Information aggregation often requires ordering of data based on different criteria, leading to the introduction of functions like Ordered Weighted Averaging (OWA) and Induced Ordered Weighted Averaging (IOWA) operators. The sorting of vector information is not trivial, and so far no induced aggregation operators have been studied in the multivalued context. We introduce the Vector Induced Ordered Weighted Averaging (VIOWA) operators, a vector extension of IOWA designed for aggregating multivalued data. We define VIOWA operators based on admissible orders, using all admissible permutations, and we explore its formulation using the best admissible permutations based on the degree of totalness. Additionally, we analyze the basic properties of these methods and their interrelations. To demonstrate the effectiveness of VIOWA operators, we apply them to multivalued time series prediction, in order to aggregate several vector-valued model predictions. Experiments on five datasets show the usefulness of VIOWA operator-based aggregation, outperforming individual models and other classical aggregation methods, highlighting its potential in multivariate settings.

This paper introduces a novel class of aggregation functions, called Sugeno-inspired aggregation functions, which are conceptually based on the Sugeno integral. The concept of fuzzy measure is rebuilt by incorporating a function designed to evaluate coalitions composed of all elements except one. This approach frames aggregation as a comparison between the value of a given element and the aggregation outcome of the coalition that excludes it. The fundamental properties of this new class of aggregation functions are investigated and their potential applications are explored. The theoretical analysis shows that Sugeno-inspired aggregation functions preserve key features of the original Sugeno integral while eliminating the need to precompute a fuzzy measure, thereby simplifying their use in practical settings. An illustrative example highlight the effectiveness of the proposed aggregation functions in evaluating clustering quality and suggest the potential for novel aggregation approaches to enhance cluster evaluation methodologies.

The 2019 coronavirus disease outbreak, caused by the severe acute respiratory syndrome type-2 virus (SARS-CoV-2), was declared a pandemic in March 2020. Since its emergence to the present day, this disease has brought multiple countries to the brink of health care collapse during several waves of the disease. One of the most common tests performed on patients is chest x-ray imaging. These images show the severity of the patient’s illness and whether it is indeed covid or another type of pneumonia. Automated assessment of this type of imaging could alleviate the time required for physicians to treat and diagnose each patient. To this end, in this paper we propose the use of Convolutional Neural Networks (CNNs) to carry out this process. The aim of this paper is twofold. Firstly, we present a pipeline adapted to this problem, covering all steps from the preprocessing of the datasets to the generation of classification models based on CNNs. Secondly, we have focused our study on the modification of the information fusion processes of this type of architectures, in the pooling layers. We propose a number of aggregation theory functions that are suitable to replace classical processes and have shown their benefits in past applications, and study their performance in the context of the x-ray classification problem. We find that replacing the feature reduction processes of CNNs leads to drastically different behaviours of the final model, which can be benefitial when prioritizing certain metrics such as precision or recall.

Overlap and grouping functions can be used to measure events in which we must consider either the maximum or the minimum lack of knowledge. The commutativity of overlap and grouping functions can be dropped out to introduce the notions of pseudo-overlap and pseudo-grouping functions, respectively. These functions can be applied in problems where distinct orders of their arguments yield different values, i.e., in non-symmetric contexts. Intending to reduce the complexity of pseudo-overlap and pseudo-grouping functions, we propose new construction methods for these functions from generalized concepts of additive and multiplicative generators. We investigate the isomorphism between these families of functions. Finally, we apply these functions in an illustrative problem using them in a time series prediction combined model using the IOWA operator to evidence that using these generators and functions implies better performance.

Choquet integral is a widely used aggregation operator on 1-D 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 article, 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 (VCIs). 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: 1) sequential image classification; 2) text classification.

Restricted equivalence functions are well-known functions to compare two numbers in the interval between 0 and 1. Despite the numerous works studying the properties of restricted equivalence functions and their multiple applications as support for different similarity measures, an extension of these functions to an n-dimensional space is absent from the literature. In this article, we present a novel contribution to the restricted equivalence function theory, allowing to compare multivalued elements. Specifically, we extend the notion of restricted equivalence functions from$L$to$L^{n}$and present a new similarity construction on$L^{n}$. proposal is tested in the context of color image anisotropic diffusion as an example of one of its many applications.

Automatic image detection is one of the most important areas in computing due to its potential application in numerous real-world scenarios. One important tool to deal with that is calledoverlap indices. They were introduced as a procedure to provide the maximum lack of knowledge when comparing two fuzzy objects. They have been successfully applied in the following fields: image processing, fuzzy rule-based systems, decision making, and computational brain interfaces. This notion ofoverlap indicesis also necessary for applications in which type-2 fuzzy sets are required. In this article, we introduce the notion oftype-$(2,k)$overlap index($k \in \lbrace 0,1,2\rbrace$) in the setting of type-2 fuzzy sets. We describe both the reasons that have led to this notion and the relationships that naturally arise among the algebraic underlying structures. Finally, we illustrate how type-$(2,k)$overlap indices can be employed in the setting of fuzzy rule-based systems when the involved objects are type-2 fuzzy sets.

A function that takes n numbers as input and outputs one number is said to be homogeneous whenever the result of multiplying each input by a certain factor λ yields the original output multiplied by that same factor.This concept has been extended by the notion of abstract homogeneity, which generalizes the product in the expression of homogeneity by a general function g and the effect of the factor λ by an automorphism.However, the effect of parameter λ remains unchanged for all the input values.In this study, we generalize further the condition of abstract homogeneity by introducing ℱ-homogeneity, which is defined with respect to a family of functions, enabling a different behavior for each of the inputs.Next, we study the properties that are satisfied by this family of functions and, moreover, we link this concept with the condition of directional monotonicity, which is a trendy property in the framework of aggregation functions.To achieve that, we generalize directional monotonicity by ℱ directional monotonicity, which is defined with respect to a family of functions ℱ and a family of vectors  .Finally, we show how the introduced concepts could be applied

In this article, we develop moderate deviation functions to measure similarity and dissimilarity among a set of given interval-valued data to construct interval-valued aggregation functions, and we apply these functions in two motor-imagery brain–computer interface (MI-BCI) systems to classify electroencephalography signals. To do so, we introduce the notion of interval-valued moderate deviation function and, in particular, we study those interval-valued moderate deviation functions, which preserve the width of the input intervals. In order to apply them in an MI-BCI system, we first use fuzzy implication operators to measure the uncertainty linked to the output of each classifier in the ensemble of the system, and then we perform the decision making phase using the new interval-valued aggregation functions. We have tested the goodness of our proposal in two MI-BCI frameworks, obtaining better results than those obtained using other numerical aggregation and interval-valued ordered weighted averaging operators, and obtaining competitive results versus some nonaggregation-based frameworks.

In this work, we generalize the notion of restricted equivalence function for type-2 fuzzy sets, leading to the notion of extended restricted equivalence functions. We also study how under suitable conditions, these new functions recover the standard axioms for restricted equivalence functions in the real setting. Extended restricted equivalence functions allow us to compare any two general type-2 fuzzy sets and to generate a similarity measure for type-2 fuzzy sets. The result of this similarity is a fuzzy set on the same referential set (i.e., domain) as the considered type-2 fuzzy set. The latter is crucial for applications such as explainable AI and decision-making, as it enables an intuitive interpretation of the similarity within the domain-specific context of the fuzzy sets. We show how this measure can be used to compare type-2 fuzzy sets with different membership functions in such a way that the uncertainty linked to type-2 fuzzy sets is not lost. This is achieved by generating a fuzzy set rather than a single numerical value. Furthermore, we also show how to obtain a numerical value for discrete referential sets.

A new concept of equivalence between intervals and the induced indistinguishability between interval-valued (IV) fuzzy sets are proposed and considered. A new notion of the degree of IV equivalence is presented where partial or linear orders and the width of intervals are involved reflecting uncertainty. Furthermore, construction methods of the considered equivalences are provided and the relation between them and other notions of IV equivalences are studied. Finally, a methodology to apply the proposed equivalences in the field of image processing is shown, along with an illustrative example.

Two of the main research lines in the theory of aggregation functions is the extension to more general domains and the relaxation of the monotonicity conditions. In this work, we discuss the state-of-the-art of the main introduced relaxed forms of monotonicity that can be found in the literature, i.e., weak, directional, ordered directional and strengthened ordered directional monotonicity. We pay special attention to the extension of a relaxed form of monotonicity to the interval-valued setting and we propose the concepts of ordered and strengthened ordered directional monotonicity for this general setting. Moreover, we study the main properties of the functions that satisfy the introduced properties and present some construction methods.

In this work we propose a new axiomatic definition of distance between interval-valued fuzzy sets which takes into account the width of the membership intervals and linear orders. We discuss some construction methods using aggregation functions which are defined in terms of admissible orders.

In this study, we discuss a new class of fuzzy subsethood measures between fuzzy sets. We propose a new definition of fuzzy subsethood measure as an intersection of other axiomatizations and provide two construction methods to obtain them. The advantage of this new approach is that we can construct fuzzy subsethood measures by aggregating fuzzy implication operators which may satisfy some properties widely studied in literature. We also obtain some of the classical measures such as the one defined by Goguen. The relationships with fuzzy distances, penalty functions, and similarity measures are also investigated. Finally, we provide an illustrative example which makes use of a fuzzy entropy defined by means of our fuzzy subsethood measures for choosing the best fuzzy technique for a specific problem.

Admissible orders have brought the structure of chains in the framework of interval-valued fuzzy sets. However, a deeper study of functions monotone with respect to admissible orders is still missing in the literature. In this work, we consider the construction of negations and strong negations on intervals with respect to admissible orders, in particular, for the Xu and Yager and lexicographical orders, as well as for those based on Kαoperators. We introduce and discuss an approach to the construction of strong negations on intervals with respect to Kα,βorders based on an arbitrary couple of strong negations defined over the standard real interval [0, 1]. The introduced strong negations have a deep impact on all fields exploiting fuzzy methods dealing with intervals, allowing to introduce complements, dual aggregations, implications, entropies, etc.