Inequalities of uncertain set with its applications
© Ning and Huang; licensee Springer. 2014
Received: 18 January 2014
Accepted: 16 April 2014
Published: 7 May 2014
An uncertain set, as a generalization of the uncertain variable, is a set-valued function on an uncertainty space. It provides theoretical foundations for uncertain inference and uncertain logic. This paper aims at providing some inequalities in the framework of uncertain set theory, including the Markov inequality, the Chebyshev inequality, the Jensen inequality, the Hölder inequality, and the Minkowski inequality. In addition, this paper applies these inequalities to the area of incomplete uncertain knowledge representation.
Probability theory, since it was founded by Kolmogorov in 1933 based on normality, nonnegativity, and countable additivity axioms, has been widely used to model the indeterminacy phenomena. A premise of applying probability is that the obtained probability distribution is close enough to the real frequency, which is ignored by many researchers. Usually, we need many samples to obtain the probability distribution via statistics, but in real life we sometimes have no sample due to economical or technological reasons. In this case, we have to invite some experts to evaluate the belief degree that a possible event occurs. For example, the bearing capacity of a bridge generally is obtained not from repeated trials but from the belief degree of bridge engineers.
The belief degree was once treated as probability distribution of a random variable, which led to some counterintuitive results. Interested readers may refer to Liu  for an example. In order to deal with belief degree, Zadeh  proposed the concept of a fuzzy set via the membership function in 1965, and Zadeh  proposed possibility theory in 1978 as a theoretical foundation of the fuzzy set. Except for Zadeh’s possibility theory, a theory of imprecise probability was introduced by Dempster  in 1967, and further elaborated by Shafer  in 1976. Essentially, imprecise probability is a set-valued mapping on a probability space. In addition, in order to deal with the belief degree, a prospect theory was proposed by Kahneman and Tversky  in 1979, and a rough set theory was founded by Pawlak  in 1982.
In order to study the belief degree, an uncertainty theory was founded by Liu  in 2007 and revised by Liu  in 2010 based on normality, duality, subadditivity, and product axioms. In order to indicate the belief degree, concepts of uncertain measure and uncertainty space were proposed. Then Gao  studied the property of continuous uncertain measure. Similar to random variable, uncertain variable was defined as a measurable function on the uncertainty space. After that, You  studied the convergence of a sequence of uncertain variables. Liu and Ha  proved the linearity of the expected value operator of uncertain variables, and Chen and Dai  proposed a maximum entropy principle for uncertain variables.
In 2010, Liu  proposed a concept of an uncertain set as a generalization of the uncertain variable. Meanwhile, he defined a membership function to describe an uncertain set, and he gave the concept of the expected value to model the size of an uncertain set. In order to derive consequences from human knowledge, Liu  proposed uncertain inference via uncertain set theory. After that, Gao et al.  extended the inference rule to the case with multiple antecedents and multiple if-then rules. In addition, Liu  presented the concepts of uncertain system and uncertain inference controller. Then Peng and Chen  proved that an uncertain system is a universal approximator. As an application, Gao  balanced an inverted pendulum by using the uncertain inference control. Except for uncertain inference, the uncertain set has also been used to uncertain logic (Liu ) as a tool to calculate the truth value of an uncertain proposition.
In 2012, Liu  recast uncertain set theory. He redefined the concept of membership function and provided the operational law of uncertain sets via inverse membership functions. Meanwhile, Liu proposed the concept of entropy to describe the divergence of an uncertain set. After that, Yao  proved that the entropy operator meets the requirement of positive linearity. As an extension, Wang and Ha  proposed the quadratic entropy for an uncertain set. In this paper, we will give some inequalities of uncertain set as well as their applications. The rest of this paper is structured as follows. The next section is intended to introduce some concepts in uncertain set theory. Then some inequalities involving a single uncertain set are given in Section 3, and some inequalities involving multiple uncertain sets are given in Section 4. After that, we apply the inequalities to the area of representing incomplete knowledge in Section 5. At last, some remarks are made in Section 6.
Uncertainty theory is a branch of mathematics based on normality, duality, subadditivity, and product axioms. So far, it has brought about many branches such as uncertain programming (Liu ), uncertain risk analysis (Liu ), uncertain process (Liu ), uncertain differential equation (Liu , Yao ), uncertain logic (Liu ), uncertain finance (Liu , Yao ), and uncertain inference control (Liu , Gao ). In this section, we will introduce some useful definitions as regards the uncertain set.
Definition 1 (Liu )
Let Γ be a nonempty set, and ℒ be a σ-algebra on Γ. A set function ℳ is called an uncertain measure if it satisfies the following axioms.
Axiom 1: (Normality) ;
Axiom 2: (Duality) for any ;
In this case, the triple is called an uncertainty space.
Besides, an axiom called product axiom was given by Liu  for the operation of uncertain sets in 2009.
where are arbitrarily chosen events from for , respectively.
Definition 2 (Liu )
They are all uncertain sets.
Definition 3 (Liu )
hold for any Borel set B of real numbers.
If an uncertain set ξ has a membership function μ, then we have . Liu  proved that a real-valued function μ is a membership function if and only if .
where is a membership function with an unknown parameter θ.
Definition 4 (Liu )
is called the inverse membership function of ξ.
A membership function is said to be regular if there exists a point such that and is unimodal about the point . If μ is a regular membership function, then the function is called the left inverse membership function, and the function is called the right inverse membership function. Note that is increasing and is decreasing with respect to α.
The inverse membership function plays an important role in the operation of independent uncertain sets.
Definition 5 (Liu )
where are arbitrarily chosen from , , respectively.
Theorem 1 (Liu )
Definition 6 (Liu )
provided that at least one of the two integral is finite.
for any real numbers a and b.
Definition 7 (Liu )
3 Inequalities of single uncertain set
Inequalities play an important role in estimating the range of a variable. The Markov inequality, the Chebyshev inequality, and the Jensen inequality have been introduced to probability theory and uncertainty theory, and they have found many applications. In this section, we will consider the Markov inequality, the Chebyshev inequality, and the Jensen inequality in the framework of uncertain set theory.
The inequality is thus verified. □
Theorem 3 (Markov inequality)
Proof Take in Theorem 2, and this inequality follows immediately. □
Theorem 4 (Chebyshev inequality)
Proof Replace the uncertain set ξ with , and take in Theorem 2. Then the inequality follows immediately. □
Theorem 5 (Jensen inequality)
Especially, when and , we have .
The inequality is thus verified. □
4 Inequalities of multiple uncertain sets
In this section, we will propose some inequalities with multiple uncertain sets in the framework of uncertain set theory.
Theorem 6 (Hölder inequality)
The inequality is thus verified. □
Theorem 7 (Minkowski inequality)
The inequality is thus verified. □
5 Applications to knowledge representation
In risk evaluation problems, representing incomplete knowledge plays a crucial role. So far, various kinds of knowledge have been considered such as expert opinions and poor statistical information, and different approaches have been proposed for representing the knowledge based on probability theory (Walley ), possibility theory (Dubois and Prade [29, 30]), or the combination of these two theories (Baudrit and Dubois ). Inspired by Baudrit and Dubois , this section will give an application of the inequalities in the area of representing incomplete uncertain knowledge.
Let be an uncertainty space, and ξ be an uncertain set on . However, only partly information of ξ is known due to technological or financial constraints, for example its expected value and variance . This section aims at providing a nested family around ξ, which contains and represents the incomplete knowledge about ξ.
In this paper, we proved some inequalities in the framework of uncertain set theory including the Markov inequality, the Chebyshev inequality, the Jensen inequality, the Hölder inequality, and the Minkowski inequality. These inequalities were applied to the area of representing incomplete uncertain knowledge.
The authors are thankful to the worthy referees for their useful suggestions. This work was supported by National Natural Science Foundation of China (Grant No. 61074193 and No. 63174082).
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