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.

Accurate number-of-clusters estimation (NCE) is a central task in many clustering applications, particularly for prototype-based k -centers methods like k -Means, which require the number of clusters k to be specified in advance. This paper presents CRB-NCE, a general cluster cohesion rule-based framework for NCE integrating three main innovations: (i) the introduction of tail ratios to reliably identify decelerations in sequences of cohesion measures, (ii) a threshold-based rule system supporting accurate NCE, and (iii) an optimization-driven approach to learn these thresholds from synthetic datasets with controlled clustering complexity. Two cohesion measures are considered: inertia (SSE) and a new, scale-invariant metric called the mean coverage index. CRB-NCE is mainly applied to derive general-purpose NCE methods, but, most importantly, it also provides an adaptable framework that enables producing specialized procedures with enhanced performance under specific conditions, such as particular clustering algorithms or overlapping cluster structures. Extensive evaluations on synthetic Gaussian datasets (both standard and high-dimensional), clustering benchmarks, and real-world datasets show that CRB-NCE methods consistently achieve robust and competitive NCE performance with efficient runtimes compared to a broad baseline of internal clustering validity indices and other NCE methods.

Aggregation theory is devoted to the fusing of several values into a unique output that summarizes the given information. Typically, the aggregation process is formalized in terms of an increasing mathematical function that maps the input values to the result, fulfilling some boundary conditions. However, this formalization can be too restrictive for some scenarios. In some cases, the inputs can be seen as observations of random variables, the aggregation result being also a random variable. In others, the aggregation process can be identified as a program that performs the aggregation rather than a mathematical function. In this direction, the concepts of aggregation of random variables and computable aggregation have been defined in the literature. This paper is devoted to the definition of computable aggregation of random variables, which are computer programs, not functions, that aggregate random variables, not numbers. Special attention is given to different possible alternatives to modelize random variables and monotonicity. The implementation of some examples is also provided.

In this paper we address the problem of fuzzy measures index calculation. On the basis of fuzzy sets, Murofushi and Soneda proposed an interaction index to deal with the relations between two individuals. This index was later extended in a common framework by Grabisch. Both indices are fundamental in the literature of fuzzy measures. Nevertheless, the corresponding calculation still presents a highly complex problem for which no approximation solution has been proposed yet. Then, using a representation of the Shapley based on orders, here we suggest an alternative calculation of the interaction index, both for the simple case of pairs of individuals, and for the more complex situation in which any set could be considered. This alternative representation facilitates the handling of these indices. Moreover, we draw on this representation to define two polynomial methods based on sampling to estimate the interaction index, as well as a method to approximate the generalized version of it. We provide some computational results to test the goodness of the proposed algorithms.

There is a wide range of papers in the literature about the explanation of machine learning models in which Shapley value is considered to measure the importance of the features in these models. We can distinguish between these which set their basis on the cooperative game theory principles, and these focused on fuzzy measures. It is important to mention that all of these approaches only provide a crisp value (or a fix point) to measure the importance of a feature in a specific model. The reason is that an aggregation process of the different marginal contributions produces a single output for each variable. Nevertheless, and because of the relations between features, we cannot distinguish the case in which we do not know all the features. To this aim, we propose a disaggregated model which allows the analysis of the importance of the features, regarding the available information. This new proposal can be viewed as a generalization of all previous measures found in literature providing a two dimensional graph which, in a very intuitive and visual way, provides this rich disaggregated information. This information may be aggregated with several aggregation functions with which obtain new measures to establish the importance of the features. Specifically, the aggregation by the sum results in the Shapley value. We also explain the characteristics of those graphics in different scenarios of the relations among features, to raise this useful information at a glance.

The concept of hierarchical structure is central when considering complex systems. On one hand, many complex systems exhibit a hierarchical structure, on the other hand, the idea of defining a hierarchical structure to cope with the complexity of the system is widely present in literature related to many different fields. Hierarchies are also present in the field of aggregation with the definition of hierarchical aggregation processes. Broadly speaking, a hierarchical system is a system formed by several components (subsystems) structured at different levels, implying a sort of ranking or ordering relation among them. Positioning in the levels could be related to many different aspects or properties of the components: priority, abstraction, granularity, specificity, precision, etc.Computable aggregations have recently been introduced as a natural approach to classical aggregation functions, in which the emphasis is placed on the program (the implementation) that makes possible the aggregation instead of simply considering the function or the algorithm that permits the aggregation process. Considered as a program that implements an aggregation process, a computable aggregation is also suitable for being interpreted in terms of a hierarchical process. In this paper, the idea of hierarchical computable aggregation is considered, exploring those situations where the aggregation process involves some intrinsic structure that can be interpreted in hierarchical terms (like priorities and veto), as well as those other situations where the hierarchical approach is mostly related to computational considerations (like recursion and parallelization). These and other types of hierarchical computable aggregation will be presented and analyzed.

The idea of computable aggregation operators was introduced as a generalization of aggregation operators, allowing the replacement of the mathematical function usually considered for aggregation, by a program that performs the aggregation process. There are different reasons to justify this extension. One of them is the interest in exploring some computational properties not directly related to the aggregation itself but to its implementation (complexity, recursivity, parallelisation, etc). Another reason, the one driving to the present paper, is the need to define a framework where the quite common process of first sampling (over a large data set) and then aggregating the sample, could be analysed as a formal aggregation process. This process does not match with the idea of an aggregation function, due to its non-deterministic nature, but could easily be adapted to that of a (non-deterministic) computable aggregation. The idea of non-deterministic aggregation requires the extension of the concept of monotonicity (a key aspect of aggregation operators) to this new framework. The present paper will explore this kind of non-deterministic aggregation processes, first from an empirical point of view and then in terms of populations, adapting the idea of monotonicity to both of them and finally defining a common framework for its analysis.

Computable aggregation operators can be seen as a generalization of aggregation operators where the mathematical function is replaced by a program that performs the aggregation process. This extension allows the introduction of new aggregation processes not feasible under the classical framework. Particularly interesting are some non-deterministic processes widely considered to merge information. However, especially in non-deterministic processes, the extension of some of the well-known concepts for aggregation operators such as monotony, is needed. In this work, a new concept of monotonicity is proposed, from a probabilistic perspective, for non-deterministic computable aggregation operators. To be consistent, the concept coincides with the classical definition in the deterministic case. In addition, some cases of interest are analysed.

In this paper we propose a new framework for community detection problems. The starting point is a n-vector which defines some evidence about the elements of a finite set. This vector is used to build an interaction measure between the n elements of the set to which it refers. This interaction measure is represented by a Sugeno λ-measure to which we make it being also a fuzzy measure. Then, we obtain the weighted graph associated with this new capacity measure. To carry on with it, we make use of the Shapley value. We also introduce the notion of extended vector fuzzy graph, which relates a graph with the capacity measure introduced in this work. Finally, we use a community detection method, based on Louvain algorithm, to search a cluster structure in the weighted graph. This partition is based on the relations among the individuals obtained from the initial vector. Let us note that in the case that there exist some connections among the elements, apart from their affinity, we can combine this extra information with that given by the vector, in order to obtain groups with highly-knit elements among which there are strong relations.

The concept of pre-aggregation function defined in [0,1]nhas been recently extended to that of generalized pre-aggregation function in the framework of a totally ordered set T with maximum and minimum value. To do so, the concept of monotonicity is transformed in that of conditioned monotonicity based on the chains in Tn, generalizing the idea of directional monotonicity. In the present paper we explore the concept of conditioned monotonicity considering some specific conditioning structures (covers, partitions and projections). On this basis we consider some situations where conditioned monotonicity ensures monotonicity. Finally we use these definitions and properties to define some pre-aggregation and aggregation functions that are applied to image preprocessing problems.

In this paper the relation between aggregation functions, algorithms and computer programs is revisited, extending the concept of recursive aggregation operator by means of the recursive computable aggregation. In particular, two different recursive computable aggregation are distinguished: on the one hand, the hard recursive computable aggregation, which appears when there is a unique recursive function generating the aggregation, and are fully related to associativity; and on the other hand, the soft recursive computable aggregation, which appears when the number of elements to be aggregated is needed. Some illustrative examples are provided.

One of the most common techniques to find the adequate weights in ordered weighted averaging (OWA) operators is based on the orness concept, where the weights are determined by maximizing the entropy (variation) for a fixed orness value. But such an entropy represents a dispersion measure for nominal variables, while weights in an OWA operator are essentially ordinal rather than nominal. Hence, in this paper, we propose a novel way to determine OWA weights based upon ordinal dispersion measures instead of an standard entropy measure. From this approach, we find an explicit formula for the weights, and we illustrate differences by means some multicriteria decision-making examples.

Inspired by the Zadeh approach to fuzzy connectives in fuzzy set theory and by some applications, we introduce and study set-based extended functions, and in particular, set-based extended aggregation functions. These functions reflect neither reordering nor repetition of input values, and, linking different arities, they introduce serious constraints for extended functions. A complete characterization of set-based extended (aggregation) functions is given, and some constructions of such functions are also proposed, including several examples.

It is common to find index construction methods based on crisp techniques and linear models. However, these procedures are not always adequate to capture the essential model of an index. Index development can be approached as a nested or hierarchical aggregation process, where part of the data can have a different level of importance, and the other part can be unstructured. In addition, the information can be studied under a fuzzy approach. This paper proposes an index construction method using the notion of consistency in aggregation operator families to ensure robustness, in addition to considering a hierarchical structure with different level of importance of input information. Two applications are presented, a measure of social and advanced uses of the Internet considering attitudinal, demographic, social and technological factors, and a measure of the child's family environment considering different aspects.

In this paper we stress the relevance of those fuzzy models that impose a couple of simultaneous views in order to represent concepts. In particular, we point out that the basic model to start with should contain at least two somehow opposite valuations plus a number of neutral concepts that are generated from the semantic relationship between those two opposites. Such a basic model should be distinguished from some other similar approaches that can be found in the literature, and that may bring some difficulties in intuition, partially because of their denomination. The general term “paired fuzzy sets” is then proposed together with the notion of sub-antonym, to be considered as a particular case of opposition relationship.

In this paper we want to stress the relevance of paired fuzzy sets, as already proposed in previous works of the authors, as a family of fuzzy sets that offers a unifying view for different models based upon the opposition of two fuzzy sets, simply allowing the existence of different types of neutrality associated to the different semantic relationships that may hold between opposite references. This scheme should be seen as a basic model for knowledge acquisition, which eventually will lead to a better understanding of the relationship of different knowledge representation models and to the acquisition of more complex valuation scales.

In this paper we present an efficient hierarchical clustering algorithm for relational data, being those relations modeled by a graph. The hierarchical clustering approach proposed in this paper is based on divisive and link criteria, to break the graph and join the nodes at different stages. We then apply this approach to a community detection problems based on the well-known edge line betweenness measure as the divisive criterium and a fuzzy similarity relation as the link criterium. We present also some computational results in some well-known examples like the Karate Zachary club-network, the Dolphins network, Les Miserables network and the Authors centrality network, comparing these results to some standard methodologies for hierarchical clustering problem, both for binary and valued graphs.

In this paper we propose an efficient and polynomial hierarchical clustering technique for unsupervised classification of items being connected by a graph. The output of this algorithm shows the cluster evolution in a divisive way, in such a way that as soon as two items are included in the same cluster they will join a common cluster until the last iteration, in which all the items belong to a singleton cluster. This output can be viewed as a fuzzy clustering in which for each alpha cut we have a standard cluster of the network. The clustering tool we present in this paper allows a hierarchical clustering of related items avoiding some unrealistic constraints that are quite often assumed in clustering problems. The proposed procedure is applied to a hierarchical segmentation problem in astronomical images.

Segmentation algorithms are well known in the field of image processing. In this work we propose an efficient and polynomial algorithm for image segmentation based on fuzzy set theory. The main difference with the classical segmentation algorithms is in the output given by the segmentation process. Since the classical output for segmentation algorithms give us the homogeneous regions in the image, our proposal is to produce an hierarchical information (in a similar way as a dendrogam does in classical clustering methods) of how the groups are formed in the image, from the initial situation in which all pixels are in the same group to the final situation in which the whole image is divided in the minimal information units.

The currently-in-use definition of aggregation function is analyzed in this paper, noting that the introduced variability in the dimension of information does not avoid some obvious dysfunctions. In particular, a potential abuse of the mathematical formalism underlies such a definition, which could lead to solve a complex concept by means of a formal mathematical expression. In this paper we propose an alternative definition making emphasis on the practical implementation of aggregation functions, taking into account the objectives and limitations observed in the application of aggregation functions within the fuzzy context.

In this work we propose an efficient and polynomial algorithm for the graph segmentation problem based on the coloring problem for graphs. The work here presented extend the algorithm published in making possible the segmentation to any class of graph (not only fuzzy-valued planar graphs) and also improving the computational complexity of the previous work.

In a previous paper, the authors proposed an alternative approach to classical dimension theory, based upon a general representation of strict preferences not being restricted to partial order sets. Without any relevant restriction, the proposed approach was conceived as a potential powerful tool for decision making problems where basic information has been modeled by means of valued binary preference relations. In fact, assuming that each decision maker is able to consistently manage intensity values for preferences is a strong assumption even when there are few alternatives being involved (if the number of alternatives is large, the same criticism applies to crisp preferences). Any representation tool, as the one proposed by the authors, will in principle play a key role in order to help decision makers to understand their preference structure. In this paper we introduce an alternative approach in order to avoid certain complexity issues of the initial proposal, allowing a close representation easier to be obtained in practice.

In this paper we address the problem of inconsistency in preference relations, pointing out the relevance of a meaningful representation in order to help decision maker to capture such inconsistencies. Dimension theory framework, despite its computational complexity, is considered here, pursuing in principle a decomposition of arbitrary preference relations in terms of linear orderings of alternatives. But we shall then stress that consistency should not be necessarily associated to a linear ordering. In this way, alternative decompositions of a preference relation can be proposed to decision maker, allowing an effective search for a useful representations of alternatives in terms of possible criteria. Such decompositions of our preference relations will then become the basis of a future decision aid model, always with the restricted aim of allowing the decision maker a better understanding of the problem. Inconsistencies may be not simply suppressed but understood, since they may contain relevant information.

Land cover analysis by means of remotely sensing images quite often suggests the existence of fuzzy classes, where no clear borders or particular shapes appear. In this paper we present an image classification aid algorithm which shows as its main output a processed image where each pixel is being colored according to the degree of similitude to their respective surrounding pixels. Such a processed image is therefore suggesting possible classes, to be implemented in a more sophisticated image classification process. A key underlying argument for this approach is the relevance of painting techniques in order to help decision makers to understand complex information relative to fuzzy image classification.