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 monthaveraged 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.

Figure 2. Rate of return of the STI.

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

Year Actual ln(Actual) Fuzzified The forecast value 95.611 4.560 A5 4.518 92.839 4.531 A5 4.518 94.750 4.551 A5 4.518 96.622 4.571 A6 4.617 88.503 4.483 A5 4.518 79.003 4.369 A4 4.221 80.103 4.383 A4 4.221 74.787 4.315 A4 4.221 69.560 4.242 A4 4.221 71.416 4.269 A4 4.221 67.625 4.214 A3 4.221 64.282 4.163 A3 4.221 63.301 4.148 A3 4.221 64.315 4.164 A3 4.221 67.143 4.207 A3 4.221 65.073 4.176 A3 4.221 61.163 4.114 A3 4.221 61.221 4.114 A3 4.221 61.043 4.112 A3 4.221 59.942 4.093 A3 4.221 60.293 4.099 A3 4.221 58.372 4.067 A3 4.221 58.736 4.073 A3 4.221 57.892 4.059 A3 4.221 59.278 4.082 A3 4.221 62.746 4.139 A3 4.221 65.467 4.182 A3 4.221 62.142 4.129 A3 4.221 76.626 4.339 A4 4.221 63.029 4.144 A3 4.221 64.728 4.170 A3 4.221 68.464 4.226 A4 4.221 70.555 4.256 A4 4.221 67.830 4.217 A4 4.221 66.696 4.200 A3 4.221 67.640 4.214 A3 4.221 69.769 4.245 A4 4.221 65.206 4.178 A3 4.221 64.669 4.169 A3 4.221 64.671 4.169 A3 4.221

Fuzzy Forecast Based on Fuzzy Time Series DOI: http://dx.doi.org/10.5772/intechopen.82843

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Step 1. First, we take the logarithm of the STI data to reduce variation and improve the forecast accuracy, letting <sup>S</sup>TIðÞ¼ <sup>~</sup><sup>t</sup> ln <sup>S</sup>TIð Þ<sup>t</sup> .

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 STIð Þt be 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<sup>00</sup> <sup>¼</sup> C C <sup>¼</sup> <sup>C</sup><sup>22</sup> <sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>42</sup>�<sup>22</sup> 2 � � � <sup>¼</sup> <sup>432</sup> , C C<sup>42</sup> <sup>2</sup> � ð Þ <sup>1</sup> � <sup>0</sup>:<sup>2</sup> � � � � � ¼ 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 ¼ ½ � DL; DU . From Table 1, Dmin ¼ 4:067 , Dmax ¼ 4:571, s ¼ 0:172, and n ¼ 42 can be obtained. Lettingα ¼ 0:05, since n is large than 30, a standard normal Z was used. Thus, Z0:<sup>05</sup> ¼ 1:645, DL ¼ Dmin� stα= ffiffiffi <sup>n</sup> <sup>p</sup> <sup>≈</sup> <sup>3</sup>:627, and DU <sup>¼</sup> <sup>D</sup>max <sup>þ</sup> stα<sup>=</sup> ffiffiffi <sup>n</sup> <sup>p</sup> <sup>≈</sup> <sup>5</sup>:011. That is, <sup>U</sup> <sup>¼</sup> ½ � <sup>3</sup>:627; <sup>5</sup>:<sup>011</sup> .

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:

$$u\_{A\_i}(\mathbf{x}) = \begin{cases} 1 & \text{for } \mathbf{x} \in [3.627 + (i - 1)(0.129), 3.627 + i(0.198)) \\ & \text{where } \mathbf{1} \le i \le m - \mathbf{1}; \\\\ 1 & \text{for } \mathbf{x} \in [3.627 + (i - 1)(0.129), 3.627 + i(0.198)] \\ & \text{where } i = m; \\\\ 0 & \text{otherwise.} \end{cases}$$

where A<sup>1</sup> = "extremely few," A<sup>2</sup> = "very few," A<sup>3</sup> = "few," A<sup>4</sup> = "some," A<sup>5</sup> = "many," A<sup>6</sup> = "very many," and A<sup>7</sup> = "extremely many." Thus, the supports are suppð Þ¼ A<sup>1</sup> ½ Þ 3:627; 3:825 , suppð Þ¼ A<sup>2</sup> ½ Þ 3:825; 4:023 , suppð Þ¼ A<sup>3</sup> ½ Þ 4:023; 4:221 , suppð Þ¼ A<sup>4</sup> ½ Þ 4:221; 4:419 , suppð Þ¼ A<sup>5</sup> ½ Þ 4:419; 4:617 , suppð Þ¼ A<sup>6</sup> ½ Þ 4:617; 4:815 , and suppð Þ¼ A<sup>7</sup> ½ � 4:815; 5:011 .

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 F tð Þ was given by F tðÞ¼ Ai when d tð Þ∈ suppð Þ Ai . Therefore, Fð Þ¼ 201501 A5, Fð Þ¼ 201502 A6, Fð Þ¼ 201503 A5, Fð Þ¼ 201504 A6, …, and Fð Þ¼ 201806 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
