A note on inequalities and critical values of fuzzy rough variables
- Mingxuan Zhao^{1},
- Jing Liu^{1} and
- Ke Wang^{1}Email author
https://doi.org/10.1186/s13660-015-0787-0
© Zhao et al. 2015
Received: 10 April 2015
Accepted: 13 August 2015
Published: 28 August 2015
Abstract
A fuzzy rough variable is defined as a rough variable on the universal set of fuzzy variables, or a rough variable taking ‘fuzzy variable’ values. In order to further discuss the mathematical properties of fuzzy rough variables, this paper extends some inequalities to the context of fuzzy rough theory based on the chance measure and the expected value operator, involving the Markov inequality, the Chebyshev inequality, the Hölder inequality, the Minkowski inequality, and the Jensen inequality. After that, linearity, monotonicity, and continuity of critical values of fuzzy rough variables are also investigated.
Keywords
1 Introduction
Fuzzy set theory has been well developed and applied in a wide variety of real problems since it was proposed in 1965 by Zadeh [1]. Kaufmann [2] first introduced fuzzy variable as a fuzzy set of real numbers to describe fuzzy phenomena. By means of a mathematical way, a fuzzy variable was defined as a function from a possibility space to the set of real numbers by Liu [3]. Now, fuzzy set theory has been proved to be an excellent tool and one of the most successful approaches to the issue of how to understand and manipulate imperfect knowledge.
On the other hand, in order to deal with vague description of objects, rough set theory was initialized by Pawlak [4] in 1982, which provides a new powerful mathematical approach to handling imperfect knowledge in the real world. A fundamental assumption in this theory is that objects are perceived through, and thus can be represented by, available information on their attributes, but such information may not be sufficient to characterize these objects exactly. One way is approximating a set by other sets. Thus a rough set may be defined by a pair of crisp sets which give the lower and upper approximations of the original set. Liu [5] defined a rough variable to be a measurable function from a rough space to the set of real numbers and gave the definition of the lower and upper approximations of the rough variable.
With the development of fuzzy set theory and rough set theory, it is generally accepted that these two theories are related but distinct and complementary with each other. Thus many researchers began to consider the combination of the two theories. For example, in the real world, sometimes it is not easy to describe a fuzzy event by a precise fuzzy set, but the lower and upper approximations of the fuzzy set of a fuzzy event can be given, which can be seen as a fuzzy rough variable. In 1990, Dubois and Prade [6] initially proposed the concept of fuzzy rough sets by constructing a pair of upper and lower approximation operators of fuzzy sets with respect to a fuzzy similarity relation by means of the t-norm Min and its dual conorm Max. A fuzzy rough variable, very different from the fuzzy rough set introduced by Dubois and Prade [6], was defined by Liu [5] as a measurable function from a rough space to the set of fuzzy variables. In other words, a fuzzy rough variable is a rough variable defined on the universal set of fuzzy variables, or a rough variable taking ‘fuzzy variable’ values. By now, the fuzzy rough theory has been studied in both theoretical and practical perspectives, for instance, the generalized definition of a fuzzy rough set [7, 8], a new definition for the lower and upper approximations [9, 10], fuzzy rough attribute (or feature) selection [11, 12], as well as its applications in data reduction and classification [13–15], complex systems monitoring [16], neural networks [17], and so on.
It is well known that there are some inequalities in probability theory such as the Markov inequality, the Chebyshev inequality, the Jensen inequality, the Hölder inequality, and the Minkowski inequality, which make an important contribution to the development of probability theory in both theories and applications. On the basis of these inequalities, in possibility and rough theory, Liu [18] proved that these analogous inequalities hold both for fuzzy variables and rough variables. Moreover, Yang and Liu [19] also proved these inequalities for fuzzy random variables. As an extension of these researches, it is necessary to study these inequalities in the context of fuzzy rough theory. Therefore, in the present paper, some inequalities are presented for fuzzy rough variables, and some properties of critical values of fuzzy rough variables are also proved.
The rest of this paper is organized as follows. In Section 2, we first review some basic knowledge of fuzzy variables, rough variables, and fuzzy rough variables involving the chance measure and the expected value operator. Some inequalities for fuzzy rough variables are presented in Section 3. Section 4 introduces definitions of the \((\gamma,\delta)\)-optimistic value and the \((\gamma,\delta)\)-pessimistic value, and it discusses the linearity, monotonicity and continuity of the critical values to explore the mathematical properties of fuzzy rough variables.
2 Preliminaries
In this section, we recall some concepts and properties of fuzzy variables, rough variables, and fuzzy rough variables, which will be applied in the following sections.
2.1 Fuzzy variable
In order to measure a fuzzy event, Zadeh [20, 21] proposed the concepts of possibility measure and necessity measure in 1978 and 1979, respectively. Subsequently, possibility theory was developed by many researchers such as Dubois and Prade [22, 23]. Liu and Liu [24] presented the concept of credibility measure in 2002 on the basis of possibility measure and necessity measure, and then a complete axiomatic foundation of credibility theory was developed by Liu [3].
Definition 1
(Liu [5])
Let Θ be a nonempty set, \(\mathscr{P}(\Theta)\) the power set of Θ, and Pos a possibility measure. The triplet \((\Theta, \mathscr{P} (\Theta),\operatorname{Pos})\) is called a possibility space. A fuzzy variable is defined as a function from a possibility space \((\Theta ,\mathscr{P}(\Theta),\operatorname{Pos})\) to the set of real numbers.
Definition 2
(Liu and Liu [24])
Example 1
Definition 3
(Liu and Gao [25])
Theorem 1
(Liu and Gao [25])
Definition 4
(Liu and Liu [24])
Example 2
Example 3
Theorem 2
(Liu [24])
2.2 Rough variable
Rough set theory, initialized by Pawlak [4], has been proved to be an excellent mathematical tool to deal with vague description of objects. In order to provide an axiomatic theory to describe rough variables, Liu [5] gave some definitions about rough set theory as follows.
Definition 5
(Liu [5])
Let Λ be a nonempty set, \(\mathscr{A}\) a σ-algebra of subset of Λ, Δ an element in \(\mathscr{A}\), and π a set function satisfying the following four axioms,
Axiom 1. \(\pi\{\Lambda\}<+\infty\);
Axiom 2. \(\pi\{\Delta\}>0\);
Axiom 3. \(\pi\{A\}\ge0\) for any \(A\in \mathscr{A}\);
Definition 6
(Liu [5])
Definition 7
(Liu [5])
Example 4
Definition 8
(Liu [5])
Example 5
Theorem 3
(Liu [3])
2.3 Fuzzy rough variable
Fuzzy rough variables have been defined in several ways. In this paper, we adopt the definition introduced by Liu [5] as follows.
Definition 9
(Liu [5])
A fuzzy rough variable is a function ξ from a rough space \((\Lambda ,\Delta, \mathscr{A},\pi)\) to the set of fuzzy variables such that \(\operatorname{Pos}\{\xi(\lambda)\in B\}\) is a measurable function of λ for any Borel set B of ℜ.
Example 6
Let \(\xi=(\rho,\rho+1,\rho+2)\) with \(\rho=([2,4],[0,6])\), where the triple \((a,b,c)\) with real numbers \(a\leq b\leq c\) denotes a triangular fuzzy variable and ρ is a rough variable, then ξ is a fuzzy rough variable.
Theorem 4
(Liu [3])
- (a)
the possibility \(\operatorname{Pos}\{\xi(\lambda)\in B\}\) is a rough variable;
- (b)
the necessity \(\operatorname{Nec}\{\xi(\lambda)\in B\}\) is a rough variable;
- (c)
the credibility \(\operatorname{Cr}\{\xi(\lambda)\in B\}\) is a rough variable.
Definition 10
(Liu [5])
Theorem 5
(Liu [5])
Definition 11
(Liu [5])
Let ξ be a fuzzy rough variable with finite expected value \(E[\xi ]\). The variance of ξ is defined as \(V[\xi]=E [(\xi-E[\xi])^{2} ]\). Then the square root of \(V[\xi]\) is called the standard deviation of ξ.
Definition 12
(Liu [5])
Definition 13
(Liu [3])
Let ξ be a fuzzy rough variable, and B a Borel set of ℜ. For any real number \(\alpha\in(0,1]\), the α-chance of a fuzzy rough event \(\xi\in B\) is defined as the value of chance at α, i.e., \(\operatorname{Ch}\{\xi\in B \}(\alpha )\), where Ch denotes the chance measure.
3 Inequalities of fuzzy rough variables
Some inequalities, including the Markov inequality, the Chebyshev inequality, the Hölder inequality, the Minkowski inequality, and the Jensen inequality, analogous to those in probability theory, have been proved to hold both for fuzzy variables and rough variables by Liu [18]. In this section, these inequalities are proved for fuzzy rough variables.
Theorem 6
Proof
On the basis of the inequality presented in Theorem 6, the well-known Markov inequality and the Chebyshev inequality in probability theory are proved in the context of fuzzy rough theory as follows, which can be seen as special cases of Theorem 6.
Theorem 7
(Markov inequality)
Proof
It is a special case of Theorem 6 when \(f(x)=|x|^{p}\). □
Theorem 8
(Chebyshev inequality)
Proof
It is a special case of Theorem 6 when the fuzzy rough variable ξ is replaced with \(\xi-E[\xi]\) and \(f(x)=x^{2}\). □
The Markov inequality gives an upper bound for the α-chance that the absolute value of a fuzzy rough variable is greater than or equal to some positive constant, whereas the Chebyshev inequality describes to what extent the values taken by a fuzzy rough variable deviate from its expected value. Both of them state important properties of a fuzzy rough variable.
For instance, the following conclusion can be deduced from the Chebyshev inequality immediately, which implies that the α-chance that the values taken by an arbitrary fuzzy rough variable with finite expected value exceed k (\(k>0\)) standard deviations away from its mean (i.e., expected value) is no more than \(\frac {1}{\alpha k^{2}}\).
Example 7
Theorem 9
(Hölder inequality)
Proof
As a special case of the Hölder inequality with \(p=q=2\), the Cauchy inequality can be obtained as follows, which is widely used for dealing with some mathematical problems.
Example 8
(Cauchy inequality)
Theorem 10
(Minkowski inequality)
Proof
Theorem 11
(Jensen inequality)
Proof
The Jensen inequality gives a lower bound for the expected value of a convex function of a fuzzy rough variable. On the basis of the Jensen inequality, some other inequalities with regard to convex functions can be proved simply and directly. For instance, from the Jensen inequality as well as the Cauchy inequality and the Minkowski inequality, some important inequalities can be further deduced as follows.
Example 9
4 Critical values of fuzzy rough variable
In this section, we first recall the concepts of the \((\gamma,\delta )\)-optimistic value and the \((\gamma,\delta)\)-pessimistic value of a fuzzy rough variable defined by Liu [5]. Then the linearity, monotonicity, and continuity of these critical values are discussed. It is shown that these mathematical properties, which have been discussed for fuzzy variables and rough variables by Liu [3] as well as for fuzzy random variables by Yang and Liu [19], are also valid for fuzzy rough variables in a similar way.
Definition 14
(Liu [5])
Theorem 12
(Liu [3])
Theorem 13
(Linearity)
- (a)
if \(c \geq0\), then \((c\xi)_{\sup}(\gamma,\delta) = c\xi_{\sup }(\gamma,\delta)\) and \((c\xi)_{\inf}(\gamma,\delta) = c\xi_{\inf}(\gamma ,\delta)\);
- (b)
if \(c < 0\), then \((c\xi)_{\sup}(\gamma,\delta)= c\xi_{\inf}(\gamma ,\delta)\) and \((c\xi)_{\inf}(\gamma,\delta) = c\xi_{\sup}(\gamma,\delta)\).
Proof
Theorem 14
(Monotonicity and continuity)
- (a)
\(\xi_{\sup}(\gamma,\delta)\) is a decreasing and left-continuous function of γ for each δ;
- (b)
\(\xi_{\sup}(\gamma,\delta)\) is a decreasing and left-continuous function of δ for each γ;
- (c)
\(\xi_{\inf}(\gamma,\delta)\) is an increasing and left-continuous function of γ for each δ;
- (d)
\(\xi_{\inf}(\gamma,\delta)\) is an increasing and left-continuous function of δ for each γ.
Proof
5 Conclusions
Based on previous study on inequalities and critical values in fuzzy set theory and rough set theory, this paper made a further study on fuzzy rough theory, and enriched the research area of this theory in the following two parts: (i) some inequalities in fuzzy rough theory were proved including the Markov inequality, the Chebyshev inequality, the Hölder inequality, the Minkowski inequality and the Jensen inequality, which are analogous to those of the fuzzy case and rough case; (ii) we explored the linearity, monotonicity and continuity of critical values of the fuzzy rough variable.
This paper discussed these inequalities as well as properties of the critical values in a theoretical way. However, the theorems and conclusions presented in this paper would also make an important contribution to practical applications of the fuzzy rough theory. Taking a decision system with fuzzy rough coefficients (e.g. [26, 27]) for instance, in order to get optimal solutions to a problem (modeled by a mathematical model with fuzzy rough coefficients), the objective functions as well as some constraints involving fuzzy rough variables should be analyzed. It is clear that the analyses of these functions and the development of a solving algorithm for the fuzzy rough model may benefit greatly from the inequalities and properties of the critical values presented in this paper.
Declarations
Acknowledgements
This work was supported in part by grants from the Ministry of Education Funded Project for Humanities and Social Sciences Research (No. 14YJC630124), the Shanghai Philosophy and Social Science Planning Project (No. 2014EGL002), and the Innovation Program of Shanghai Municipal Education Commission (No. 14YS005).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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