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 [5–12].

Step 1. Let d tð Þ be the data under consideration and let F tð Þ be 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 <sup>C</sup><sup>∗</sup> <sup>¼</sup> C C<sup>k</sup> <sup>2</sup> <sup>þ</sup> Cn�<sup>k</sup> <sup>2</sup> . <sup>C</sup><sup>λ</sup> <sup>¼</sup> <sup>C</sup><sup>n</sup> <sup>2</sup> � ð Þ <sup>1</sup> � <sup>λ</sup> .

Step 2. Determine the universe of discourse U ¼ ½ � DL; DU .

Figure 1. Procedure of the proposed model.

Definition 19 [12]. Set up new triangular fuzzy numbers by S = (min

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

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

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

3. Procedure of fuzzy time series forecasting

Time Series Analysis - Data, Methods, and Applications

stable state, where the critical region is <sup>C</sup><sup>∗</sup> <sup>¼</sup> C C<sup>k</sup>

ing steps [8], illustrated as a flowchart in Figure 1 [5–12].

Step 2. Determine the universe of discourse U ¼ ½ � DL; DU .

! ). 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 satis-

This section proposes a method to forecast the long-term predictive significance level by Chou. The stepwise procedure of the proposed method consists the follow-

Step 1. Let d tð Þ be the data under consideration and let F tð Þ be fuzzy time series.

<sup>2</sup> <sup>þ</sup> Cn�<sup>k</sup>

<sup>2</sup> . <sup>C</sup><sup>λ</sup> <sup>¼</sup> <sup>C</sup><sup>n</sup> <sup>2</sup> � ð Þ <sup>1</sup> � <sup>λ</sup> 

Following Definition 11, a difference test is performed to determine whether stability of the information. Recursion is performed until the information is in a

max

Figure 1.

28

Procedure of the proposed model.

fying the following:

, ^ d tð Þ,

d tð Þ; and

.

d tð Þ.

d tð Þ;

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

$$u\_{A\_i}(\mathbf{x}) = \begin{cases} 1 & \text{for } \mathbf{x} \in \left[ D\_L + (i - 1) \frac{D\_U - D\_L}{m}, D\_L + \frac{i(D\_U - D\_L)}{m} \right) \\ & \text{where } 1 \le i \le m - 1; \\\ 1 & \text{for } \mathbf{x} \in \left[ D\_L + (i - 1) \frac{D\_U - D\_L}{m}, D\_L + \frac{i(D\_U - D\_L)}{m} \right] \\ & \text{where } i = m; \\\ 0 & \text{otherwise.} \end{cases}$$

Step 4. Then, F tðÞ¼ Ai if d tð Þ∈ suppð Þ Ai , where suppð Þ� denotes the support. Step 5. Derive the transition rule from period t � 1 to t and denote it as F tð Þ! � 1 F tð Þ. Aggregate all transition rules. Let the set of rules be R ¼ ri ri : Pi ! Qi f j g.

Step 6. The value of d tð Þ can be predicted using the fuzzy time series F tð Þ as follows. Let T tðÞ¼ rj d tð Þ∈ supp Pj � �; where rj ∈ R � � � � be the set of rules fired by d tð Þ, where supp Pj � � is the support of Pj. Let supp Pj � � be the median of supp Pj � �. The predicted value of d tð Þ is <sup>∑</sup>rj <sup>∈</sup> T tð Þ �<sup>1</sup> suppð Þ Qj j j T tð Þ �<sup>1</sup> .

Step 7. The long-term predictive value interval for d tð Þ is given as (min , max ! ). Step 8. Set up new triangular fuzzy numbers by <sup>Δ</sup>S = (min ,^ d tð Þmax ! ) . Step 9. Defuzzify S to be ΔS.
