Article

Linguistic quantifiers modeled by interval-valued intuitionistic Sugeno integrals1

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

Ying's model of linguistic quantifiers based on Sugeno integral is generalized to interval-valued intuitionistic Sugeno integral, the truth value of a quantified proposition is evaluated by using interval-valued intuitionistic Sugeno integral. Some logical properties of linguistic quantifiers in this model are discussed, and some application examples in uncertainty decision making and linguistic summarization of data are presented.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... Definition 1 [21,22] Let U be an ordinary set, F (U ) denote the set of all fuzzy sets on U , A, B ⊆ U , and S: Let U be an nonempty finite set, and A, B ⊆ U . Then, we can prove that ...
Article
Full-text available
The main purpose of this paper is to establish a type of quantitative model by using the contangent similarity function in the three‐valued Łukasiewicz propositional logic system Ł3. We introduce the concepts of the cotangent similarity degree, cotangent pseudo‐distance and cotangent truth degree of the propositions, together with their basic properties in Ł3. We investigate the relationship between the cotangent truth degree and contangent pseudo‐distance, and prove the continuity of the logical connectives ¬,∨ and → in the Ł3 logical metric space. We propose a graded reduction method and three types of graded reasoning frameworks on the propositions set F(S), and provide several examples and basic properties of it.
... The concept of an intuitionistic fuzzy (IF for short) set, initiated by Atanassov [15][16][17], is another important tool for dealing with imperfect and imprecise information. Compared with Zadeh's fuzzy set, an IF set is more objective than a fuzzy set to describe the vagueness of data, because IF set gives both a membership and a non-membership degree of which an element belongs to a set [18][19][20][21]. As an important generalization of IF set, Atanassov and Stoeva defined intuitionistic L-fuzzy (ILF for short ) set in 1984, which actually is an IF set based on residuated lattice L. ...
Article
Full-text available
By introducing the concepts of intuitionistic L-fuzzy β-covering and intuitionistic L-fuzzy β-neighborhood, we define three kinds of intuitionistic L-fuzzy β-covering rough set models. The basic properties of those intuitionistic L-fuzzy β-covering rough set models are investigated. Moreover, we define the other three kinds of intuitionistic L-fuzzy β-covering rough set models by using the former three models. Finally, we present the matrix representations of the newly defined lower and upper approximation operators so that the calculation of lower and upper approximations of subsets can be converted into operations on matrices.
... Many extensions of Zadeh's fuzzy set, such as vague set, intuitionistic fuzzy, hesitant fuzzy set, etc., have been proposed and been utilized management decision and engineering science problems [6][7][8][9][10][11][12]. Fuzzy integral, first introduced by Sugeno in 1974, is an important analytical tool to measure uncertain information [13][14][15][16][17][18]. ...
... Fuzzy measurements and fuzzy integrals, which were originally introduced by Sugeno in 1974 [1], are important analytical methods of measuring uncertain information [2][3][4]. The Sugeno integral has been applied to many fields, such as management decision-making and control engineering [5][6][7][8][9]. Because of the special integral operator of the Sugeno integral, it is limited in many practical problems. ...
Article
Full-text available
Integral inequalities play critical roles in measure theory and probability theory. Given recent profound discoveries in the field of fuzzy set theory, fuzzy inequality has become a hot research topic in recent years. For classical Sandor type inequality, this paper intends to extend the Sugeno integral. Based on the (s,m)-convex function in the second sense, a new Sandor type inequality is proposed for the Sugeno integral. Examples are given to verify the conclusion of this paper.
... Among many applications of the Sugeno integral we shall mention only few. Chen et al. [9] proposed a fusion recognition scheme based on nonlinear decision fusion, Seyedzadeh et al. [31] presented a new RGB color image encryption using keystream generator, Zhang and Zheng [37] generalized Ying's model of linguistic quantifiers, Nemmour and Chibani [19] proposed a new support vector mixture, used to evaluate the interaction of scale factors and to compare the energy performance of buildings in different scale factors [16]. Wang and Klir [35] and Pap [20] have given a general overview on fuzzy measures and fuzzy integration theory. ...
Conference Paper
The classical Jensen inequality for concave function \(\varphi \) is adapted for the Sugeno integral using the notion of the subdifferential. Some examples in the framework of the Lebesgue measure to illustrate the results are presented.
... Seyedzadeh et al. [30] presented a new RGB color image encryption using keystream generator based on Sugeno integral. Zhang [36] generalized Ying's model of linguistic quantifiers based on Sugeno integral to interval-valued intuitionistic Sugeno integral and evaluated the truth value of a quantified proposition by using interval-valued intuitionistic Sugeno integral. Nemmour and Chibani [18] proposed a new support vector mixture in which Sugeno integral is used as a gater to remove the time complexity induced by conventional gaters such as artificial neural networks. ...
Article
In this paper, the classical Jensen inequalities for concave function φ, i.e., φ(∫f(x)dμ)⩾∫φ(f)dμandφ(∑i=1nλixi)⩾∑i=1nλiφ(xi), are adapted for the Sugeno integral using the notion of the supergradient. Moreover, we give some modifications of previous results of Román-Flores et al. concerning Jensen-type inequalities for Sugeno integral. Some examples in the framework of the Lebesgue measure and counting measure to illustrate the results are presented.
Conference Paper
Full-text available
Respiratory rate can often be extracted from electrocardiogram (ECG) or (photoplethysmogram) PPG by analyzing its' beat morphological feature or the variability of time interval. In this paper, the respiratory rates were derived from ECG and PPG signals by using two algorithms based on beat morphology and time interval, respectively. The four groups of estimation results were evaluated by comparing with the reference respiratory rate collected using a piezoelectric sensor. In the in-situ experiments, ECG, PPG and reference respiratory signals were collected from ten subjects. To evaluate the performances of each algorithm at different respiratory rates, the subjects were asked to breathe at the natural rhythm, at the controlled respiratory rates and hold the breath during the experiment. The results demonstrated the method based on ECG peak feature was more accurate and stable among these methods.
Article
Full-text available
Linguistic summarization (LS) is a data mining or knowledge discovery approach to extract patterns from databases. Many authors have used this technique to generate summaries like “Most senior workers have high salary,” which can be used to better understand and communicate about data; however, few of them have used it to generate IF-THEN rules like “IF X is large and Y is medium, THEN Z is small,” which not only facilitate understanding and communication of data but can also be used in decision-making. In this paper, an LS approach to generate IF-THEN rules for causal databases is proposed. Both type-1 and interval type-2 fuzzy sets are considered. Five quality measures-the degrees of truth, sufficient coverage, reliability, outlier, and simplicity-are defined. Among them, the degree of reliability is especially valuable for finding the most reliable and representative rules, and the degree of outlier can be used to identify outlier rules and data for close-up investigation. An improved parallel coordinates approach for visualizing the IF-THEN rules is also proposed. Experiments on two datasets demonstrate our LS and rule visualization approaches. Finally, the relationships between our LS approach and the Wang-Mendel (WM) method, perceptual reasoning, and granular computing are pointed out.
Conference Paper
Full-text available
Linguistic summarization (LS) is a data mining or knowledge discovery approach to extract patterns from databases. It has been studied by many researchers; however, none of them has used it to generate IF-THEN rules, which can be added to a knowledge base for better understanding of the data, or be used in Perceptual Reasoning to infer the outputs for new scenarios. In this paper LS using IF-THEN rules is proposed. Five quality measures for such summaries are defined. Among them, the degree of usefulness is especially valuable for finding the most reliable and representative rules, and the degree of outlier can be used to identify outlier rules and data. An example verifies the effectiveness of our approach. The relationship between LS and the Wang-Mendel method is also discussed.
Article
Full-text available
In this paper, we generalize Ying's model of linguistic quantifiers [M.S. Ying, Linguistic quantifiers modeled by Sugeno integrals, Artificial Intelligence, 170 (2006) 581-606] to intuitionistic linguistic quantifiers. An intuitionistic linguistic quantifier is represented by a family of intuitionistic fuzzy-valued fuzzy measures and the intuitionistic truth value (the degrees of satisfaction and non-satisfaction) of a quantified proposition is calculated by using intuitionistic fuzzy-valued fuzzy integral. Description of a quantifier by intuitionistic fuzzy-valued fuzzy measures allows us to take into account differences in understanding the meaning of the quantifier by different persons. If the intuitionistic fuzzy linguistic quantifiers are taken to be linguistic fuzzy quantifiers, then our model reduces to Ying's model. Some excellent logical properties of intuitionistic linguistic quantifiers are obtained including a prenex norm form theorem. A simple example is presented to illustrate the use of intuitionistic linguistic quantifiers.
Book
The manipulation of databases is an integral part of a world which is becoming increasingly and pervasively information-focused. This book puts forward a suggestion to advocate preference queries and fuzzy sets as a central concern in database queries and offers an important contribution to the design of intelligent information systems. It provides a comprehensive study on fuzzy preference queries in the context of relational databases. Preference queries, a recent hot topic in database research, provide a basis for rank-ordering the items retrieved, which is especially valuable for large sets of answers. This book aims to show that fuzzy set theory constitutes a highly expressive framework for modeling preference queries. It presents a study of the algorithmic aspects related to the evaluation of such queries in order to demonstrate that this framework offers a good trade-off between expressivity and efficiency. Numerous examples and proofs are liberally and lucidly demonstrated throughout, and greatly enhance the detailed theoretical aspects explored in the book. Researchers working in databases will greatly benefit from this comprehensive and up-to-date study of fuzzy preference queries, and it will also become an invaluable reference point for postgraduate students interested in advanced database techniques.
Article
Extends the concept and theory of Sugeno integral and fuzzy measure, presents a new method for multi-attribute decision making under interval intuitionistic fuzzy environment. The concept of interval intuitionistic fuzzy-valued fuzzy measure and interval intuitionistic fuzzy-valued Sugeno integral are introduced, and the relationship between the extension and original fuzzy measure and Sugeno integral are represented. Some properties and calculating methods of the extended Sugeno integral are given. A multi-attribute decision making method is established in which interval intuitionistic fuzzy-valued Sugeno integral is taken as its aggregation function. An illustrative example is provided to verify the new method and to demonstrate its feasibility and practicality.
Article
Based on first-order fuzzy logic system K* and Ying's framework for linguistic quantifiers modeled by Sugeno integrals, a many-sorted first-order logic system IMTL*Q is constructed. Applying the triple I method, some fuzzy reasoning forms with linguistic quantifiers are investigated.
Article
The lattice-valued fuzzy measure, null-additivity, pseudo-null- additivity, autocontinuity and pseudo-autocontinuity of lattice-valued set functions were introduced, and some of their properties were discussed by the first author [Proc. Int. Conf. Inform. Syst., Dalian/China, 898-901 (1992)]. In this paper, a new concept – a lattice- valued fuzzy integral – is introduced, and some properties of lattice- valued fuzzy integrals are given, and a series of convergence theorems for a sequence of lattice-valued fuzzy integrals are proved.
Article
In real multi-attribute group decision making, there exist interaction phenomena among decision making attributes and preference of experts, thus an intuitionistic fuzzy-valued integral method for multiple attribute group decision making is investigated. Based on intuitionistic fuzzy-valued measures, an intuitionistic fuzzy-valued Sugeno integral operator is presented. And some properties are analyzed. The intuitionistic fuzzy-valued Sugeno integral operator is applied to deal with multiple attribute group decision making problems. An illustrative example is provided to verify the new method and to demonstrate its feasibility and practicality.
Book
Fuzzy measure theory, the subject of this text, is an offspring of classical measure theory. The latter has its roots in metric geometry, which is characterized by assigning numbers to lengths, areas, or volumes. In antiquity, this assignment process, or measurement, was first conceived simply as a comparison with a standard unit. Soon, however, the problem of incommensurables (exemplified by the problem of measuring the length of the diagonal of a square whose sides each measure one unit) revealed that measurement is more complicated than this simple, intuitively suggestive process. It became clear that measurement must inevitably involve infinite sets and infinite processes.
Article
This article investigates new score and accuracy functions for ranking interval-valued intuitionistic fuzzy numbers (IVIFNs). The novelty of these functions is that they allow the comparison of IVIFNs by taking into account of the decision makers' attitudinal character. The new attitudinal expected score and accuracy functions extend Xu and Chen's score and accuracy degree functions, and verify the following set of properties: (1) boundedness; (2) monotonicity; (3) commutativity; and (4) symmetry. These novel functions are used to propose a total order on the set of IVIFNs, and to develop an interval-valued intuitionistic fuzzy multi-attribute decision making selection process in which the final result depends on the decision maker's risk attitude. In addition, a ranking sensitivity analysis with respect to the risk attitude is provided.
Conference Paper
The notion of intuitionistic fuzzy Choquet integrals is introduced. Then, some nice properties, including component-wise decomposition property, duality property, a simple form of the discrete expression, of intuitionistic fuzzy Choquet integrals are studied. As an application of intuitionistic fuzzy Choquet integrals, Cui and Li's model (L.C.Cui, Y.M.Li, Linguistic quantifiers based on Choquet integrals, International Journal of Approximate Reasoning, 48 (2008) 559-582.) of linguistic quantifiers is generalized to intuitionistic linguistic quantifiers. Description of a quantifier by intuitionistic fuzzy measures allows us to take into account differences in understanding the meaning of the quantifier by different persons. Some elegant logical properties of linguistic quantifiers are derived within this approach.
Article
A social network analysis (SNA) trust-consensus based group decision making model with interval-valued fuzzy reciprocal preference relation (IFRPR) is investigated. The main novelty of this model is that it determines the importance degree of experts by combining two reliable resources: trust degree (TD) and consensus level (CL). To do that, an interval-valued fuzzy SNA methodology to represent and model trust relationship between experts and to compute the trust degree of each expert is developed. The multiplicative consistency property of IFRPR is also investigated, and the consistency indexes for the three different levels of an IFRPR are defined. Additionally, similarity indexes of IFRPR are defined to measure the level of agreement among the group of experts. The consensus level is derived by combining both the consistency index and similarity index, and it is used to guide a feedback mechanism to support experts in changing their opinions to achieve a consensus solution with a high degree of consistency. Finally, a quantifier guided non-dominance possibility degree (QGNDPD) based prioritisation method to derive the final consensus-trust based solution is proposed.
Article
In this paper the concept of an ordered weighted average (OWA) operator is extended to any complete lattice endowed with a t-norm and a t-conorm. In the case of a complete distributive lattice it is shown to agree with a particular case of the discrete Sugeno integral. As an application, we show several ways of aggregating closed intervals by using OWA operators. In a complementary way, the notion of generalized Atanassov's operators is weakened in order to be extended to intervals contained in any lattice. This new approach allows us to build a kind of binary aggregation functions for complete lattices, including OWA operators.
Article
In this article, the ordered weighted aggregation operator and hybrid aggregation operator are developed for aggregating interval-valued intuitionistic preference information. Interval-valued intuitionistic judgment matrix and its score matrix and accuracy matrix are defined. Some of their desirable properties are investigated in detail. The relationships among interval-valued intuitionistic judgment matrix, intuitionistic judgment matrix, and complement judgment matrix, are discussed. On the basis of the arithmetic aggregation operator and hybrid aggregation operator, an approach to group decision making with interval-valued intuitionistic judgment matrices is given. Finally, a practical example is provided to verify the effectiveness of the developed approach.
Article
This is a subsequent paper of [9]. By using the concepts of fuzzy number fuzzy measures [9] and fuzzy-valued functions [10], a theory of fuzzy integrals of fuzzy-valued functions with respect to fuzzy number fuzzy measures is built up. So far, it is a more general one following Sugeno's [5].
Article
A generalization of the notion of intuitionistic fuzzy set is given in the spirit of ordinary interval valued fuzzy sets. The new notion is called interval valued intuitionistic fuzzy set (IVIFS). Here we present the basic preliminaries of IVIFS theory.
Article
The so-called linguistic summaries of databasesare the semi-natural language sentences that enable distilling th e most relevant information from large numbers of tuples, and present it in the human consistent forms. Recently, the methods of constructing and evaluating lin- guistic summaries have been based on Zadeh's fuzzy sets, whi ch represent uncertain data. The main aim of the paper is to enhance and generalize the Yager's approach to linguistic summarization of data. This enhancement is based on interval-valued fuzzy sets. The newly presented methods enable handling fuzzy concepts, whose membership degrees are not given by real values explicitly, but are approximated by intervals in (0, 1). From now on, the Yager's approach can be viewed as a special case of the method presented in this paper. Finally, illustrative examples ar e presented. Keywords: interval-valued fuzzy set, interval-valued linguistic va ri- able, interval-valued linguistic quantifier, linguistic s ummary of database, interval-valued linguistic summary of database.
Article
This paper is situated in the area of flexible queries addressed to regular relational databases. Some works have been carried out in the past years in order to design languages allowing the expression of queries involving preferences through the use of fuzzy predicates. In the relational setting, extensions of the relational algebra as well as of SQL-like languages have been proposed and a wide range of fuzzy queries has been made available. However, the use of conditions involving an aggregate function applying to a fuzzy set is not yet possible except for the cardinality (count) in the context of the so-called fuzzy quantified statements. The objective of this paper is to investigate how (and under which conditions) other aggregate functions (such as the maximum…) could be applied to fuzzy sets in a flexible query.
Article
We present basic ideas and perspectives related to the use of fuzzy logic for the derivation of linguistic summaries of data sets (databases in practice). We concentrate on the issue of how to measure the goodness of a linguistic summary. We advocate the use of an interactive approach in which the user indicates first the class of linguistic summaries of interest (basically, by specifying attributes relation between which he or she is interested in) by using a fuzzy querying interface to a database. Finally, we present an implementation for deriving linguistic summaries of a sales database at a computer retailer, and show how the linguistic summaries obtained can be useful for supporting decisions by the business owner.
Article
We present basic ideas and perspectives related to the use of fuzzy logic for the derivation of linguistic summaries of data (databases). We concentrate on the issue of how to measure the goodness of a linguistic summary, and on how to embed data summarization within the fuzzy querying environment, for an effective and efficient implementation. In particular, we propose how to efficiently implement Kacprzyk and Yager’s new quality indicators of linguistic summaries to derive summaries via Kacprzyk and Zadrożny’s fuzzy querying add-on. Finally, we present an implementation for deriving linguistic summaries of a sales database at a computer retailer, and show how the linguistic summaries obtained can be useful for supporting decisions of the business owner.
Article
We point out some relevant issues that are related to the computing-with-words (CWW) paradigm and argue for an urgent need for a new, nontraditional look at the area, since the traditional approach has resulted in very valuable theoretical research results. However, there is no proper exposure and recognition in other areas to which CWW belongs and can really contribute, notably natural-language processing (NLP), in general, and natural-language understanding (NLU) and natural-language generation (NLG), in particular. First, we present crucial elements of CWW, in particular Zadeh's protoforms, and indicate their power and stress a need to develop new tools to handle more modalities. We argue that CWW also has a high implementation potential and present our approach to linguistic data(base) summaries, which is a very intuitive and human-consistent natural-language-based knowledge-discovery tool. Special emphasis is on the use of Zadeh's protoform (prototypical form) as a general form of a linguistic data summary. We present an extension of our interactive approach, which is based on fuzzy logic and fuzzy database queries, to implement such linguistic summaries. In the main part of the paper, we discuss a close relation between linguistic summarization in the sense considered and some basic ideas and solutions in NLG, thus analyzing possible common elements and an opportunity to use developed tools, as well as some inherent differences and difficulties. Notably, we indicate a close relation of linguistic summaries that are considered to be some type of an extended template-based, and even a simple phrase-based, NLG system and emphasize a possibility to use software that is available in these areas. An important conclusion is also an urgent need to develop new protoforms, thus going beyond the classical ones of Zadeh. For illustration, we present an implementation for a sales database in a computer retailer, thereby showing the power of linguistic summaries, as well - - as an urgent need for new types of protoforms. Although we use linguistic summaries throughout, our discussion is also valid for CWW in general. We hope that this paper-which presents our personal view and perspective that result from our long-time involvement in both theoretical work in broadly perceived CWW and real-world implementations-will trigger a discussion and research efforts to help find a way out of a strange situation in which, on one hand, one can clearly see that CWW is related to words (language) and computing and, hence, should be part of broadly perceived mainstream computational linguistics, which lack tools to handle imprecision. These tools can be provided by CWW. Yet, CWW is practically unknown to these communities and is not mentioned or cited, and---reciprocally---even the top people in CWW do not refer to the results that are obtained in these areas. We hope that our paper, for the benefit of both the areas, will help bridge this gap that results from a wrong and dangerous fragmentation of break science.
Article
The generic term fuzzy quantifier is employed in this paper to denote the collection of quantifiers in natural languages whose representative elements are: several, most, much, not many, very many, not very many, few, quite a few, large number, small number, close to five, approximately ten, frequently, etc. In our approach, such quantifiers are treated as fuzzy numbers which may be manipulated through the use of fuzzy arithmetic and, more generally, fuzzy logic.A concept which plays an essential role in the treatment of fuzzy quantifiers is that of the cardinality of a fuzzy set. Through the use of this concept, the meaning of a proposition containing one or more fuzzy quantifiers may be represented as a system of elastic constraints whose domain is a collection of fuzzy relations in a relational database. This representation, then, provides a basis for inference from premises which contain fuzzy quantifiers. For example, from the propositions “Most U's are A's” and “Most A's are B's,” it follows that “Most2U's are B's,” where most2 is the fuzzy product of the fuzzy proportion most with itself.The computational approach to fuzzy quantifiers which is described in this paper may be viewed as a derivative of fuzzy logic and test-score semantics. In this semantics, the meaning of a semantic entity is represented as a procedure which tests, scores and aggregates the elastic constraints which are induced by the entity in question.
Article
An extension of TOPSIS, a multi-criteria interval-valued intuitionistic fuzzy decision making technique, to a group decision environment is investigated, where inter-dependent or interactive characteristics among criteria and preference of decision makers are taken into account. To get a broad view of the techniques used, first, some operational laws on interval-valued intuitionistic fuzzy values are introduced. Based on these operational laws, a generalized interval-valued intuitionistic fuzzy geometric aggregation operator is proposed which is used to aggregate decision makers’ opinions in group decision making process. In addition, some of its properties are discussed. Then Choquet integral-based Hamming distance between interval-valued intuitionistic fuzzy values is defined. Combining the interval-valued intuitionistic fuzzy geometric aggregation operator with Choquet integral-based Hamming distance, an extension of TOPSIS method is developed to deal with a multi-criteria interval-valued intuitionistic fuzzy group decision making problems. Finally, an illustrative example is used to illustrate the developed procedures.
Article
This paper, based on the fuzzy measures and fuzzy integrals given by Sugeno, first defines interval number fuzzy measures (INF-measures) and fuzzy number fuzzy measures (FNF-measures), and the fuzzy integral of function with respect to INF-measures are given. Then it defines the fuzzy integral of functions with respect to FNF-measures, and the properties and convergence theorems are obtained. All these are generalizations of Sugeno's works.
Article
Nonadditive measure is a generalization of additive probability measure. Sugeno integral is a useful tool in several theoretical and applied statistics which has been built on non-additive measure. Integral inequalities play important roles in classical probability and measure theory. The classical Berwald integral inequality is one of the famous inequalities. This inequality turns out to have interesting applications in information theory. In this paper, Berwald type inequality for the Sugeno integral based on a concave function is studied. Several examples are given to illustrate the validity of this inequality. Finally, a conclusion is drawn and a problem for further investigations is given.
Article
Since quantifiers have the ability of summarizing the properties of a class of objects without enumerating them, linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning. This paper introduces a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures, and the truth value of a quantified proposition is evaluated by using Sugeno's integral. This framework allows us to have some elegant logical properties of linguistic quantifiers. We compare carefully our new model of quantification and other approaches to linguistic quantifiers. A set of criteria for linguistic quantification was proposed in the previous literature. The relationship between these criteria and the results obtained in the present paper is clarified. Some simple applications of the Sugeno's integral semantics of quantifiers are presented.
Article
In this paper, (1) the concepts of lattice-valued fuzzy measure (with no valuation property) and lower (resp. upper) lattice-valued fuzzy integral are proposed, which give the unified discription to the fuzzy measures and fuzzy integrals studied by Delgado and Moral, Qiao, Ralescu, Adams, Sugeno, Wang, and Zhang; (2) some asymptotic structural characteristics of lattice-valued fuzzy measures are introduced, and some relations between them are given; (3) some concepts of convergences for lattice-valued functions are defined, and Riesz' theorem, Egoroff's theorems and Lebesgue's theorem for lattice-valued measurable function are proved; (4) the monotone increasing (resp. decreasing) convergence theorem and almost (resp. pseudo almost) everywhere convergence theorrem for lower (resp. upper) lattice-valued fuzzy integral are shown under some weak conditions.
Article
In this paper we introduce some relations and operations of interval-valued intuitionistic fuzzy numbers and define some types of matrices, including interval-valued intuitionistic fuzzy matrix, interval-valued intuitionistic fuzzy similarity matrix and interval-valued intuitionistic fuzzy equivalence matrix. We study their properties, develop a method based on distance measure for group decision making with interval-valued intuitionistic fuzzy matrices and, finally, provide an illustrative example.
Article
An inequality related to Minkowski type for the Sugeno integral on abstract spaces is studied in a rather general form. Some previous results on Chebyshev type inequality obtained by the authors are generalized. Several examples are given to illustrate the validity of this inequality. The conditions such that this inequality becomes an equality are also discussed. Finally, conclusions and some problems for further investigations are included.
Article
Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set, the top element of the lattice to the entire universe and satisfy the property of monotonicity. If the lattice is complete then a lattice-valued fuzzy integral of Sugeno type, with similar properties such as the Sugeno integral in its original form, can be introduced in a natural way. The main result of the paper is a componentwise decomposition theorem of an L-valued fuzzy integral to its L-valued fuzzy integrals components, where L is a complete lattice with negation and is organized as a complete lattice too. This result is useful to obtain the properties of L-valued fuzzy integrals from the properties of L-valued fuzzy integrals and to calculate in a simple way the values of some integrals from the values of the components. The important case L = [0, 1], when L becomes the lattice of the intuitionistic fuzzy values is distinctly discussed. An idea of application to synthetic evaluation of objects is also suggested.
Article
Let (X,A,μ) be a finite nonadditive measure space and M be the set of all finite measurable functions on X. The topology on M, which is determined by the Choquet integral with respect to μ, is investigated. The relationship between this topology and the one determined by the Sugeno integral is examined. Some interesting examples are included.
Article
The introduction of linguistic quantifiers has provided an important tool to model a large number of issues in intelligent systems. Ying [M.S. Ying, Linguistic quantifiers modeled by Sugeno integrals, Artificial Intelligence 170 (2006) 581–606] recently introduced a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures, and the truth value of a quantified proposition is evaluated by using Sugeno’s integral. Representing linguistic quantifiers by fuzzy measures, this paper evaluates linguistic quantified propositions by the Choquet integral. Some elegant logical properties of linguistic quantifiers are derived within this approach, including a prenex normal form theorem stronger than that in Ying’s model. In addition, our Choquet integral approach to the evaluation of quantified statements is compared with others, in particular with Ying’s Sugeno integral approach.
Article
We introduce a new approach to the summarization of data based upon the theory of fuzzy subsets. This new summarization allows for a linguistic summary of the data and is useful for both numeric and nonnumeric data. It summarizes the data in terms of three values: a summarizer, a quantity in agreement, and a truth value. We also discuss a procedure for investigating the informativeness of a summary.
Article
This paper introduces an application of type-2 fuzzy sets in data linguistic summarization. The original approach by Yager (1982) based on representing natural language statements via type-1, i.e., the Zadeh fuzzy sets, is generalized with type-2 fuzzy sets applied as models of linguistically expressed quantities and/or properties of objects. Type-2 sets extend the known summarization procedures by handling fuzzy values stored in databases, and allow to represent a linguistic term via a few different membership functions (e.g., provided by different experts), which makes the method more general and human-consistent. Furthermore, quality measures for type-2 summaries are discussed in order to evaluate the informativeness of the messages generated. Finally, two prototype applications are presented and the success of the new method is discussed.
Model of linguistic quantifiers based on general Sugeno integrals
  • Zheng
The first-order fuzzy logic (I)
  • Ying