Open Access
d-XC integrals: on the generalization of the expanded form of the Choquet integral by restricted dissimilarity functions and their applications
(IEEE, 2022) Wieczynski, Jonata; Fumanal Idocin, Javier; Lucca, Giancarlo; Borges, Eduardo N.; Da Cruz Asmus, Tiago; Emmendorfer, Leonardo R.; Bustince Sola, Humberto; Pereira Dimuro, Graçaliz; Automática y Computación; Automatika eta Konputazioa; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Restricted dissimilarity functions (RDFs) were introduced to overcome problems resulting from the adoption of the standard difference. Based on those RDFs, Bustince et al. introduced a generalization of the Choquet integral (CI), called d-Choquet integral, where the authors replaced standard differences with RDFs, providing interesting theoretical results. Motivated by such worthy properties, joint with the excellent performance in applications of other generalizations of the CI (using its expanded form, mainly), this paper introduces a generalization of the expanded form of the standard Choquet integral (X-CI) based on RDFs, which we named d-XC integrals. We present not only relevant theoretical results but also two examples of applications. We apply d-XC integrals in two problems in decision making, namely a supplier selection problem (which is a multi-criteria decision making problem) and a classification problem in signal processing, based on motor-imagery brain-computer interface (MI-BCI). We found that two d-XC integrals provided better results when compared to the original CI in the supplier selection problem. Besides that, one of the d-XC integrals performed better than any previous MI-BCI results obtained with this framework in the considered signal processing problem.
Embargo
Discrete fuzzy integrals and their applications
(2025) Wieczynski, Jonata; López Molina, Carlos; Pereira Dimuro, Graçaliz; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
This doctoral thesis introduces new generalizations of discrete fuzzy integrals with three distinct applications. We introduce four new families of discrete fuzzy integrals, including the d-XChoquet integral, the dCF -integrals, and the extended families of discrete Choquet and Sugeno integrals. These new generalizations are designed to provide better flexibility and performance in applications. In addition, we investigate the theoretical properties of these new integral generalizations, such as monotonicity. Furthermore, extensive experiments are conducted to compare the performance of the new, proposed, integrals with existing ones in three distinct applications: in classification, decision-making analysis, and in brain signal processing. The results demonstrate that the proposed generalizations perform better than other discrete fuzzy integrals in most applications.
Open Access
General admissibly ordered interval-valued overlap functions
(CEUR Workshop Proceedings (CEUR-WS.org), 2021) Da Cruz Asmus, Tiago; Pereira Dimuro, Graçaliz; Sanz Delgado, José Antonio; Wieczynski, Jonata; Lucca, Giancarlo; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Overlap functions are a class of aggregation functions that measure the verlapping degree between two values. They have been successfully applied in several problems in which associativity is not required, such as classification and image processing. Some generalizations of overlap functions were proposed for them to be applied in problems with more than two classes, such as 𝑛- dimensional and general overlap functions. To measure the overlapping of interval data, interval-valued overlap functions were defined, and, later, they were also generalized in the form of 𝑛-dimensional and general interval-valued overlap functions. In order to apply some of those concepts in problems with interval data considering the use of admissible orders, which are total orders that refine the most used partial order for intervals, 𝑛-dimensional admissibly ordered interval-valued overlap functions were recently introduced, proving to be suitable to be applied in classification problems. However, the sole construction method presented for this kind of function do not allow the use of the well known lexicographical orders. So, in this work we combine previous developments to introduce general admissibly ordered interval-valued overlap functions, while also presenting different construction methods and the possibility to combine such methods, showcasing the flexibility and adaptability of this approach, while also being compatible with the lexicographical orders.
Open Access
Applying d-XChoquet integrals in classification problems
(IEEE, 2022) Wieczynski, Jonata; Lucca, Giancarlo; Borges, Eduardo N.; Emmendorfer, Leonardo R.; Ferrero Jaurrieta, Mikel; Pereira Dimuro, Graçaliz; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Several generalizations of the Choquet integral have been applied in the Fuzzy Reasoning Method (FRM) of Fuzzy Rule-Based Classification Systems (FRBCS's) to improve its performance. Additionally, to achieve that goal, researchers have searched for new ways to provide more flexibility to those generalizations, by restricting the requirements of the functions being used in their constructions and relaxing the monotonicity of the integral. This is the case of CT-integrals, CC-integrals, CF-integrals, CF1F2-integrals and dCF-integrals, which obtained good performance in classification algorithms, more specifically, in the fuzzy association rule-based classification method for high-dimensional problems (FARC-HD). Thereafter, with the introduction of Choquet integrals based on restricted dissimilarity functions (RDFs) in place of the standard difference, a new generalization was made possible: the d-XChoquet (d-XC) integrals, which are ordered directional increasing functions and, depending on the adopted RDF, may also be a pre-aggregation function. Those integrals were applied in multi-criteria decision making problems and also in a motor-imagery brain computer interface framework. In the present paper, we introduce a new FRM based on the d-XC integral family, analyzing its performance by applying it to 33 different datasets from the literature.
Open Access
Application of the Sugeno integral in fuzzy rule-based classification
(Elsevier, 2024-09-27) Wieczynski, Jonata; Lucca, Giancarlo; Borges, Eduardo N.; Urío Larrea, Asier; López Molina, Carlos; Bustince Sola, Humberto; Pereira Dimuro, Graçaliz; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Fuzzy Rule-Based Classification System (FRBCS) is a well-known technique to deal with classification problems. Recent studies have considered the usage of the Choquet integral and its generalizations (e.g.: 𝐶𝑇 -integral, 𝐶𝐹 - Integral and 𝐶𝐶-integral) to enhance the performance of such systems. Such fuzzy integrals were applied to the Fuzzy Reasoning Method (FRM) to aggregate the fired fuzzy rules when classifying new data. However, the Sugeno integral, another well-known aggregation operator, obtained good results in other applications, such as brain–computer interfaces. These facts led to the present study, in which we consider the Sugeno integral in classification problems. That is, the Sugeno integral is applied in the FRM of a widely used FRBCS, and its performance is analyzed over 33 different datasets from the literature, also considering different fuzzy measures. To show the efficiency of this new approach, the results obtained are also compared with previous studies that involved the application of different aggregation functions. Finally, we perform a statistical analysis of the application.
Open Access
Constructing interval-valued fuzzy material implication functions derived from general interval-valued grouping functions
(IEEE, 2022) Pereira Dimuro, Graçaliz; Santos, Helida; Da Cruz Asmus, Tiago; Wieczynski, Jonata; Pinheiro, Jocivania; Bedregal, Benjamin; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
Grouping functions and their dual counterpart, overlap functions, have drawn the attention of many authors, mainly because they constitute a richer class of operators compared to other types of aggregation functions. Grouping functions are a useful theoretical tool to be applied in various problems, like decision making based on fuzzy preference relations. In pairwise comparisons, for instance, those functions allow one to convey the measure of the amount of evidence in favor of either of two given alternatives. Recently, some generalizations of grouping functions were proposed, such as (i) the n-dimensional grouping functions and the more flexible general grouping functions, which allowed their application in n-dimensional problems, and (ii) n-dimensional and general interval-valued grouping functions, in order to handle uncertainty on the definition of the membership functions in real-life problems. Taking into account the importance of interval-valued fuzzy implication functions in several application problems under uncertainty, such as fuzzy inference mechanisms, this paper aims at introducing a new class of interval-valued fuzzy material implication functions. We study their properties, characterizations, construction methods and provide examples.
Open Access
Application and comparison of CC-integrals in business group decision making
(Springer, 2022) Wieczynski, Jonata; Lucca, Giancarlo; Borges, Eduardo N.; Pereira Dimuro, Graçaliz; Lourenzutti, Rodolfo; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Optimized decisions is required by businesses (analysts) if they want to stay open. Even thought some of these are from the knowhow of the managers/executives, most of them can be described mathematically and solved (semi)-optimally by computers. The Group Modular Choquet Random Technique for Order of Preference by Similarity to Ideal Solution (GMC-RTOPSIS) is a Multi-Criteria Decision Making (MCDM) that was developed as a method to optimize the later types of problems, by being able to work with multiple heterogeneous data types and interaction among different criteria. On the other hand the Choquet integral is widely used in various fields, such as brain-computer interfaces and classification problems. With the introduction of the CC-integrals, this study presents the GMC-RTOPSIS method with CC-integrals. We applied 30 different CC-integrals in the method and analyzed its results using 3 different methods. We found that by modifying the decisionmaking method we allow for more flexibility and certainty in the choosing process.
Open Access
CC-separation measure applied in business group decision making
(SciTePress, 2021) Wieczynski, Jonata; Lucca, Giancarlo; Borges, Eduardo N.; Pereira Dimuro, Graçaliz; Lourenzutti, Rodolfo; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In business, one of the most important management functions is decision making The Group Modular Choquet Random TOPSIS (GMC-RTOPSIS) is a Multi-Criteria Decision Making (MCDM) method that can work with multiple heterogeneous data types. This method uses the Choquet integral to deal with the interaction between different criteria. The Choquet integral has been generalized and applied in various fields of study, such as imaging processing, brain-computer interface, and classification problems. By generalizing the so-called extended Choquet integral by copulas, the concept of CC-integrals has been introduced, presenting satisfactory results when used to aggregate the information in Fuzzy Rule-Based Classification Systems. Taking this into consideration, in this paper. we applied 11 different CC-integrals in the GMC-RTOPSIS. The results demonstrated that this approach has the advantage of allowing more flexibility and certainty in the choosing process by giving a higher separation between the first and second-ranked alternatives.
Open Access
dCF-integrals: generalizing CF-integrals by means of restricted dissimilarity functions
(IEEE, 2022) Wieczynski, Jonata; Lucca, Giancarlo; Pereira Dimuro, Graçaliz; Borges, Eduardo N.; Sanz Delgado, José Antonio; Da Cruz Asmus, Tiago; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa, PJUPNA1926
The Choquet integral (CI) is an averaging aggregation function that has been used, e.g., in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems (FRBCSs) and in multicriteria decision making in order to take into account the interactions among data/criteria. Several generalizations of the CI have been proposed in the literature in order to improve the performance of FRBCSs and also to provide more flexibility in the different models by relaxing both the monotonicity requirement and averaging conditions of aggregation functions. An important generalization is the CF -integrals, which are preaggregation functions that may present interesting nonaveraging behavior depending on the function F adopted in the construction and, in this case, offering competitive results in classification. Recently, the concept of d-Choquet integrals was introduced as a generalization of the CI by restricted dissimilarity functions (RDFs), improving the usability of CIs, as when comparing inputs by the usual difference may not be viable. The objective of this article is to introduce the concept of dCF -integrals, which is a generalization of CF -integrals by RDFs. The aim is to analyze whether the usage of dCF -integrals in the FRM of FRBCSs represents a good alternative toward the standard CF -integrals that just consider the difference as a dissimilarity measure. For that, we consider six RDFs combined with five fuzzy measures, applied with more than 20 functions F . The analysis of the results is based on statistical tests, demonstrating their efficiency. Additionally, comparing the applicability of dCF -integrals versus CF -integrals, the range of the good generalizations of the former is much larger than that of the latter.
Open Access
Application of the Sugeno integral in fuzzy rule-based classification
(Springer, 2022) Wieczynski, Jonata; Lucca, Giancarlo; Borges, Eduardo N.; Pereira Dimuro, Graçaliz; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Fuzzy Rule-Based Classification System (FRBCS) is a well known technique to deal with classification problems. Recent studies have considered the usage of the Choquet integral and its generalizations to enhance the quality of such systems. Precisely, it was applied to the Fuzzy Reasoning Method (FRM) to aggregate the fired fuzzy rules when classify new data. On the other side, the Sugeno integral, another well known aggregation operator, obtained good results when applied to brain-computer interfaces. Those facts led to the present study in which we consider the Sugeno integral in classification problems. That is, the Sugeno integral is applied in the FRM of a widely used FRBCS and its performance is analyzed over 33 different datasets from the literature. In order to show the efficiency of this new approach, the obtained results are also compared to past studies involving the application of different aggregation functions. Finally, we perform a statistical analysis of the application.