Weighted Moving Averages for a Series of Fuzzy Numbers Based on Nonadditive Measures with σ − λ Rules and Choquet Integral of Fuzzy-Number-Valued Function

. The aim of this study is to generalize moving average by means of Choquet integral. First, by employing nonadditive measures with δ − λ rules, the calculation of the moving average for a series of fuzzy numbers can be transformed into Choquet integration of fuzzy-number-valued function under discrete case. Meanwhile, the Choquet integral of fuzzy number and Choquet integral of fuzzy number vector are deﬁned. Finally, some properties are investigated by means of convolution formula of Choquet integral. It shows that the results obtained in this paper extend the previous conclusions.


Introduction
e concept of nonadditive measures was originally proposed by Sugeno [1]. It replaces additivity in classical additive measures with monotonicity and can be regarded as an extension of classical additive measures. Indeed, nonadditive measures can be used to describe interdependent or interactive characteristics of information in practical applications. e Choquet integral, initiated by Choquet [2], provides a mechanism to integrate function on the basis of nonadditive measures and is a powerful technique to address interdependence and interaction among information. In fact, the Choquet integral [2] with respect to nonadditive measures has successful application in pattern recognition [3], decision-making [4][5][6][7], information fusion [8][9][10], economic theory [11], and so on.
Another key mathematical structure to cope with imperfect or imprecise information is a fuzzy set, developed by Zadeh [12]. Fuzzy numbers [13], a specific format of fuzzy sets, are utilized to express values in practical situation where the exact values may not be determined because of lack or imperfection of information [14].
at is, fuzzy numbers take into account the fact that all phenomena in the physical universe have a degree of inherent uncertainty and have been used as a way of modeling uncertain and incomplete systems. Fuzzy numbers have been investigated intensively by research studies [15][16][17] from various aspects since it was introduced.
Motivated by the ability of Choquet integral with respect to nonadditive measures in handling interaction among information and the merit of fuzzy number in depicting uncertainty, it is of both theoretical and practical importance to combine them together and apply the combination to moving average. In this work, we want to give more insight into issues connected with the weighted moving averages for a series of fuzzy numbers based on nonadditive measures with σ − λ rules by the new tools, Choquet integral and fuzzy number. is is a new contribution to our previous work [18], in which the moving average for a series of fuzzy numbers based on nonadditive measures with σ − λ rules is proposed and discussed. e aim of this paper is to show that the calculation of the moving average for a series of fuzzy numbers can be transformed into Choquet integration of fuzzy-number-valued function under discrete case. Meanwhile, the Choquet integral of fuzzy number and Choquet integral of fuzzy number vector are defined. Finally, some properties are investigated by means of the convolution formula of Choquet integral. e structure of this paper is as follows. In Section 2, we review some basic concepts and properties about nonadditive measure with σ − λ rules and fuzzy numbers. And the definition of product between a nonnegative matrix and fuzzy number vector is given to make our analysis possible.
In Section 3, it shows that the calculation of the moving average for a series of fuzzy numbers can be transformed into Choquet integration of fuzzy-number-valued function under discrete case. Meanwhile, the Choquet integral of fuzzy number and Choquet integral of fuzzy number vector are defined and their properties are investigated by means of the convolution formula of Choquet integral. e paper ends with conclusion in Section 4.

Preliminaries
In this section, some basic notations and concepts of HFLTS and DTRS are briefly reviewed. roughout this study, R m denotes the m-dimension real Euclidean space and R + � (0, ∞).
Definition 1 (see [1,19,20]). Let X denote a nonempty set and A, a σ− algebra on the X. A set function μ is referred to as a regular fuzzy measure if Definition 2 (see [1,19,20]). g λ is called a fuzzy measure based on σ − λ rules if it satisfies Particularly, if λ � 0, then g λ is a classic probability measure.
A regular fuzzy measure μ is called Sugeno measure based on σ − λ rules if μ satisfies σ − λ rules, briefly denoted as g λ . e fuzzy measure denoted in this paper is Sugeno measure.

Remark 1.
In Definition 2, if n � 2, then Remark 2. If X is a finite set, for any subset A of X, then Remark 3 (see [19]). If X is a finite set, then the parameter λ of a regular Sugeno measure based on σ − λ rules is determined by the following equation: Let g λ be a fuzzy measure satisfying σ − λ rules. Denoting A � x 1 , x 2 , . . . , x m ∈ A, f: A ⟶ R be real-valued function, and then, the Choquet integral of f on A is defined as follows [1]: where is obtained from the following recurrence relation: Let Ã (x) ∈Ẽ, r ∈ (0, 1] and [Ã] r � {x ∈ R: u Ã (X) ≥ r}. Ã satisfies the following: (1) Ã is a normal fuzzy set, i.e., an x 0 ∈ R exists such that u Ã (x 0 ) � 1 (2) Ã is a convex fuzzy set, i.e., u Ã (λx + (1 − λ) y) ≥ min {u Ã (x), u Ã (y)} for any x, y ∈ R and λ ∈ (0, 1] (3) Ã is a upper semicontinuous fuzzy set where � A denotes the closure of A en, Ã is called a fuzzy number. We useẼ to denote the fuzzy number space [21].
It is clear that each x ∈ R can be considered as a fuzzy number Ã defined by Given any two fuzzy numbers Ã 1 , Ã 2 , k, k 1 L¬k 2 ≥ 0, the operational rules are as follows: Lemma 1 (see [21][22][23] where r * denotes the fuzzy set whose membership function is a constant function r.
Let A, B ∈ E and k ∈ R; the addition and scalar conduct are defined by Lemma 2 (see [21][22][23] Definition 3 (see [24]). A triangle fuzzy number Ã is a fuzzy number with piecewise linear membership function Ã defined by x − a l a m − a l , a l ≤ x ≤ a m , a n − x a n − a m , a m < x ≤ a n , 0, otherwise, which can be indicated as a triplet (a l , a m , a n ). Given any two triangle fuzzy numbers 1 ) and x j � (x j − δ j,1 , x j , x j + δ j,1 )) and k ≥ 0, the operational rules are as follows: Definition 4 (see [18]). Given a nonnegative matrix P � [p ij ] and a fuzzy-number vector X, if P ∈ R m×m + and X � [x 1 , x 2 , . . . , x m ] T ∈ E m (the T denotes the conjugate transpose of a vector or a matrix.), then the product of P and X is defined as follows:

Weighted Moving Averages for Fuzzy Numbers Based on a Nonadditive Measure with σ − λ Rules and Choquet Integral of Fuzzy-Number-Valued Function
Definition 5 (see [18]).
en, the weighted moving averages for fuzzy numbers based on a nonadditive measure with σ − λ rules is defined as follows: where n > m.
and g λ be fuzzy measures satisfying δ − λ rules.
. ., m, and A m+1 � ∅. en, for fuzzy number x n (n > m), the Choquet integral of x n (n > m) with respect to fuzzy measure g λ on A is defined as follows: Similarly, for vector X n � [x n , x n+1 , . . . , x n+m− 1 ] T (n > m), the Choquet integral of X n with respect to fuzzy measure g λ on A is defined as follows: Remark 4. Accordingly, if x n is a triangle fuzzy number, then the Choquet integral of fuzzy number x n (n > m) with respect to fuzzy measure g λ on A is defined as follows: where x n � (x n − δ n,1 , x n , x n + δ n,2 ).

Proof
(1) According to Definition 6, we know that us, we have (2) According to Definition 6, we can obtain en, by the expression of (C) A X n+1 dg λ in (1), we have (3) By (2), we know that Since P is an invertible matrix, we have (4) By using eorem 2 in Reference [18], we note that lim n⟶∞ P n exists and Combining (3), it follows that Take limit of the above equation, we obtain e proof is complete.
the Choquet integral of X − n (r) with respect to fuzzy measure g λ on A is defined as follows:

Journal of Function Spaces
Also, the Choquet integral of X − n (r) with respect to fuzzy measure g λ on A is defined by where P is the same matrix in eorem 1, we have (2) (3) especially, if g λ (A 1 ) − g λ (A 2 ) > 0 and n − t > m, then where e � m i�1 e k � [1, 1, . . . , 1] T ∈ R m×1 and e k is the ith standard unit column vector:

Proof
(1) According to Definition 6, we know that Furthermore, us, we have 6 Journal of Function Spaces (2) According to Definition 7, we can obtain en, by the expression of (C) A X − n+1 (r)dg λ in (1), we have (3) By (2), we know that Since P is an invertible matrix, we have (4) By using eorem 2 in Reference [18], we note that lim n⟶∞ P n exists and Combining (3), it follows that Taking limit of the above equation, we obtain e proof is complete.
and g λ be a fuzzy measure satisfying δ − λ rules. Denote A i � {t i , t i+1 , . . ., t m }, i � 1, 2, . . ., m, and A m+1 � ∅, and t be a positive real number. If x i is a triangle fuzzy number, and we have Journal of Function Spaces (3)
e proof is complete.
□ Example 1. We choose the same example in Reference [18] to illustrate our study and make comparison. Given a closing stock price system over 5 days, the closing prices of each day are denoted as x i , (x 1 , x 2 , . . . , x 5 ) ∈ E 5 , and every x i is a triangle fuzzy number, . . ., 5, and A 6 � ∅. e value and weight of each x i , i � 1, 2, . . ., 5, are shown in Table 1. en, we can obtain the closing stock price over 10 days and some relevant results. According to Remark 3 in Reference [18], we can obtain By Definition 6 and Remark 4, the Choquet integral of x 6 with respect to fuzzy measure g λ on A is determined as follows: Similarly, we can also calculate the Choquet integral of x n , n � 7, 8, 9, 10, with respect to fuzzy measure g λ on A, as shown in Table 2.
And according to Definition 6 and eorem 4, the Choquet integral of fuzzy number vector X 6 � [x 6 , x 7 , . . . , x 10 ] T with respect to fuzzy measure g λ on A is determined as follows: Journal of Function Spaces is article is a complement of our previous work [18]; namely, the method presented in this article can be regarded as a generalization of the previous method [18]. at is, the calculation of the moving average for a series of fuzzy numbers in [18] is transformed into Choquet integration of fuzzy-number-valued function under discrete case in this work. More specifically, compared with our previous work in Reference [18], we introduce the new concepts: the Choquet integral of fuzzy number and the Choquet integral of fuzzy number vector, containing m elements needed to make forecasting of the m + 1 th element. ese new concepts provide a possibility to dealing with the moving average from vector integral, which could describe the moving average of time series in a more intuitive perspective using an important mathematical tool.
Meanwhile, when the data degenerate into distinct data and the nonadditive measure degenerates into probability measure, our method will degenerate into the classical moving weighted average method. erefore, this method is the extension of the classical method. In this paper, we consider the mutual influence and connection of time nodes, while in the classical method, time nodes are independent of each other. Moreover, the classical time series cannot deal with problems of natural language assignment, Internet language assignment, qualitative description, etc. So, the advantage of this method is obvious.

Conclusion
In this paper, on the combination of Choquet integral and fuzzy number, the Choquet integral of fuzzy number and Choquet integral of fuzzy number vector are defined. And it shows that the calculation of the moving average for a series of fuzzy numbers can be transformed into Choquet integration of fuzzy-number-valued function under discrete case. Subsequently, the Choquet integral of fuzzy number and Choquet integral of fuzzy number vector are defined, respectively. Finally, by means of the convolution formula of Choquet integral, some properties of the Choquet integral of fuzzy number and Choquet integral of fuzzy number vector are also investigated.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.