfor all x∈dt. The set dtis said to be an interval higher if dthas an upper interval. A number, max, is the maximum of dtif max is an upper interval for dtand max∈dt.

Definition 14 [8]. Let dt⊆R. The least upper interval of dtis a number max→satisfying:

1. max→is an upper interval for dtsuch that x≤max→for all x∈dtand

2. max→is the least upper interval for dt, that is, x≤bfor all x∈dt⇒max→≤b.

Definition 15 [8]. Let dtbe a set of real numbers dt⊆R. A lower interval for dtis a number b such that x≥bfor all x∈dt. The set dtis said to be an interval below if dthas a lower interval. A number, min, is the minimum of dtif min is a lower interval for dtand min∈dt.

Definition 16 [8]. Let dt⊆R. The least lower interval of dtis a number min←satisfying:

1. min←is a lower interval for dtsuch that x≥min←for all x∈dtand.

2. min←is the least lower interval for dt, that is, x≥bfor all x∈dt⇒min←≤b.

Definition 17 [8]. The long-term predictive value interval (min←, max→) is called the static long-term predictive value interval.

Definition 18 [2]. Let Ai=αiβiγi,i=1,2,…,n,be n triangular fuzzy numbers. By using the graded mean integration representation (GMIR) method, the GMIR value PAiof Aiis PAi=αi+4βi+γi/6. PAiand PAjare the GMIR values of the triangular fuzzy numbers Aiand Aj, respectively.

Definition 19 [12]. Set up new triangular fuzzy numbers by S = (min←, d^t, max→). After GMIR transformation, S becomes a real number ΔS. This is called the long-term significance level with fuzzy time series. The ΔS is a real number satisfying the following:

1. ΔS is called a long-term significance level up, only if: ΔS > d^t;

2. ΔS is called a long-term significance level down, only if: ΔS < d^t; and

3. ΔS is called a long-term significance level stable, only if: ΔS = d^t.

3. Procedure of fuzzy time series forecasting

This section proposes a method to forecast the long-term predictive significance level by Chou. The stepwise procedure of the proposed method consists the following steps [8], illustrated as a flowchart in Figure 1 [5, 6, 7, 8, 9, 10, 11, 12].

Step 1. Let dtbe the data under consideration and let Ftbe fuzzy time series. Following Definition 11, a difference test is performed to determine whether stability of the information. Recursion is performed until the information is in a stable state, where the critical region is C∗=CC2k+C2n−k>Cλ=C2n×1−λ.

Step 2. Determine the universe of discourse U=DLDU.

Step 3. Define Aiby letting its membership function be as follows:

Step 4. Then, Ft=Aiif dt∈suppAi, where supp⋅denotes the support.

Step 5. Derive the transition rule from period t−1to tand denote it as Ft−1→Ft. Aggregate all transition rules. Let the set of rules be R=riri:Pi→Qi.

Step 6. The value of dtcan be predicted using the fuzzy time series Ftas follows. Let Tt=rjdt∈suppPjwhererj∈Rbe the set of rules fired by dt, where suppPjis the support of Pj. Let suppPj¯be the median of suppPj. The predicted value of dtis ∑rj∈Tt−1suppQjTt−1¯.

Step 7. The long-term predictive value interval for dtis given as (min←, max→).

Step 8. Set up new triangular fuzzy numbers by ΔS = (min←,d^tmax→) .

Step 9. Defuzzify S to be ΔS.

4. Numerical example of Shipping and Transportation Index in Taiwan

In this study, the Shipping and Transportation Index (STI) in Taiwan is used for a numerical example. The STI reflects the spot rates of the Taiwan Stock Exchange Corporation. The STI data are sourced from the Taiwan Stock Exchange Corporation [23], the historical data for which is defined here as the STI, and month-averaged data for the period between January, 2015, and June, 2018, was collected.

Over these 42 data points, the analysis produces an average of 4.226, with a standard deviation of 0.172, maximum value of 4.571, and minimum value of 4.067. These descriptive statistics show that the STI has largely remained at the 1124.70 level. As shown in Figure 2, its current rate of return is negative.

The following steps in the procedure are performed when using fuzzy time series to analyze STI.

Step 1. First, we take the logarithm of the STI data to reduce variation and improve the forecast accuracy, letting STIt˜=lnSTIt.

Step 2. Maintaining stationary data while forecasting helps to improve the forecast quality; therefore, we conduct a stationary test on the STI data. For fuzzy time series, a fuzzy trend test can measure whether the STI’s fuzzy trend moves upward or downward. Using this fuzzy trend test, the STI data can be converted into a stationary series. If the original STI data exhibited a fuzzy trend, it can be eliminated by taking the difference. We then repeat the test after taking the first difference to measure if the STI data exhibits a fuzzy trend. If a fuzzy trend is again observed, then we take the second difference, and so on.

Letting STItbe the historical data under consideration and fuzzy time series, a difference test is used (following Definition 11) to understand whether the stability of the information. Recursion is performed until the information is determined to be stable. Once the region C''=CC=C222+C242−22=432<CC242×1−0.2=688.8, the STI data are considered in a stable state and are not rejected.

Step 3. According to the interval setting of the STI data, we define the upper and lower bounds, which facilitate dividing the linguistic value intervals later. From Definition 10, the discourse U=DLDU. From Table 1, Dmin=4.067, Dmax=4.571, s=0.172, and n=42can be obtained. Lettingα=0.05, since nis large than 30, a standard normal Z was used. Thus, Z0.05=1.645, DL=Dmin−stα/n≈3.627, and DU=Dmax+stα/n≈5.011. That is, U=3.6275.011.

Step 4. After defining the upper and lower bounds of the STI data in Step 3, we can define the SCFI range by determining the membership function as well as the linguistic values. We can also define the range of the subinterval for each linguistic value, assuming that the following linguistic values are under consideration: extremely few, very few, few, some, many, very many, and extremely many. According to Definition 11, the supports of fuzzy numbers that represent these linguistic values are given as follows:

where A1= “extremely few,” A2= “very few,” A3= “few,” A4= “some,” A5= “many,” A6= “very many,” and A7= “extremely many.” Thus, the supports are suppA1=3.6273.825, suppA2=3.8254.023, suppA3=4.0234.221, suppA4=4.2214.419, suppA5=4.4194.617, suppA6=4.6174.815, and suppA7=4.8155.011.

Step 5. According to the subinterval setting of each linguistic value, we classified each historical dataset of the STI into its corresponding interval to measure the value corresponding to the linguistic value for each interval. The fuzzy time series Ftwas given by Ft=Aiwhen dt∈suppAi. Therefore, F201501=A5, F201502=A6, F201503=A5, F201504=A6, …, and F201806=A3. Table 1 shows the comparison between the actual SCFI data and the fuzzy enrollment data.

Step 6. We apply fuzzy theory to define the corresponding value for the intervals of the STI data, arrange the corresponding method for the STI data, and integrate the changes from all the rules to determine the rules for the STI. The transition rules are derived from Table 1. For example, F201501→F201502is A5→A5. Table 2 shows all transition rules obtained from Table 1.

Step 7. We calculate each rule by determining all the rules of the STI, and the calculation results can be used to forecast future values. Table 1 shows the forecasting results from 201001 to 201806.

Step 8. The calculated STI rules can define the intervals of the STI data; using these intervals, we can determine the variation in future long-term intervals. The long-term predictive value interval for the STI is given as (3.726, 4.913). Thus, the long-term predictive interval for the STI is given as (41.506, 136.022). Therefore, the current long-term S STI is bounded by this interval. According to Step 8, the fuzzy STI of 201501 shown in Table 1 is A₅, and from Table 2, we can see that the rules are the fuzzy logical relationships in Rule 8 of Table 2, in which the current state of fuzzy logical relationships is A₃. Thus, the 201806 STI predictive value is 41.506.

Step 9. Letting defuzzified S be ΔS, the STI 201806 forecast value based on our investigation is 68.090, and its trading range is between 41.506 and 136.002. Thus, the new triangular fuzzy numbers by S = (41.506, 68.090, 136.002). Thus, the defuzzified S is ΔS = 74.981, and ΔS = 74.981 > d^t= 68.090. ΔS is called a long-term significance level up.

The result shows that based on the long-term significance level, the STI is currently oversold. This result and the risk–reward ratio are both related within the group. We used Table 1 data in our analysis according to the root mean square percentage error (R.M.S.P.E.) method, with an average prediction error of 1.708%. Figure 3 shows the forecast visitor arrivals determined through fuzzy time series analysis and the actual STI values. Based on the fuzzy time series results, the average STI is estimated to be 68.090 in 201806 (Figure 3).

5. Conclusions and future work

In this article, a long-term predictive value interval model is developed for forecasting the STI. This model facilitates minimizing the uncertainties associated with fuzzy numbers. The method is examined by forecasting the STI by using data from which ΔS = 74.981 and ΔS > d^tis obtained. For index returns, the current rate of return is negative and its volatility is increasing. The long-term predictive significance level of the STI is at the ΔS level; the STI should thus exhibit extreme volatility.

The current model for the STI 201806 forecast level deviates insignificantly from the actual values for an average of 68.090 and is within the group; the prediction error does not exceed 1.708% of the significance level. By constructing a fuzzy time series forecasting model for the STI with an error of less than 1.708%, with the traditional fuzzy time excluded from the single-point forecast comparison, this model provides a long-term predictive significance level.

Furthermore, the proposed method can be computerized. Thus, by improving fuzzy linguistic assessments as well as the evaluation of fuzzy time series, decision makers can automatically obtain the final long-term predictive significance level.

The STI used in this chapter is used as a forecasting example. If you predict that the future will rise, you can use the buying strategy. For example, if the index returns in the future, you can use the selling strategy.

The four functions of management are mainly four functions: planning, organization, leadership and control. The fuzzy time series mode used in this chapter can be applied to controlled projects to compare and correct whether the re-executed work meets expectations. If you meet expectations, re-plan the original settings.

Acknowledgments

This chapter is extended and revised the article “An improved fuzzy time series theory with applications in the Shanghai containerized freight index”.