Nhung Cao

How to Incorporate Excluding Features in Fuzzy Relational Compositions and What for

The aim of this paper is, first, to recall fuzzy relational compositions (products) and, to introduce an idea, how the so-called excluding features could be incorporated into the theoretical background. Apart from rather natural definitions, we provide readers with a theoretical investigation that provides and answer to a rather natural question, under which conditions, in terms of the underlying algebraic structures, the proposed incorporation of excluding features preserves the same properties as the incorporation in the classical relational compositions. The positive impact of the incorporation on reducing the suspicions provided by the basic “circlet” composition without losing the possibly correct suspicion, as in the case of the use of the Bandler-Kohout products, is demonstrated on an example.

Extensions of fuzzy relational compositions based on generalized quantifiers

Abstract The aim of this paper is to extend the fuzzy relational compositions motivated by their applications to medical diagnosis problem by Bandler and Kohout. These compositions employ two quantifiers, the universal quantifier and the existential quantifier. As there exists a big gap between these two quantifiers, that may be appropriately filled in by intermediate generalized quantifiers, we take this as a motivation for our study. In particular, we introduce the concept of fuzzy relational compositions based on generalized quantifiers. Furthermore, we show that the properties that are typically valid for standard fuzzy relational compositions are also valid for the compositions based on generalized quantifiers, yet sometimes in a weaken form. Apart from fuzzy relational compositions, the use of generalized quantifiers is also applied to images and preimages of fuzzy sets under fuzzy relations.

Excluding features in fuzzy relational compositions

Incorporation of the excluding features in fuzzy relational systems is proposed.Three alternatives ways of the incorporation are introduced.Their coincidence is investigated.Theoretical properties are studied.Application potential and impact is demonstrated on a real biological problem (Odonata identification). The aim of this paper is, first, to recall fuzzy relational compositions (products) and, to introduce an idea, how excluding features could be incorporated into the theoretical background. Apart from definitions, we provide readers with a theoretical investigation. This investigation addresses two natural questions. Firstly, under which conditions (in which underlying algebraic structures) the given three natural approaches to the incorporation of excluding symptoms coincide. And secondly, under which conditions, the proposed incorporation of excluding features preserves the same natural and desirable properties similar to those preserved by fuzzy relational compositions.The positive impact of the incorporation on reducing the suspicions provided by the basic circlet composition without losing the possibly correct suspicion is demonstrated on a real taxonomic identification (classification) of Odonata. Here, we demonstrate how the proposed concept may eliminate the weaknesses provided by the classical fuzzy relational compositions and, at the same time, compete with powerful machine learning methods. The aim of the demonstration is not to show that proposed concept outperforms classical approaches, but to show, that its potential is strong enough in order to complete them or in order to be combined with them and to use its different nature.

Incorporation of Excluding Features in Fuzzy Relational Compositions Based on Generalized Quantifiers

The concepts of incorporation of excluding features in fuzzy relational compositions and the compositions based on generalized quantifiers are useful tools for improving relevance and precision of the suspicion provided by the standard fuzzy relational compositions initial studied by Willis Bandler and Ladislav Kohout. They are independently extended from the standard compositions. However, it may become a very effective tool if they are used together. Taking this natural motivation leads us to introduce the concept of incorporation of excluding features in fuzzy relational compositions based on generalized quantifiers. Most of valid properties preserved for the two mentioned approaches will be proved for the new concept as well. Furthermore, an illustrative example will be presented for showing the usefulness of the approach.

Fuzzy Relation Equations with Fuzzy Quantifiers

In this paper, we follow the previous works on fuzzy relation compositions based on fuzzy quantifiers and we introduce systems of fuzzy relation equations stemming from compositions based on fuzzy quantifiers. We address the question, whether such systems under some specific conditions may become solvable, and we provide a positive answer. Based on the computational forms of the compositions using fuzzy quantifiers, we explain a way of getting solutions of the systems. In addition to showing some new properties and theoretical results, we provide readers with illustrative examples.

Fuzzy relational compositions based on grouping features

Fuzzy relational compositions play a crucial role in fundamentals of fuzzy mathematics as well as in distinct application areas. Recent studies introduce distinct generalizations, e.g., incorporation of excluding features or the use of generalized quantifiers. No matter the huge positive potential of these approaches, we demonstrate on a real example, that some limitations even for these extensions may be encountered if the features are constructed in a certain specific yet very natural way. However, these limitations can be overcome by a further improvement if a sort of grouping of features is applied. In particular, we study the compositions of fuzzy relations based on partitioned universe of features and then the combination of both above mentioned extensions will be applied and experimentally validated. The results are compared to the original ones provided in the investigation of excluding features without any implementation of generalized quantifiers nor with grouping of the features.

Non-preservation of chosen properties of fuzzy relational compositions based on fuzzy quantifiers

Fuzzy relational compositions based on fuzzy quantifiers naturally do not preserve all the properties that are preserved for “standard” fuzzy relational compositions and, in many cases, the property is preserved only in a weaker form. For example, the associativity, that is preserved in the standard case derived from the universal and the existential quantifiers, generally does not hold for the case of compositions based on fuzzy quantifiers. However, is it the case that only the standard quantifiers lead to the preservation of such properties? Without any restriction on the shape of the fuzzy relations, the answer is positive.

Compositions of Partial Fuzzy Relations

The aim of this contribution is to study compositions of partial fuzzy relational compositions, i.e., of fuzzy relations with membership degrees not defined on the whole universe. This is motivated by the possibility of existence of the relationships which are “undefined”, “unknown”, “meaningless”, “non-applicable”, “irrelevant”, etc. We introduce definitions for the new concept based on suitable operations used in the framework of the partial fuzzy set theory. The preservations of well-known interesting properties of compositions are studied for the compositions of partial fuzzy relations as well. An illustrative example is provided.

Fuzzy Relational Compositions Can Be Useful for Customers Credit Scoring in Financial Industry

Fuzzy relational compositions is an important topic in fuzzy mathematics and many researchers have applied that in various fields which the classification problem was more and less accounted for the significant part. Related to this problem, in this paper, we will show that fuzzy relational compositions assist in evaluating customers creditability (credit scoring) which is one of the most important problems in the financial industry. The purpose is to classify a given customer into two classes of accepted or rejected and to help loan officers to make a better decision. We will illustrate an experimental example with initial values provided by an bank expert and use LFL R-package as the practical tool to calculate the compositions for our application. The concept of so-called generalized quantifiers and excluding features incorporating in the compositions will be employed as well.

On the Use of Subproduct in Fuzzy Relational Compositions Based on Grouping Features

Fuzzy relational compositions have been extended and studied from distinct perspectives, and their use on the classification problem has been already demonstrated too. One of the recent approaches foreshadowed the positive influence of the so-called grouping features. When this improvement is being applied, the universe of features is partitioned into a number of groups of features and then the relevant composition is applied. The use of the concept was demonstrated on the real classification of Odonata (dragonflies). This paper shows that the Bandler-Kohout subproduct may appropriately serve as the chosen compositions in order to obtain an effective tool. The concepts of excluding features and generalized quantifiers will be employed in the constructed method as well. Some interesting properties will be introduced and a real example of the influence of the new concept will be provided.