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

Gong, Zengtai; Lei, Wenjing; Liu, Kun; Qin, Na

doi:https://doi.org/10.1155/2020/3013648

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 3013648 | https://doi.org/10.1155/2020/3013648

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

Zengtai Gong,^1,2Wenjing Lei,³Kun Liu,⁴and Na Qin²

Academic Editor: Alberto Fiorenza

Received12 Oct 2019

Revised02 Jan 2020

Accepted24 Jan 2020

Published30 Mar 2020

Abstract

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.

1. Introduction

The 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. The 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 [4–7], information fusion [8–10], 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]. That 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 [15–17] 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. This 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. The 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.

The 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. The paper ends with conclusion in Section 4.

2. Preliminaries

In this section, some basic notations and concepts of HFLTS and DTRS are briefly reviewed. Throughout 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 σ− algebra on the X. A set function μ is referred to as a regular fuzzy measure if(1)μ (∅) = 0(2)μ (X) = 1(3)For every A and such that A ⊆ B, μ (A) ≤ μ (B)

Definition 2. (see [1, 19, 20]). _λ is called a fuzzy measure based on σ − λ rules if it satisfieswhere , , and A_i ∩ A_j = ∅ for all i, j = 1, 2, …and i ≠ j.
Particularly, if λ = 0, then _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 _λ. The 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 _λ be a fuzzy measure satisfying σ − λ rules. Denoting be real-valued function, and then, the Choquet integral of f on A is defined as follows [1]:where A_i = {x_i, x_i+1, …, x_m}, i = 1, 2, …, m, and f (x₁) ≤ f (x₂) ≤ f (x_m), i = 1, 2, …, m.
Let _λ ({x_i}) = , i = 1, 2, …, m; then, _λ (A_i) 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 x₀ ∈ R exists such that u_Ã (x₀) = 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(4) is compact, where Ā denotes the closure of AThen, Ã is called a fuzzy number. We use Ẽ to denote the fuzzy number space [21].
It is clear that each x ∈ R can be considered as a fuzzy number Ã defined byGiven any two fuzzy numbers Ã₁, Ã₂, k, k₁¬k₂ ≥ 0, the operational rules are as follows:(1)k (Ã₁ + Ã₂) = kÃ₁ + kÃ₂(2)k₁ (k₂Ã₁) = (k₁k₂) Ã₁(3)

Lemma 1 (see [21–23]). For a fuzzy set Ã, it satisfies the following equation:where r denotes the fuzzy set whose membership function is a constant function r.

Let ; the addition and scalar conduct are defined byrespectively, where [Ã]^r = {x: u_Ã (x) ≥ r} = [A⁻(r), A⁺(r)], for any r ∈ (0, 1].

Lemma 2 (see [21–23]). If Ã ∈ Ẽ, then(1)[Ã]^r is a nonempty bounded closed interval for any r ∈ (0, 1].(2) where 0 ≤ r₁ ≤ r₂ ≤ 1.(3)If r_n > 0 and {r_n} converge increasingly to r ∈ (0, 1], then

Conversely, if for any r ∈ [0, 1], there exists satisfying (1)–(3), then there exists a unique Ã ∈ Ẽ such that [Ã]^r = A^r, r ∈ (0, 1] and .

Definition 3. (see [24]). A triangle fuzzy number Ã is a fuzzy number with piecewise linear membership function Ã defined bywhich can be indicated as a triplet (a_l, a_m, a_n).
Given any two triangle fuzzy numbers and k ≥ 0, the operational rules are as follows:(1)(2)

Definition 4. (see [18]). Given a nonnegative matrix P = [p_ij] and a fuzzy-number vector , if and (the ^T denotes the conjugate transpose of a vector or a matrix.), then the product of P and X is defined as follows:

3. 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]). Let , (t₁, t₂, …, t_m) ∈ R^m, and _λ be fuzzy measures satisfying δ − λ rules. Denote A_i = {t_i, t_i+1, …, t_m}, i = 1, 2, …, m, and A_m+1 = ∅. Then, the weighted moving averages for fuzzy numbers based on a nonadditive measure with σ − λ rules is defined as follows:where n > m.

Definition 6. Let , (t₁, t₂, …, t_m) ∈ R^m, and _λ be fuzzy measures satisfying δ − λ rules. Let A_i = {t_i, t_i+1, …, t_m}, i = 1, 2, …, m, and A_m+1 = ∅. Then, for fuzzy number , the Choquet integral of with respect to fuzzy measure _λ on A is defined as follows:Similarly, for vector , the Choquet integral of with respect to fuzzy measure _λ on A is defined as follows:

Remark 4. Accordingly, if is a triangle fuzzy number, then the Choquet integral of fuzzy number with respect to fuzzy measure _λ on A is defined as follows:where .

Theorem 1. Let , (t₁, t₂, …, t_m) ∈ R^m, and _λ be fuzzy measures satisfying δ − λ rules. Denote A_i = {t_i, t_i+1, …, t_m}, i = 1, 2, …, m, A_m+1 = ∅, and t be a positive real number. Then, for vector andwe have(1)(2)(3) especially, if _λ (A₁) − _λ (A₂) > 0 and n − t > m, then(4)If gcd {i ∈{1, 2, …, m}: _λ (A_i) − _λ (A_i+1) > 0} = 1, then exists and where and e_k is the ith standard unit column vector:

Proof. (1)According to Definition 6, we know that Thus, we have(2)According to Definition 6, we can obtain Then, by the expression of in (1), we have(3)By (2), we know that Since P is an invertible matrix, we have(4)By using Theorem 2 in Reference [18], we note that exists and Combining (3), it follows that Take limit of the above equation, we obtainThe proof is complete.

Definition 7. For vector andthe Choquet integral of with respect to fuzzy measure _λ on A is defined as follows:Also, the Choquet integral of with respect to fuzzy measure _λ on A is defined by

Theorem 2. Let , (t₁, t₂, …, t_m) ∈ R^m, and _λ be the 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 the positive real number. Then, for vectorwhere P is the same matrix in Theorem 1, we have(1)(2)(3) especially, if _λ (A₁) − _λ (A₂) > 0 and n − t > m, then(4)If gcd {i ∈ {1, 2, …, m}: _λ (A_i) − _λ (A_i+1) > 0} = 1, then exists and where and e_k is the ith standard unit column vector:

Proof. (1)According to Definition 6, we know that Furthermore, Thus, we have(2)According to Definition 7, we can obtain Then, by the expression of in (1), we have (3) By (2), we know that Since P is an invertible matrix, we have(4)By using Theorem 2 in Reference [18], we note that exists and Combining (3), it follows that Taking limit of the above equation, we obtainThe proof is complete.

Theorem 3. Let , (t₁, t₂, …, t_m) ∈ R^m, and _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. For vectorwe have(1)(2)(3) especially, if _λ (A₁) − _λ (A₂) > 0 and n − t > m, then(4)If gcd {i ∈ {1, 2, …, m}: _λ (A_i) − _λ (A_i+1) > 0} = 1, then exists and where and e_k is the ith standard unit column vector:

Proof. Theorem 1 implies.

Theorem 4. Let , (t₁, t₂, …, t_m) ∈ R^m, and _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 is a triangle fuzzy number, and , i = 1, 2, …, then forwe have(1)(2)(3)

Proof. (1)By Remark 4, we know that the Choquet integral of fuzzy number with respect to fuzzy measure _λ on A of is where . Combining Theorem 1 (1), we can obtain the above equation.(2)According to Theorem 3 in Reference [18], we know that if is a triangle fuzzy number and , i = 1, 2, …, then and by Remark 4 and Theorem 3, we have(3)(2) implies.The 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 , , and every is a triangle fuzzy number, , i = 1, 2, …, 5. Suppose (t₁, t₂, …, t₅) ∈ R⁵, A_i = {t_i, t_i+1, …, t₅}, i = 1, 2, …, 5, and A₆ = ∅. The value and weight of each , i = 1, 2, …, 5, are shown in Table 1. Then, we can obtain the closing stock price over 10 days and some relevant results.
According to Remark 3 in Reference [18], we can obtainBy Definition 6 and Remark 4, the Choquet integral of with respect to fuzzy measure _λ on A is determined as follows:Similarly, we can also calculate the Choquet integral of , with respect to fuzzy measure _λ on A, as shown in Table 2.
And according to Definition 6 and Theorem 4, the Choquet integral of fuzzy number vector with respect to fuzzy measure _λ on A is determined as follows:This 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]. That 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. These 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. Therefore, 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.

4. 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

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (61763044).

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Copyright © 2020 Zengtai Gong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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