The concept of unavoidable features in fuzzy relational compositions☆
Introduction
The impressively interesting area of fuzzy relational compositions [1] was elaborated by numerous researchers, in particular by Bandler and Kohout, Sanchez, Pedrycz, Di Nola, Sessa, Kerre, De Baets, Bělohlávek and by many others. For interested readers, we introduced only some of the crucial references being aware that the full list only of the most important ones would outreach the abilities of this article [2], [3], [4], [5], [6]. One of the main advantages of this area was the ability to mathematically formulate relationships expressed transparently in natural language and thus, reaching the primary and initial goal of fuzzy systems as proposed by Zadeh. The unquestionable advantages of such white-boxes are however, in the recent years, suppressed by the major focus on accuracy provided by black-box data-driven models, usually using machine learning principles. The goal of this contribution is not to criticize this paradigm shift. This article attempts to comprehend it and challenge it by enhanced fuzzy relational-based approaches to tasks typically solved by machine learning techniques, e.g., classification tasks.
Although usual machine learning and other data-driven black-box approaches are possessing several weaknesses, for instance its difficult implementation in the cases of insufficient data or the lack of explainability [7] of such systems, they usually dominate in accuracy, generality, and efficiency. Moreover, there are several approaches trying to heal the Achilles heel of such techniques in the situations where complementary white-box techniques could be used instead. So, we may observe distinct approaches to improve the explainability [8] of such systems or distinct techniques helping to overcome potential the lack of training data by their augmentation [9]. It is worth mentioning that such approaches are strongly domain-related and while they can be very efficient for computer vision, they may face problems in, e.g., medical data analyses with the lack of patient records.
We are convinced that favoring the data-driven machine learning approaches is a natural consequence of often insufficient accuracy of white-boxes. Indeed, such systems were not primarily designed to optimize the accuracy but rather to involve transparency and interpretability [10] and we could argue that white-box approaches could offer more than black-boxes in these aspects. However, comparable accuracy is unquestionably a demand that has to be met. Thus, instead of downgrading the black-box approaches, we should attempt to improve the accuracy white-boxes to become competitive. In this article, we follow this task and focus on expert knowledge based classifiers constructed as combinations of fuzzy relational compositions. We will show, that this improvement does not have to be reached only via technical optimization by distinct tools but often just by employing a crucial idea that mimics a real expert decision-making, which is in practice exactly what is done by powerful machine learning methods.
Recall, that fuzzy relational compositions, that have rich origins in the investigations by Bandler and Kohout from early 1980 s, have been extended in several directions [11], [12]. We recall mainly the concept of excluding features [13] that was successfully applied to a classification of biological samples (in particular dragonflies) into species. This approach does not require mathematically as complicated apparatus as, e.g., the employment of generalized quantifiers [12]. But the positive impact was stronger. Its main idea consisted in building simple compositions and combining them in such a way, that an object is never assigned to a class if it carries an element that is excluding for the given class. Though such an idea is very simple and natural, it improved significantly the results obtained by standard compositions dealing only with connecting features and not focusing on the excluding ones.
In this paper, we follow this approach based on appropriate combinations of simple compositions that express potentially powerful pieces of information. Compared to the concept of excluding features, the proposed concept of unavoidable features is semantically opposite. It ensures that an object is assigned only to classes, whose all unavoidable features are carried by the given object. This idea mimics how humans think when assigning the most natural (most typical or most expected) classes to given objects. Indeed, the unavoidable features cannot be omitted, no-one would assign an object to a class only because of observing that the object carries auxiliary features of the class. And such a requirement to carry unavoidable features would be necessarily mirrored in labeling the data by an expert. And consequently, mimicked by any data-driven method. As we will show, knowledge-based system may boldly compete with the machine learning ones even in accuracy, assuming that they will employ the same paradigms. In this case, if the knowledge-based method will employ the idea of unavoidable features.
The structure of the paper is following. Section 2 provides readers with preliminaries related to fuzzy relational compositions, Section 3 introduces the concept of unavoidable features, Section 4 brings an exhaustive study on the preservation of mathematical properties by the proposed composition, and Section 5 provides readers with illustrative experiments justifying the suggested approach.
Section snippets
Algebra of operations
In order to set up algebra of operations that we will use in the whole paper, we fix our choice to a complete residuated lattice defined on the unit interval . For some specific cases, we will narrow our focus on complete MV-algebra defined on the same support, for more details about the structures of truth-values, we refer readers to [14], [15].
Definition 1 An algebra is a residuated lattice if is a lattice with the least and the greatest element is a commutative
Employing features that are unavoidable
The semantics of the four basic fuzzy relational compositions of two binary fuzzy relations, namely , , , and , provide an inspiring insight into the expressive power of such constructions. However, it is clear, that in their original form, they can hardly compete in terms of accuracy with recent methods. Nevertheless, we are convinced that a significant improvement in the accuracy is just a matter of enhancing the models using the related fuzzy relational constructions. The use of
Properties
This section provides an investigation of the properties of the newly proposed fuzzy relational compositions that employ the concept of unavoidable features. Unless explicitly stated, all the operations will be from a complete residuated lattice . Let be non-empty universes and let the symbols for the intersection and the union will denote the Gödel variants of the operations.
Experiment with dragonfly classification
This section provides readers with an illustrative experiment demonstrating the impact of the unavoidable features in fuzzy relational compositions. Naturally, we consider an experiment that has been already used for justifying the concept of excluding features [13] and that has been used for testing distinct approaches within a part of a long-term project connecting citizen-science, biodiversity and modern technologies. This project led to the development of a cell-phone application [26] that
Conclusions
Obviously, the experimental part cannot clearly answer what is better, whether the data-driven classifiers or the explainable fuzzy relational models based on expert knowledge. First of all, due to the limited number of experiments for which we have the expert data stored in , and at disposal, and secondly, due to the above-mentioned different interpretation of the numbers in provided by methods of distinct nature. Even if the number of experiments increased substantially, we are convinced
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (39)
- et al.
Semantics of implication operators and fuzzy relational products
Int. J. Man-Mach. Stud.
(1980) - et al.
Fuzzy relational compositions
Fuzzy Sets and Systems
(1993) Sup-t-norm and inf-residuum are one type of relational product: unifying framework and consequences
Fuzzy Sets and Systems
(2012)Human knowledge in constructing ai systems — Neural logic networks approach towards an explainable ai
Procedia Comput. Sci.
(2018)- et al.
Relational compositions in fuzzy class theory
Fuzzy Sets and Systems
(2009) - et al.
Extensions of fuzzy relational compositions based on generalized quantifiers
Fuzzy Sets and Systems
(2018) - et al.
Excluding features in fuzzy relational compositions
Expert Syst. Appl.
(2017) Fuzzy relational equations with generalized connectives and their applications
Fuzzy Sets and Systems
(1983)- et al.
L-fuzzy quantifiers of type determined by fuzzy measures
Fuzzy Sets and Systems
(2009) - et al.
Type fuzzy quantifiers determined by fuzzy measures. part II. permutation and isomorphism invariances
Fuzzy Sets and Systems
(2014)
AI-enabled recruiting: What is it and how should a manager use it?
Business Horizons
Missing values and dragonfly operations in fuzzy relational compositions
Internat. J. Approx. Reason.
Fuzzy Relational Systems: Foundations and Principles
Fuzzy Relation Equations and Their Applications to Knowledge Engineering
A bibliometric analysis of the explainable artificial intelligence research field
Return of the devil in the details: Delving deep into convolutional nets
Interpretability of linguistic variables: A formal account
Kybernetika
Cited by (7)
Immediate consequences operator on generalized quantifiers
2023, Fuzzy Sets and SystemsGeneralized quantifiers in formal concept analysis
2022, Journal of Computational and Applied MathematicsCitation Excerpt :The natural generalization of universal and existential quantifiers to fuzzy sets and fuzzy logic is given by the infimum and supremum operators [1–6], which also inherits this drawback. In order to solve the gap between both notions, generalized quantifiers [7–13] have been introduced. These new operators are intermediate quantifiers between the universal and existential quantifiers, which can model fuzzy notions such as “Most” or “Many”, providing less strict quantifiers to the applications.
lfl: An R package for linguistic fuzzy logic
2022, Fuzzy Sets and SystemsCitation Excerpt :This, jointly with other inferences (Takagi-Sugeno-Kang or Ishibuchi's method), makes frbs a powerfull and rich package for distinct purposes, such as regression, data-driven classification, control, or decision-making. Owing to mention other differences, lfl contains complete and very recent group of fuzzy relational composition calculus including the most recent techniques [46,30]. These structures can be used in the inference processing fuzzy inputs as well as in distinct, e.g., expert-driven, classification or decision-making tasks.
EVKA—Fuzzy Modelling Based System for the Decision-Making Support of Community Workers
2023, Social Science Computer ReviewDynamic Evaluation of Fuzzy Compositions
2023, IEEE International Conference on Fuzzy SystemsComposition Models of Fuzzy Relations Considering Importance Levels of Features∗
2022, Proceedings - International Conference on Knowledge and Systems Engineering, KSE