Open Access
Pre-aggregation functions: construction and an application
(IEEE, 2015) Lucca, Giancarlo; Sanz Delgado, José Antonio; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Mesiar, Radko; Kolesárová, Anna; Bustince Sola, Humberto; Automática y Computación; Automatika eta Konputazioa
In this work we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build pre-aggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators like the maximum and the Choquet integral.
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.
Open Access
Neuro-inspired edge feature fusion using Choquet integrals
(Elsevier, 2021) Marco Detchart, Cedric; Lucca, Giancarlo; López Molina, Carlos; Miguel Turullols, Laura de; Pereira Dimuro, Graçaliz; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
It is known that the human visual system performs a hierarchical information process in which early vision cues (or primitives) are fused in the visual cortex to compose complex shapes and descriptors. While different aspects of the process have been extensively studied, such as lens adaptation or feature detection, some other aspects, such as feature fusion, have been mostly left aside. In this work, we elaborate on the fusion of early vision primitives using generalizations of the Choquet integral, and novel aggregation operators that have been extensively studied in recent years. We propose to use generalizations of the Choquet integral to sensibly fuse elementary edge cues, in an attempt to model the behaviour of neurons in the early visual cortex. Our proposal leads to a fully-framed edge detection algorithm whose performance is put to the test in state-of-the-art edge detection datasets.
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
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
Fuzzy integrals for edge detection
(Springer, 2023) Marco Detchart, Cedric; Lucca, Giancarlo; Pereira Dimuro, Graçaliz; Da Cruz Asmus, Tiago; López Molina, Carlos; Borges, Eduardo N.; Rincón Arango, Jaime Andrés; Julian, Vicente; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In this work, we compare different families of fuzzy integrals in the context of feature aggregation for edge detection. We analyze the behaviour of the Sugeno and Choquet integral and some of its generalizations. In addition, we study the influence of the fuzzy measure over the extracted image features. For testing purposes, we follow the Bezdek Breakdown Structure for edge detection and compare the different fuzzy integrals with some classical feature aggregation methods in the literature. The results of these experiments are analyzed and discussed in detail, providing insights into the strengths and weaknesses of each approach. The overall conclusion is that the configuration of the fuzzy measure does have a paramount effect on the results by the Sugeno integral, but also that satisfactory results can be obtained by sensibly tuning such parameter. The obtained results provide valuable guidance in choosing the appropriate family of fuzzy integrals and settings for specific applications. Overall, the proposed method shows promising results for edge detection and could be applied to other image-processing tasks.
Open Access
Additively generated (a,b)-implication functions*
(IEEE, 2023) Santos, Helida; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Paiva, Rui; Lucca, Giancarlo; Moura, Bruno; Cruz, Anderson; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Some problems involving classification through neural networks are known to use inputs out of the scope of the unit interval. Therefore, defining operations on arbitrary closed real intervals may be an interesting strategy to tackle this issue and enhance those application environments. In this paper we follow the ideas already discussed in the literature regarding (a,b)-fusion functions, and (a,b)-negations, to provide a new way to construct implication functions. The main idea is to construct an operator using additively generated functions that preserve the properties required by implication functions.
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
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
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.