**2. Preliminaries**

## *2.1. Fluctuation Time Series*

**Definition 1.** *Let {Vt*|*t* = *1, 2,* ... *, T} be a stock time series, where T is the number of observations. Then, {Ut*|*t* = *2, 3,* ... *, T} is called a fluctuation time series, where Ut* = *Vt* − *Vt*−*<sup>1</sup> (t* = *2, 3,* ... *, T).*

## *2.2. Information Entropy of the mth-Order Fluctuation in a Time Series*

Information entropy (IE) [27] was proposed as a measurement of event uncertainty. The amount of information can be expressed as a function of event occurrence probability. The general formula for information entropy is:

$$E = -\sum\_{t=1}^{N} p(\mathbf{x}\_t) \log \mathbf{2} \left( p(\mathbf{x}\_t) \right) \tag{1}$$

where *p(*·*)* is the probability function of a set of *N* events. In addition, the information entropy must satisfy the following conditions: *N t*=1 *p*(*xt*) = 1, 0 < *p*(*xt*) < 1. The information entropy is always positive.

According to the fuzzy set definition by Zadeh [28], each number in a time series can be fuzzified by its membership function of a fuzzy set *L* = *L*1, *L*2, ... , *Lg*, which can be regarded as an event in a time series. For example, when *g* = 5, it might represent a set of linguistic event variants as: *L* = {*L*1, *L*2, *L*3, *L*4, *L*5} = {*very low*, *low*, *equal*, *high*, *very high*}, etc.

**Definition 2.** *Let F(t* − *1), F(t* − *2),* ... *, F(t* − *m) be fuzzy sets of the mth-order fluctuation time series {Ut*|*t* = *m* + *1, m* + *2,* ... *, T}. Let pUt(L1), pUt (L2), pUt (L3), pUt (L4), and pUt(L5) be the probabilities of the occurrence of the linguistic variants L1, L2, L3, L4, and L5 for F(t* − *1), F(t* − *2),* ... *, F(t* − *m). The information entropy of the mth-order fluctuation is defined as:*

$$E(\mathcal{U}\_t) = -\sum\_{n=1}^{\mathcal{S}} p\_{\mathcal{U}t}(L\_n) \log\_2(p\_{\mathcal{U}t}(L\_n)) \tag{2}$$

*where g* = *5, E*(*Ut*) *is the information entropy of the mth-order fluctuation at point t in the fluctuation time series {Ut*|*t* = *m* + *1, m* + *2,* ... *, T}.*

## *2.3. Neutrosophic Fluctuation Time Series*

**Definition 3.** *(Smarandache [12]) Let W be a space of points (objects), with a generic element in W denoted by w. A neutrosophic set A in W is characterized by a truth-membership function TA(w), am indeterminacy-membership function IA(w), and a falsity-membership function FA(w). The functions TA(w), IA(w), and FA(w) are real*

*standard or nonstandard subsets of ]0*<sup>−</sup>*,1*<sup>+</sup>*[, where* 0− = 0 − ε*,* 1<sup>+</sup> = 1 + ε*,* ε > 0 *is an infinitesimal number. There is no restriction on the sum of TA(w), IA(w), and FA(w).*

**Definition 4.** *Let {Ut*|*t* = *2, 3,* ... *, T} be a fluctuation time series of a stock time series as defined in Definition 1. A number Ut in U is characterized by an upward-trend function T(Ut), a fluctuation-inconsistency function I(Ut), and downward-trend function F(Ut), which can be correspondingly mapped to the truth-membership, indeterminacy-membership, and falsity-membership dimension of a neutrosophic set, respectively. The upwardtrend function T(Ut) and downward-function F(Ut) are defined according to the number Ut shown as follows:*

$$T(\mathcal{U}\_{l}) = \begin{cases} 0, & \mathcal{U}\_{l} \le m\_{1} \\ f\_{1}(\mathcal{U}\_{l}, m\_{1}, m\_{2}), & m\_{1} \le \mathcal{U}\_{l} \le m\_{2} \\ 1, & \text{otherwise} \end{cases} \qquad F(\mathcal{U}\_{l}) = \begin{cases} 1, & \mathcal{U}\_{l} \le o\_{1} \\ f\_{2}(\mathcal{U}\_{l}, o\_{1}, o\_{2}), & o\_{1} \le \mathcal{U}\_{l} \le o\_{2} \\ 1, & \text{otherwise} \end{cases} \tag{3}$$

*where mjand oj(j* = *1, 2) are parameters according to the fluctuation time series.*

The fluctuation-inconsistency function *I*(*Ut*) can be represented by the information entropy *E*(*Ut*) as defined in Equation (2).

Thus, a fluctuation time series {*Ut*|*t* = 1, 2, 3, ... , *T*} can be represented by a neutrosophic fluctuation time series {*Xt*|*t* = *m* + 1, *m* + 2, ... , *T*}, where *Xt* = (*T*(*Ut*), *I*(*Ut*), *F*(*Ut*)) is a neutrosophic set.

## *2.4. Neutrosophic Logical Relationship*

**Definition 5.** *Let {Xt*|*t* = *1, 2, 3,* ... *, T} be a fluctuation time series. If there exists a relation R(t, t* + *1), such that:*

$$X\_{t+1} = X\_t \diamond R(t, t+1) \tag{4}$$

*where* ◦ *is a max–min composition operator, Xt*+*<sup>1</sup> is said to be derived from Xt, denoted by the neutrosophic logical relationship (NLR) Xt* → *Xt*+*1. Xt and Xt*+*<sup>1</sup> are called the left-hand side (LHS) and the right-hand side (RHS) of the NLR, respectively. Xt*+*<sup>1</sup> can also represented by Dt. Therefore, Xt* → *Xt*+*<sup>1</sup> can also be represented by Xt* → *Dt.*

The Jaccard index, also known as the Jaccard similarity coefficient, is used to compare similarities and differences between finite sample sets [29]. The larger the Jaccard similarity value, the higher the similarity.

**Definition 6.** *Let Xt, Xj be two NSs. The Jaccard similarity between Xt and Xj in vector space can be expressed as follows:*

$$f\left(\mathbf{X}\_{l\prime},\mathbf{X}\_{\mathbf{j}}\right) = \frac{T\_{\mathbf{X}\_l}T\_{\mathbf{X}\_{\mathbf{j}}} + I\_{\mathbf{X}\_l}I\_{\mathbf{X}\_{\mathbf{j}}} + F\_{\mathbf{X}\_l}F\_{\mathbf{X}\_{\mathbf{j}}}}{\left(T\_{\mathbf{X}\_l}\right)^2 + \left(I\_{\mathbf{X}\_l}\right)^2 + \left(F\_{\mathbf{X}\_l}\right)^2 + \left(I\_{\mathbf{X}\_{\mathbf{j}}}\right)^2 + \left(F\_{\mathbf{X}\_{\mathbf{j}}}\right)^2 - \left(T\_{\mathbf{X}\_l}T\_{\mathbf{X}\_{\mathbf{j}}} + I\_{\mathbf{X}\_l}I\_{\mathbf{X}\_{\mathbf{j}}} + F\_{\mathbf{X}\_l}F\_{\mathbf{X}\_{\mathbf{j}}}\right)}}\tag{5}$$

## *2.5. Aggregation Operator for NLRs*

**Definition 7.** *Let X* = {*X*1, *X*2, ... , *Xt*, ... , *Xn*}*, D* = {*D*1, *D*2, ... , *Dt*, ... , *Dn*} *be the LHSs and RHSs of a group of NLRs, respectively. The Jaccard similarities between Xt (t* = *1, 2,* ... *, n) and Xj are SXi*,*<sup>j</sup> (i* = *1, 2,* ... *, n), respectively. The corresponding Dj can be calculated by an aggregation operator [30] as:*

$$T\_{D\_j} = \frac{\sum\_{t=1}^{n} S\_{X\_{t,j}} \times T\_{D\_t}}{\sum\_{t=1}^{n} S\_{X\_{t,j}}},\\ I\_{D\_j} = \frac{\sum\_{t=1}^{n} S\_{X\_{t,j}} \times I\_{D\_t}}{\sum\_{t=1}^{n} S\_{X\_{t,j}}},\tag{6}$$

*According to the definition of NLR, Dj can be represented by Xj*+*1*.
