Why λ-additive (fuzzy) measures?

2019

Information Sciences

In this study, the general formula for λ-additive measure of union of n sets is introduced. Here, it is demonstrated that the well-known Poincaré formula of probability theory may be viewed as a limit case of our general formula. Moreover, it is also explained how this novel formula along with an alternatively parameterized λ-additive measure can be applied in theory of belief-and plausibility measures, and in theory of rough sets.

Section: Accepted Manuscriptmentioning

confidence: 99%

“…Although there are many theoretical and practical articles (see, e.g. [11,12,2,1,18]) that discuss the λ-additive measure, its properties and its applicability, there are no papers dealing with the general form of λ-additive measure of union of n sets. Our study seeks to fill this gap.…”

Section: Introductionmentioning

confidence: 99%

The general Poincaré formula for λ-additive measures

2019

Information Sciences

“…One of the most widely applied classes of monotone measures is the class of λ-additive measures (Sugeno λ-measures) (see, e.g. [21,11,12,2,1,17]). [21].…”

Section: Introductionmentioning

confidence: 99%

“…where P(X) denotes the power set of X. Our proof in [4] is based on the fact that Q λ is representable [2]; that is, one has Q λ = h λ • µ for a uniquely determined additive measure µ : P(X) → [0, 1], where h λ : [0, 1] → [0, 1] is a strictly increasing bijection given via…”

Section: Introductionmentioning

confidence: 99%

An Elementary Proof of the General Poincaré Formula for λ-additive Measures

2019

Acta Cybern

In a previous paper of ours [4], we presented the general formula for lambda-additive measure of union of n sets and gave a proof of it. That proof is based on the fact that the lambda-additive measure is representable. In this study, a novel and elementary proof of the formula for lambda-additive measure of the union of n sets is presented. Here, it is also demonstrated that, using elementary techniques, the well-known Poincare formula of probability theory is just a limit case of our general formula.

“…It is an acknowledged fact that the λ-additive measure (Sugeno λ-measure) (Sugeno 1974) is one of the most widely applied monotone measures (fuzzy measure). The usefulness, versatility and applicability of λ-additive measures have inspired numerous theoretical and practical researches since Sugeno's original results were published in 1974 (see, e.g., Magadum and Bapat 2018; Mohamed and Xiao 2003;Chiţescu 2015;Chen et al 2016;Singh 2018 The aim of the present study is twofold. On the one hand, we will revisit the λ-additive measure and give a state-of-theart summary of its most important properties.…”

Section: Introductionmentioning

confidence: 99%

The $$\lambda $$-additive measure in a new light: the $$Q_{\nu }$$ measure and its connections with belief, probability, plausibility, rough sets, multi-attribute utility functions and fuzzy operators

2019

Soft Comput

The aim of this paper is twofold. On the one hand, the λ-additive measure (Sugeno λ-measure) is revisited, and a stateof-the-art summary of its most important properties is provided. On the other hand, the so-called ν-additive measure as an alternatively parameterized λ-additive measure is introduced. Here, the advantages of the ν-additive measure are discussed, and it is demonstrated that these two measures are closely related to various areas of science. The motivation for introducing the ν-additive measure lies in the fact that its parameter ν ∈ (0, 1) has an important semantic meaning as it is the fix point of the complement operation. Here, by utilizing the ν-additive measure, some well-known results concerning the λ-additive measure are put into a new light and rephrased in more advantageous forms. It is discussed here how the ν-additive measure is connected with the belief-, probability-and plausibility measures. Next, it is also shown that two ν-additive measures, with the parameters ν 1 and ν 2 , are a dual pair of belief-and plausibility measures if and only if ν 1 + ν 2 = 1. Furthermore, it is demonstrated how a ν-additive measure (or a λ-additive measure) can be transformed to a probability measure and vice versa. Lastly, it is discussed here how the ν-additive measures are connected with rough sets, multi-attribute utility functions and certain operators of fuzzy logic. Keywords Belief • Probability • Plausibility • λ-Additive measure • Rough sets • Multi-attribute utility functions Definition 2 The function g : P(X) → [0, 1] is a monotone measure on the finite set X iff g satisfies the following requirements: (1) g(∅) = 0, g(X) = 1 (2) if B ⊆ A, then g(B) ≤ g(A) for any A, B ∈ P(X) (monotonicity). Note that the monotone measures given by Definitions 1 and 2 are known as fuzzy measures, which were originally defined by Choquet (1954) and Sugeno (1974). 2.1 Some examples of monotone measures 2.1.1 Dirac measure Definition 3 The function δ x 0 : P(X) → [0, 1] is a Dirac measure on the set X , iff ∀A ∈ P(X): δ x 0 (A) = 1, if x 0 ∈ A 0, otherwise. 2.1.2 Probability measure Definition 4 Let Σ be a σ-algebra over the set X. Then, the function Pr : Σ → [0, 1] is a probability measure on the space (X , Σ) iff Pr satisfies the following requirements: