**3. Research Methodology**

In this section, we will introduce a neutrosophic forecasting model for time series based on first-order state and information entropy of high-order fluctuation. The detailed steps are shown as follow steps and in Figure 1.

**Figure 1.** The flowchart of the neutrosophic forecasting model.

#### *3.1. Step 1: Using Neutrosophic Fluctuation Sets to Describe a Time Series*

Let {*Vt*|*t* = 1, 2, 3, ... , *T*} be a stock index time series and {*Ut*|*t* = 2, 3, ... , *T*} be its fluctuation time series, where *Ut* = *Vt* − *Vt*−<sup>1</sup> *(t* = 2, 3, ... , *T)*. Then, we can calculate *len* = *T t*=2 |*Ut*| *T*−1 , which is the benchmark for interval division when calculating membership. Let {*Xt*|*t* = *m*, *m* + 1, *m* + 2, ... , *T*} be the *m*th-order neutrosophic expression of fluctuation time series {*U*t|*t* = 2, 3, ... , *T*}. The conversion rules for the truth-membership *TXt*and falsity-membership *FXt*of *Xt* are defined as follows:

$$T\_{X\_{\ell}} = \begin{cases} 0, & \text{l} \text{l}\_{\ell} \le -0.5 \text{l} \text{en} \\ \frac{\text{l}\_{\ell}}{3/2 \times \text{len}} + \frac{1}{3}, & -0.5 \times \text{len} \le \text{l} \text{l}\_{\ell} \le \text{len} \\ 1, & \text{l} \text{l}\_{\ell} \ge \text{len} \end{cases} \qquad F\_{X\_{\ell}} = \begin{cases} 1, & \text{l} \text{l}\_{\ell} \le -\text{len} \\ \frac{-\text{l} \text{l}\_{\ell}}{3/2 \times \text{len}} + \frac{1}{3}, & -\text{len} \le \text{l}\_{\ell} \le 0.5 \times \text{len} \\ 0, & \text{l} \text{l}\_{\ell} \ge 0.5 \text{len} \end{cases} \tag{7}$$

#### *3.2. Step 2: Using Information Entropy to Represent the Complexity of Historical Fluctuations*

*{Ut*|*t* = 1, 2, 3, ... , *T*} can be fuzzified according to a linguistic set *L* = {*l*1, *l*2, *l*3, *l*4, *l*5}. Specifically, *l*1 = [ *U*min , <sup>−</sup>1.5×*len*), *l*2 = [−1.5×*len* , <sup>−</sup>0.5×*len*), *l*3 = [−0.5×*len* , 0.5×*len*), *l*4 = [0.5×*len* , 1.5×*len*), and *l*5 = [1.5 × *len* , *<sup>U</sup>*max). The conversion rule for the indeterminacy-membership *IXt* is defined as follows:

$$I\_{\mathcal{X}\_l} = -\sum\_{n=1}^{\mathcal{S}} p\_{\mathcal{X}\_l}(L\_n) \log\_2(p\_{\mathcal{X}\_l}(L\_n)) \tag{8}$$

where *g* = *5*, *pXt*(*Ln*) indicates the probability of occurrence of the label *ln* in the past *m* days.

## *3.3. Step 3: Establishing Logical Relationships for Training Data*

According to Definition 5, NLRs were established as a training dataset.

#### *3.4. Step 4: Calculating the Similarities between Current Data and Training Data*

According to Definition 6, similarities between current data and training data were calculated. Let *t* be the current data of the point. *SXt*,*<sup>j</sup>* is the similarity of NFTS between the current point *t* and training data *j*.

#### *3.5. Step 5: Forecasting Neutrosophic Value Using the Aggregation Operator*

According to Definition 7, the future neutrosophic fluctuation number *Xt*+<sup>1</sup> can be generated based on the training dataset and the similarities with *Xt*. In order to eliminate very low similarity data, valid NLRs satisfy *SXt*,*<sup>j</sup>*≥ *w*- .

#### *3.6. Step 6: Deneutrosophication for the Neutrosophic Fluctuation Set and Calculating the Forecasted Value*

Calculating the expected value of the forecasted neutrosophic set *Xt*+1, the forecasted fluctuation value can be calculated by:

$$V\_{t+1}' = (T\_{X\_{t+1}} - F\_{X\_{t+1}}) \times len + \mathbf{V}\_t \tag{9}$$
