Open Access
An empirical study on supervised and unsupervised fuzzy measure construction methods in highly imbalanced classification
(IEEE, 2020) Uriz Martín, Mikel Xabier; Paternain Dallo, Daniel; Bustince Sola, Humberto; Galar Idoate, Mikel; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
The design of an ensemble of classifiers involves the definition of an aggregation mechanism that produces a single response obtained from the information provided by the classifiers. A specific aggregation methodology that has been studied in the literature is the use of fuzzy integrals, such as the Choquet or the Sugeno integral, where the associated fuzzy measure tries to represent the interaction existing between the classifiers of the ensemble. However, defining the big number of coefficients of a fuzzy measure is not a trivial task and therefore, many different algorithms have been proposed. These can be split into supervised and unsupervised, each class having different learning mechanisms and particularities. Since there is no clear knowledge about the correct method to be used, in this work we propose an experimental study for comparing the performance of eight different learning algorithms under the same framework of imbalanced dataset. Moreover, we also compare the specific fuzzy integral (Choquet or Sugeno) and their synergies with the different fuzzy measure construction methods.
Open Access
d-Choquet integrals: Choquet integrals based on dissimilarities
(Elsevier, 2020) Bustince Sola, Humberto; Mesiar, Radko; Fernández Fernández, Francisco Javier; Galar Idoate, Mikel; Paternain Dallo, Daniel; Altalhi, A. H.; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Takáč, Zdenko; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa, PJUPNA13
The paper introduces a new class of functions from [0,1]n to [0,n] called d-Choquet integrals. These functions are a generalization of the 'standard' Choquet integral obtained by replacing the difference in the definition of the usual Choquet integral by a dissimilarity function. In particular, the class of all d-Choquet integrals encompasses the class of all 'standard' Choquet integrals but the use of dissimilarities provides higher flexibility and generality. We show that some d-Choquet integrals are aggregation/pre-aggregation/averaging/functions and some of them are not. The conditions under which this happens are stated and other properties of the d-Choquet integrals are studied.
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
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
A generalization of the Choquet integral defined in terms of the Mobius transform
(IEEE, 2020) Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Horanská, Lubomíra; Mesiar, Radko; Stupñanová, Andrea; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In this article, we propose a generalization of the Choquet integral, starting fromits definition in terms of the Mobius transform. We modify the product on R considered in the Lovasz extension form of the Choquet integral into a function F, and we discuss the properties of this new functional. For a fixed n, a complete description of all F yielding an n-ary aggregation function with a fixed diagonal section, independent of the considered fuzzy measure, is given, and several particular examples are presented. Finally, all functions F yielding an aggregation function, independent of the number n of inputs and of the considered fuzzy measure, are characterized, and related aggregation functions are shown to be just the Choquet integrals over the distorted inputs.
Open Access
Funções de agregação baseadas em integral de Choquet aplicadas em redimensionalização de imagens
(Universidade Passo Fundo, 2019) Bueno, Jéssica C. S.; Dias, Camila A.; Pereira Dimuro, Graçaliz; Borges, Eduardo N.; Botelho, Silvia S. C.; Mattos, Viviane L. D. de; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
The increasing data volume, coupled with the high complexity of these data, has generated the need to develop increasingly efficient knowledge extraction techniques, both in computational cost and precision. Most of the problems that are addressed by these techniques have complex information to be identified. For this, machine learning methods are used, where these methods use a variety of functions inside the different steps that are employed in their architectures. One of these consists in the use of aggregation functions to resize images. In this context, a study of aggregation functions based on the Choquet integral is presented, where the main feature of Choquet integral, in comparison with other aggregation functions, resides in the fact that it considers, through the fuzzy measure, the interaction between the elements to be aggregated. Thus, an evaluation study of the performance of the standard Choquet integral functions is presented (Choquet integral based on Copula in relation to the maximum and average functions) looking for results that may be better than the usual applied aggregation functions. The results of such comparisons are promising when evaluated through measures of image quality.
Open Access
Enhancing LSTM for sequential image classification by modifying data aggregation
(IEEE, 2021) Takáč, Zdenko; Ferrero Jaurrieta, Mikel; Horanská, Lubomíra; Krivonakova, Nada; Pereira Dimuro, Graçaliz; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Recurrent Neural Networks (RNN) model sequential information and are commonly used for the analysis of time series. The most usual operation to fuse information in RNNs is the sum. In this work, we use a RNN extended type, Long Short-Term Memory (LSTM) and we use it for image classification, to which we give a sequential interpretation. Since the data used may not be independent to each other, we modify the sum operator of an LSTM unit using the n-dimensional Choquet integral, which considers possible data coalitions. We compare our methods to those based on usual aggregation functions, using the datasets Fashion-MNIST and MNIST.
Open Access
VCI-LSTM: Vector choquet integral-based long short-term memory
(IEEE, 2022) Ferrero Jaurrieta, Mikel; Takáč, Zdenko; Fernández Fernández, Francisco Javier; Horanská, Lubomíra; Pereira Dimuro, Graçaliz; Montes Rodríguez, Susana; Díaz, Irene; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Choquet integral is a widely used aggregation operator on one-dimensional and interval-valued information, since it is able to take into account the possible interaction among data. However, there are many cases where the information taken into account is vectorial, such as Long Short-Term Memories (LSTM). LSTM units are a kind of Recurrent Neural Networks that have become one of the most powerful tools to deal with sequential information since they have the power of controlling the information flow. In this paper, we first generalize the standard Choquet integral to admit an input composed by $n$-dimensional vectors, which produces an $n$-dimensional vector output. We study several properties and construction methods of vector Choquet integrals. Then, we use this integral in the place of the summation operator, introducing in this way the new VCI-LSTM architecture. Finally, we use the proposed VCI-LSTM to deal with two problems: sequential image classification and text classification.
Open Access
Aggregation functions based on the Choquet integral applied to image resizing
(Atlantis Press, 2019) Bueno, Jéssica C. S.; Dias, Camila A.; Pereira Dimuro, Graçaliz; Santos, Helida; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
The rising volume of data and its high complexity has brought the need of developing increasingly efficient knowledge extraction techniques, which demands efficiency both in computational cost and in accuracy. Most of problems that are handled by these techniques has complex information to be identified. So, machine learning methods are frequently used, where a variety of functions can be applied in the different steps that are employed in their architecture. One of them is the use of aggregation functions aiming at resizing images. In this context, we introduce a study of aggregation functions based on the Choquet integral, whose main characteristic in comparison with other aggregation functions is that it considers, through fuzzy measure, the interaction between the elements to be aggregated. Thus, our main goal is to present an evaluation study of the performance of the standard Choquet integral the and copula-based generalization of the Choquet integral in relation to the maximum and mean functions, looking for results that may be better than the aggregation functions commonly applied. The results of such comparisons are promising, when evaluated through image quality metrics.
Open Access
Dissimilarity based choquet integrals
(Springer, 2020) Bustince Sola, Humberto; Mesiar, Radko; Fernández Fernández, Francisco Javier; Galar Idoate, Mikel; Paternain Dallo, Daniel; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In this paper, in order to generalize the Choquet integral, we replace the difference between inputs in its definition by a restricted dissimilarity function and refer to the obtained function as d-Choquet integral. For some particular restricted dissimilarity function the corresponding d-Choquet integral with respect to a fuzzy measure is just the ‘standard’ Choquet integral with respect to the same fuzzy measure. Hence, the class of all d-Choquet integrals encompasses the class of all 'standard' Choquet integrals. This approach allows us to construct a wide class of new functions, d-Choquet integrals, that are possibly, unlike the 'standard' Choquet integral, outside of the scope of aggregation functions since the monotonicity is, for some restricted dissimilarity function, violated and also the range of such functions can be wider than [0, 1], in particular it can be [0, n].