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

Authors: Zengtai Gong; Wenjing Lei; Kun Liu; Na Qin
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The aim of this study is to generalize moving average by means of Choquet integral. First, by employing nonadditive measures withδ − 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.


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## Body

## 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 withσ − 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 Section2, 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,Rm 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(1)
μ (∅) = 0(2)
μ (X) = 1(3)
For everyA and B∈A such that A ⊆ B, μ (A) ≤ μ (B)Definition 2.
(see [1, 19, 20]). gλ is called a fuzzy measure based on σ − λ rules if it satisfies(1)gλ∪i=1∞Ai=1λ∏i=1∞1+λgλAi−1,λ≠0,∑i=1∞gλAi,λ=0,where λ∈−1/supμ,∞∪0, Ai⊂A, and Ai ∩ Aj = ∅ for all i, j = 1, 2, …and i ≠ j.
Particularly, ifλ = 0, then gλ is a classic probability measure.
A regular fuzzy measureμ is called Sugeno measure based on σ − λ rules if μ satisfies σ − λ rules, briefly denoted as gλ. The fuzzy measure denoted in this paper is Sugeno measure.Remark 1.
In Definition2, if n = 2, then(2)μA∪B=μA+μB+λμAμB,λ≠0,μA+μB,λ=0.Remark 2.
IfX is a finite set, for any subset A of X, then(3)gλA=1λ∏x∈A1+λgλx−1,λ≠0,∑i=1∞gλx,λ=0.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:(4)∏i=1n1+λgλi=1+λ.
Letgλ be a fuzzy measure satisfying σ − λ rules. Denoting A=x1,x2,…,xm∈A,f:A⟶ℝ be real-valued function, and then, the Choquet integral of f on A is defined as follows [1]:(5)c∫Afdgλ=∑i=1mfxigλAi−gλAi+1,where Ai = {xi, xi+1, …, xm}, i = 1, 2, …, m, and f (x1) ≤ f (x2)  ≤  f (xm), i = 1, 2, …, m.
Letgλ ({xi}) = gi, i = 1, 2, …, m; then, gλ (Ai) is obtained from the following recurrence relation:(6)gλAm=gλxm=gm,gλAi=gλAi+1+λgigλAi+1,1≤i<m.
LetÃ (x) ∈ Ẽ, r ∈ (0, 1] and [Ã]r = {x ∈ R: uÃ (X) ≥ r}. Ã satisfies the following:(1)
Ã is a normal fuzzy set, i.e., an x0 ∈ R exists such that uÃ (x0) = 1(2)
Ã is a convex fuzzy set, i.e., uÃ (λx + (1 − λ) y) ≥ min{uÃ (x), uÃ (y)} for any x, y ∈ R and λ ∈ (0, 1](3)
Ã is a upper semicontinuous fuzzy set(4)
A0=X∈R:uA˜x>0¯=∪r∈0,1Ar¯ is compact, where Ā denotes the closure of A
Then,Ã is called a fuzzy number. We use Ẽ to denote the fuzzy number space [21].
It is clear that eachx ∈ R can be considered as a fuzzy number Ã defined by(7)uA˜x=1,x=A,0,otherwise.
Given any two fuzzy numbersÃ1, Ã2, k, k1L¬k2 ≥ 0, the operational rules are as follows:(1)
k (Ã1 + Ã2) = kÃ1 + kÃ2(2)
k1 (k2Ã1) = (k1k2) Ã1(3)
k1+k2A˜1=k1A˜1+k2A˜1Lemma 1 (see [21–23]).
For a fuzzy setÃ, it satisfies the following equation:(8)A˜=∪r∈0,1r∗∩A˜r,where r∗ denotes the fuzzy set whose membership function is a constant function r.LetA˜,B˜∈E˜andk∈ℝ; the addition and scalar conduct are defined by(9)A˜+B˜r=A˜r+B˜r,kA˜r=kA˜r,respectively, where [Ã]r = {x: uÃ (x) ≥ r} = [A−(r), A+(r)], for any r ∈ (0, 1].Lemma 2 (see [21–23]).
IfÃ ∈ Ẽ, then(1)
[Ã]r is a nonempty bounded closed interval for any r ∈ (0, 1].(2)
A˜r1⊃A˜r2 where 0 ≤ r1 ≤ r2 ≤ 1.(3)
Ifrn > 0 and {rn} converge increasingly to r ∈ (0, 1], then(10)∩n=1∞A˜rn=A˜r.Conversely, if for anyr ∈ [0, 1], there exists Br⊂ℝ satisfying (1)–(3), then there exists a unique Ã ∈ Ẽ such that [Ã]r = Ar, r ∈ (0, 1] and A˜0=∪r∈0,1A˜r¯⊂B0.Definition 3.
(see [24]). A triangle fuzzy number Ã is a fuzzy number with piecewise linear membership function Ã defined by(11)uA˜x=x−alam−al,al≤x≤am,an−xan−am,am<x≤an,0,otherwise,which can be indicated as a triplet (al, am, an).
Given any two triangle fuzzy numbersx˜i=xi−δi,1,xi,xi+δi,1andx˜j=xj−δj,1,xj,xj+δj,1 and k ≥ 0, the operational rules are as follows:(1)
x˜i+x˜j=xi−δi,1+xj−δj,1,xi+xj,xi+δi,1+xj+δj,1(2)
k⋅x˜i=kxi−kδi,1,kxi,kxi+kδi,2Definition 4.
(see [18]). Given a nonnegative matrix P = [pij] and a fuzzy-number vector X˜, if P∈R+m×m and X˜=x˜1,x˜2,…,x˜mT∈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:(12)PX˜n−1=∑j=1mpijx˜j⋮∑j=1mpmjx˜j.


## 3. Weighted Moving Averages for Fuzzy Numbers Based on a Nonadditive Measure withσ − λ Rules and Choquet Integral of Fuzzy-Number-Valued Function

Definition 5.
(see [18]). Let x˜1,x˜2,…,x˜m∈E˜m, (t1, t2, …, tm) ∈ Rm, and gλ be fuzzy measures satisfying δ − λ rules. Denote Ai = {ti, ti+1, …, tm}, i = 1, 2, …, m, and Am+1 = ∅. Then, the weighted moving averages for fuzzy numbers based on a nonadditive measure with σ − λ rules is defined as follows:(13)x˜n=gλA1−gλA2x˜n−m+gλA2−gλA3x˜n−m+1+⋯+gλAm−gλAm+1x˜n−1,where n > m.Definition 6.
Letx˜1,x˜2,…,x˜m∈E˜m, (t1, t2, …, tm) ∈ Rm, and gλ be fuzzy measures satisfying δ − λ rules. Let Ai = {ti, ti+1, …, tm}, i = 1, 2, …, m, and Am+1 = ∅. Then, for fuzzy number x˜nn>m, the Choquet integral of x˜nn>m with respect to fuzzy measure gλ on A is defined as follows:(14)C∫Ax˜ndgλ=∑i=1mx˜n−m+i−1gλAi−gλAi+1.
Similarly, for vectorX˜n=x˜n,x˜n+1,…,x˜n+m−1Tn>m, the Choquet integral of X˜n with respect to fuzzy measure gλ on A is defined as follows:(15)C∫AX˜ndgλ=C∫Ax˜ndgλ,C∫Ax˜n+1dgλ,…,C∫Ax˜n+m−1dgλT.Remark 4.
Accordingly, ifx˜n is a triangle fuzzy number, then the Choquet integral of fuzzy number x˜nn>m with respect to fuzzy measure gλ on A is defined as follows:(16)C∫Ax˜ndgλ=∑i=1mxn−m+i−1−δn−m+i−1,1gλAi−gλAi+1,xn−m+i−1gλAi−gλAi+1,xn−m+i−1+δn−m+i−1,2gλAi−gλAi+1=∑i=1mxn−m+i−1−δn−m+i−1,1gλAi−gλAi+1,∑i=1mxn−m+i−1gλAi−gλAi+1,∑i=1mxn−m+i−1+δn−m+i−1,2gλAi−gλAi+1,where x˜n=xn−δn,1,xn,xn+δn,2.Theorem 1.
Letx1˜,x2˜,…,xm˜∈E˜m, (t1, t2, …, tm) ∈ Rm, and gλ be fuzzy measures satisfying δ − λ rules. Denote Ai = {ti, ti+1, …, tm}, i = 1, 2, …, m, Am+1 = ∅, and t be a positive real number. Then, for vector X˜n=x˜n,x˜n+1,…,x˜n+m−1Tn>m and(17)P=010⋯0001⋯0⋮⋯⋯⋯⋮000⋯1gλA1−gλA2gλA2−gλA3gλA3−gλA4⋯gλAm−gλAm+1,we have(1)(18)C∫AX˜ndgλ=∑i=1mx˜n−m+i−1gλAi−gλAi+1∑i=1mx˜n−m+igλAi−gλAi+1∑i=1mx˜n−m+i+1gλAi−gλAi+1⋯∑i=1mx˜n+i−2gλAi−gλAi+1.(2)(19)C∫AX˜n+1dgλ=P⋅C∫AX˜ndgλ=P2⋅C∫AX˜n−1dgλ=⋯=Pn−m−1⋅C∫AX˜m+2dgλ=Pn−m⋅C∫AX˜m+1dgλ.(3)(20)C∫AX˜n+tdgλ=Pt⋅C∫AX˜ndgλ,especially, ifgλ (A1) − gλ (A2) > 0 and n − t > m, then(21)C∫AX˜n−tdgλ=P−t⋅C∫AX˜ndgλ.(4)
If gcd {i ∈{1, 2, …, m}: gλ (Ai) − gλ (Ai+1) > 0} = 1, then limn⟶∞C∫AX˜ndgλ exists and(22)limn⟶∞C∫AX˜ndgλ=limn⟶∞Pn−1⋅C∫AX˜m+1dgλ=eaTaTe⋅C∫AX˜1dgλ=erT⋅C∫AX˜1dgλ,wheree=∑i=1mek=1,1,…,1T∈ℝm×1 and ek is the ith standard unit column vector:(23)a=a1,a2,…,amT,b=b1,b2,…,bmT,ak=∑i=1kgλAi−gλAi+1,bk=aTekaTe=ak∑i=1mai=gλA1−gλAk+1mgλA1−∑i=2mgλAi,k=1,2,3,…,m.Proof.
(1)
According to Definition6, we know that
(24)
C
∫
A
x
˜
n
d
g
λ
=
∑
i
=
1
m
x
˜
n
−
m
+
i
−
1
g
λ
A
i
−
g
λ
A
i
+
1
,
C
∫
A
x
˜
n
+
1
d
g
λ
=
∑
i
=
1
m
x
˜
n
−
m
+
i
g
λ
A
i
−
g
λ
A
i
+
1
,
…
,
C
∫
A
x
˜
n
+
m
−
1
d
g
λ
=
∑
i
=
1
m
x
˜
n
+
i
−
2
g
λ
A
i
−
g
λ
A
i
+
1
,
C
∫
A
X
˜
n
d
g
λ
=
C
∫
A
x
˜
n
d
g
λ
,
C
∫
A
x
˜
n
+
1
d
g
λ
,
…
,
C
∫
A
x
˜
n
+
m
−
1
d
g
λ
T
.
Thus, we have
(25)
C
∫
A
X
˜
n
d
g
λ
=
∑
i
=
1
m
x
˜
n
−
m
+
i
−
1
g
λ
A
i
−
g
λ
A
i
+
1
∑
i
=
1
m
x
˜
n
−
m
+
i
g
λ
A
i
−
g
λ
A
i
+
1
∑
i
=
1
m
x
˜
n
−
m
+
i
+
1
g
λ
A
i
−
g
λ
A
i
+
1
⋯
∑
i
=
1
m
x
˜
n
+
i
−
2
g
λ
A
i
−
g
λ
A
i
+
1
.
(2)
According to Definition6, we can obtain
(26)
P
⋅
C
∫
A
X
˜
n
d
g
λ
=
∑
i
=
1
m
x
˜
n
−
m
+
i
g
λ
A
i
−
g
λ
A
i
+
1
∑
i
=
1
m
x
˜
n
−
m
+
i
+
1
g
λ
A
i
−
g
λ
A
i
+
1
∑
i
=
1
m
x
˜
n
−
m
+
i
+
2
g
λ
A
i
−
g
λ
A
i
+
1
⋯
∑
i
=
1
m
x
˜
n
+
i
−
1
g
λ
A
i
−
g
λ
A
i
+
1
.
Then, by the expression ofC∫AX˜n+1dgλ in (1), we have
(27)
C
∫
A
x
˜
n
+
1
d
g
λ
=
P
⋅
C
∫
A
X
˜
n
d
g
λ
.
(3)
By (2), we know that
(28)
C
∫
A
X
˜
n
+
t
d
g
λ
=
P
t
⋅
C
∫
A
X
˜
n
d
g
λ
.
SinceP is an invertible matrix, we have
(29)
C
∫
A
X
˜
n
−
t
d
g
λ
=
P
−
t
⋅
C
∫
A
X
˜
n
d
g
λ
.
(4)
By using Theorem2 in Reference [18], we note that limn⟶∞Pn exists and
(30)
lim
n
⟶
∞
P
n
=
e
a
T
a
T
e
=
e
r
T
.
Combining (3), it follows that
(31)
C
∫
A
X
˜
n
+
t
d
g
λ
=
P
t
⋅
C
∫
A
X
˜
n
d
g
λ
.
Take limit of the above equation, we obtain
(32)
lim
n
⟶
∞
C
∫
A
X
˜
n
d
g
λ
=
lim
n
⟶
∞
P
n
−
1
⋅
C
∫
A
X
˜
1
d
g
λ
=
e
a
T
a
T
e
⋅
C
∫
A
X
˜
1
d
g
λ
=
e
r
T
⋅
C
∫
A
X
˜
1
d
g
λ
.
The proof is complete.Definition 7.
For vectorX˜n=x˜n,x˜n+1,…,x˜n+m−1Tn>m and(33)X˜n−r=x˜n−r,x˜n+1−r,…,x˜n+m−1−rT,X˜n+r=x˜n+r,x˜n+1+r,…,x˜n+m−1+rT,the Choquet integral of X˜n−r with respect to fuzzy measure gλ on A is defined as follows:(34)C∫AX˜n−rdgλ=C∫Ax˜n−rdgλ,C∫Ax˜n+1−rdgλ,…,C∫Ax˜n+m−1−rdgλT.
Also, the Choquet integral ofX˜n−r with respect to fuzzy measure gλ on A is defined by(35)C∫AX˜n+rdgλ=C∫Ax˜n+rdgλ,C∫Ax˜n+1+rdgλ,…,C∫Ax˜n+m−1+rdgλT.Theorem 2.
Letx˜1,x˜2,…,x˜m∈E˜m, (t1, t2, …, tm) ∈ Rm, and gλ be the fuzzy measure satisfying δ − λ rules. Denote Ai = {ti, ti+1, …, tm}, i = 1, 2, …, m, and Am+1 = ∅, and t be the positive real number. Then, for vector(36)X˜n−r=x˜n−r,x˜n+1−r,…,x˜n+m−1−rT,where P is the same matrix in Theorem 1, we have(1)(37)C∫AX˜n−rdgλ=∑i=1mx˜n−m+i−1−rgλAi−gλAi+1∑i=1mx˜n−m+i−rgλAi−gλAi+1∑i=1mx˜n−m+i+1−rgλAi−gλAi+1⋯∑i=1mx˜n+i−2−rgλAi−gλAi+1.(2)(38)C∫AX˜n+1−rdgλ=P⋅C∫AX˜n−rdgλ=P2⋅C∫AX˜n−1−rdgλ=⋯=Pn−m−1⋅C∫AX˜m+2−rdgλ=Pn−m⋅C∫AX˜m+1−rdgλ.(3)(39)C∫AX˜n+t−rdgλ=Pt⋅C∫AX˜n−rdgλ,especially, ifgλ (A1) − gλ (A2) > 0 and n − t > m, then(40)C∫AX˜n−t−rdgλ=P−t⋅C∫AX˜n−rdgλ.(4)
If gcd {i ∈ {1, 2, …, m}: gλ (Ai) − gλ (Ai+1) > 0} = 1, then limn⟶∞C∫AX˜n−rdgλ exists and(41)limn⟶∞C∫AX˜n−rdgλ=limn⟶∞Pn−1⋅C∫AX˜1−rdgλ=eaTaTe⋅C∫AX˜1−rdgλ=ebT⋅C∫AX˜1−rdgλ,wheree=∑i=1mek=1,1,…,1T∈Rm×1 and ek is the ith standard unit column vector:(42)a=a1,a2,…,amT,b=b1,b2,…,bmT,ak=∑i=1kgλAi−gλAi+1,bk=aTekaTe=ak∑i=1mai=gλA1−gλAk+1mgλA1−∑i=2mgλAi,k=1,2,3,…,m.Proof.
(1)
According to Definition6, we know that
(43)
C
∫
A
x
˜
n
−
r
d
g
λ
=
∑
i
=
1
m
x
˜
n
−
m
+
i
−
1
−
r
g
λ
A
i
−
g
λ
A
i
+
1
,
C
∫
A
x
˜
n
+
1
−
r
d
g
λ
=
∑
i
=
1
m
x
˜
n
−
m
+
i
−
r
g
λ
A
i
−
g
λ
A
i
+
1
,
…
,
C
∫
A
x
˜
n
+
m
−
1
−
r
d
g
λ
=
∑
i
=
1
m
x
˜
n
+
i
−
2
−
r
g
λ
A
i
−
g
λ
A
i
+
1
.
Furthermore,
(44)
C
∫
A
X
˜
n
−
r
d
g
λ
=
C
∫
A
x
˜
n
−
r
d
g
λ
,
C
∫
A
x
˜
n
+
1
−
r
d
g
λ
,
…
,
C
∫
A
x
˜
n
+
m
−
1
−
r
d
g
λ
T
.
Thus, we have
(45)
C
∫
A
X
˜
n
−
r
d
g
λ
=
∑
i
=
1
m
x
˜
n
−
m
+
i
−
1
−
r
g
λ
A
i
−
g
λ
A
i
+
1
∑
i
=
1
m
x
˜
n
−
m
+
i
−
r
g
λ
A
i
−
g
λ
A
i
+
1
∑
i
=
1
m
x
˜
n
−
m
+
i
+
1
−
r
g
λ
A
i
−
g
λ
A
i
+
1
⋯
∑
i
=
1
m
x
˜
n
+
i
−
2
−
r
g
λ
A
i
−
g
λ
A
i
+
1
.
(2)
According to Definition7, we can obtain
(46)
P
⋅
C
∫
A
X
˜
n
−
r
d
g
λ
=
∑
i
=
1
m
x
˜
n
−
m
+
i
−
r
g
λ
A
i
−
g
λ
A
i
+
1
∑
i
=
1
m
x
˜
n
−
m
+
i
+
1
−
r
g
λ
A
i
−
g
λ
A
i
+
1
∑
i
=
1
m
x
˜
n
−
m
+
i
+
1
+
2
−
r
g
λ
A
i
−
g
λ
A
i
+
1
⋯
∑
i
=
1
m
x
˜
n
+
i
−
1
−
r
g
λ
A
i
−
g
λ
A
i
+
1
.
Then, by the expression ofC∫AX˜n+1−rdgλ in (1), we have
(47)
C
∫
A
X
˜
n
+
1
−
r
d
g
λ
=
P
⋅
C
∫
A
X
˜
n
−
r
d
g
λ
.
(3) By (2), we know that
(48)
C
∫
A
X
˜
n
+
t
−
r
d
g
λ
=
P
t
⋅
C
∫
A
X
˜
n
−
r
d
g
λ
.
SinceP is an invertible matrix, we have
(49)
C
∫
A
X
˜
n
−
t
−
r
d
g
λ
=
P
−
t
⋅
C
∫
A
X
˜
n
−
r
d
g
λ
.
(4)
By using Theorem2 in Reference [18], we note that limn⟶∞Pn exists and
(50)
lim
n
⟶
∞
P
n
=
e
a
T
a
T
e
=
e
r
T
.
Combining (3), it follows that
(51)
C
∫
A
X
˜
n
+
t
−
r
d
g
λ
=
P
t
⋅
C
∫
A
X
˜
n
−
r
d
g
λ
.
Taking limit of the above equation, we obtain
(52)
lim
n
⟶
∞
C
∫
A
X
˜
n
−
r
d
g
λ
=
lim
n
⟶
∞
P
n
−
1
⋅
C
∫
A
X
˜
1
−
r
d
g
λ
=
e
a
T
a
T
e
⋅
C
∫
A
X
˜
1
−
r
d
g
λ
=
e
b
T
⋅
C
∫
A
X
˜
1
−
r
d
g
λ
.
The proof is complete.Theorem 3.
Letx˜1,x˜2,…,x˜m∈E˜m, (t1, t2, …, tm) ∈ Rm, and gλ be a fuzzy measure satisfying δ − λ rules. Denote Ai = {ti, ti+1, …, tm}, i = 1, 2, …, m, and Am+1 = ∅, and t be a positive real number. For vector(53)X˜n+r=x˜n+r,x˜n+1+r,…,x˜n+m−1+rT,we have(1)(54)C∫AX˜n+rdgλ=∑i=1mx˜n−m+i−1+rgλAi−gλAi+1∑i=1mx˜n−m+i+rgλAi−gλAi+1∑i=1mx˜n−m+i+1+rgλAi−gλAi+1⋯∑i=1mx˜n+i−2+rgλAi−gλAi+1.(2)(55)C∫AX˜n+1−rdgλ=P⋅C∫AX˜n−rdgλ=P2⋅C∫AX˜n−1+rdgλ=⋯=Pn−m−1⋅C∫AX˜m+2+rdgλ=Pn−m⋅C∫AX˜m+1+rdgλ.(3)(56)C∫AX˜n+t+rdgλ=Pt⋅C∫AX˜n+rdgλ,especially, ifgλ (A1) − gλ (A2) > 0 and n − t > m, then(57)C∫AX˜n−t+rdgλ=P−t⋅C∫AX˜n+rdgλ.(4)
If gcd {i ∈ {1, 2, …, m}: gλ (Ai) − gλ (Ai+1) > 0} = 1, then limn⟶∞C∫AX˜n+rdgλ exists and(58)limn⟶∞C∫AX˜n+rdgλ=limn⟶∞Pn⋅C∫AX˜m+1+rdgλ=eaTaTe⋅C∫AX˜m+1+rdgλ=ebT⋅C∫AX˜m+1+rdgλ.wheree=∑i=1mek=1,1,…,1T∈ℝm×1 and ek is the ith standard unit column vector:(59)a=a1,a2,…,amT,b=b1,b2,…,bmT,ak=∑i=1kgλAi−gλAi+1,bk=aTekaTe=ak∑i=1mai=gλA1−gλAk+1mgλA1−∑i=2mgλAi,k=1,2,3,…,m.Proof.
Theorem1 implies.Theorem 4.
Letx˜1,x˜2,…,x˜m∈E˜m, (t1, t2, …, tm) ∈ Rm, and gλ be a fuzzy measure satisfying δ − λ rules. Denote Ai = {ti, ti+1, …, tm}, i = 1, 2, …, m, and Am+1 = ∅, and t be a positive real number. If x˜i is a triangle fuzzy number, and xi˜=xi−δi,1,xi,xi+δi,2, i = 1, 2, …, then for(60)X˜n=x˜n,x˜n+1,…,x˜n+m−1T,we have(1)(61)C∫AX˜ndgλ=∑i=1mxn−m+i−1−δn−m+i−1,1gλAi−gλAi+1,∑i=1mxn−m+i−1gλAi−gλAi+1,∑i=1mxn−m+i−1+δn−m+i−1,2gλAi−gλAi+1,∑i=1mxn−m+i−δn−m+i,1gλAi−gλAi+1,∑i=1mxn−m+igλAi−gλAi+1,∑i=1mxn−m+i+δn−m+i,2gλAi−gλAi+1,…,∑i=1mxn+i−2−δn+i−2,1gλAi−gλAi+1,∑i=1mxn+i−2gλAi−gλAi+1,∑i=1mxn+i−2+δn+i−2,2gλAi−gλAi+1T.(2)(62)C∫AX˜n−rdgλ=∑i=1mδn−m+i−1,1r+xn−m+i−1−δn−m+i−1,1∑i=1mδn−m+i,1r+xn−m+i−δn−m+i,1∑i=1mδn−m+i+1,1r+xn−m+i+1−δn−m+i+1,1⋯∑i=1mδn+i−2,1r+xn+i−2−δn+i−2,1.(3)(63)C∫AX˜n+rdgλ=∑i=1m−δn−m+i−1,2r+xn−m+i−1+δn−m+i−1,2∑i=1m−δn−m+i,2r+xn−m+i+δn−m+i,2∑i=1m−δn−m+i+1,2r+xn−m+i+1+δn−m+i+1,2⋯∑i=1m−δn+i−2,2r+xn+i−2+δn+i−2,2.Proof.
(1)
By Remark4, we know that the Choquet integral of fuzzy number x˜nn>m with respect to fuzzy measure gλ on A of is(64)C∫Ax˜ndgλ=∑i=1mxn−m+i−1−δn−m+i−1,1gλAi−gλAi+1,∑i=1mxn−m+i−1gλAi−gλAi+1,∑i=1mxn−m+i−1+δn−m+i−1,2gλAi−gλAi+1,wherex˜n=x˜n−δn,1,x˜n,x˜n+δn,2. Combining Theorem 1 (1), we can obtain the above equation.(2)
According to Theorem3 in Reference [18], we know that if x˜i is a triangle fuzzy number and xi˜=xi−δi,1,xi,xi+δi,2, i = 1, 2, …, then(65)X˜n−r=x˜n−r,x˜n+1−r,…,x˜n+m−1−rT=δn,1r+xn−δn,1,δn+1,1r+xn+1−δn+1,1,…,δn+m−1,1r+xn+m−1−δn+m−1,1T,and by Remark4 and Theorem 3, we have(66)C∫AX˜n−rdgλ=∑i=1mδn−m+i−1,1r+xn−m+i−1−δn−m+i−1,1∑i=1mδn−m+i,1r+xn−m+i−δn−m+i,1∑i=1mδn−m+i+1,1r+xn−m+i+1−δn−m+i+1,1⋯∑i=1mδn+i−2,1r+xn+i−2−δn+i−2,1.(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 x˜i, x˜1,x˜2,…,x˜5∈E˜5, and every x˜i is a triangle fuzzy number, x˜i=xi−δi,1,xi,xi+δi,2, i = 1, 2, …, 5. Suppose (t1, t2, …, t5) ∈ R5, Ai = {ti, ti+1, …, t5}, i = 1, 2, …, 5, and A6 = ∅. The value and weight of each x˜i, 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 Remark3 in Reference [18], we can obtain(67)gλA1=1,gλA2=0.88,gλA3=0.65,gλA4=0.33,gλA5=0.175,gλA6=0.
By Definition6 and Remark 4, the Choquet integral of x˜6 with respect to fuzzy measure gλ on A is determined as follows:(68)x˜6=C∫Ax˜6dgλ=22.04,23.04,24.04.
Similarly, we can also calculate the Choquet integral ofx˜n,n=7,8,9,10, with respect to fuzzy measure gλ on A, as shown in Table 2.
And according to Definition6 and Theorem 4, the Choquet integral of fuzzy number vector X˜6=x˜6,x˜7,…,x˜10T with respect to fuzzy measure gλ on A is determined as follows:(69)C∫AX˜6dgλ=C∫Ax˜6dgλ,C∫Ax˜7dgλ,…,C∫Ax˜10dgλT=22.04,23.04,24.04,22.76,23.76,24.76,22.72,23.72,24.72,22.5,23.5,24.5,22.45,23.45,24.45T.
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 + 1th 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.Table 1
Closing stock prices over 5 days.
Day
Closing stock price
g
λ
1
(19, 20, 21)
0.1
2
(21, 22, 23)
0.2
3
(23, 24, 25)
0.3
4
(24, 25, 26)
0.15
5
(22, 23, 24)
0.175Table 2
Closing stock prices over 10 days.
Day
Closing stock price
g
λ
1
(19, 20, 21)
0.1
2
(21, 22, 23)
0.2
3
(23, 24, 25)
0.3
4
(24, 25, 26)
0.15
5
(22, 23, 24)
0.175
6
(22.04, 23.04, 24.04)
7
(22.76, 23.76, 24.76)
8
(22.72, 23, 72, 24.72)
3 9
(22.5, 23.5, 24.5)
10
(22.45, 23.45, 24.45)


## 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.



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