While deep learning has achieved significant success across various fields, the lack of transparency in decision-making remains a major challenge. To enhance interpretability, we propose an Inclusion-Exclusion Integral Neural Network (IEINN), a novel neural network model based on a non-linear integral defined with respect to a non-additive measure. This model is particularly suitable for systems where monotonic relationships exist between inputs and outputs. However, training without enforcing monotonicity constraints may lead to overfitting and difficulties in interpreting learned parameters. To address this, we introduce a monotonicity-preserving regularization term to ensure learned parameters remain consistent with non-additive measure theory. Experimental results across multiple datasets demonstrate that our approach enhances both model interpretability and predictive stability.

The Möbius transform has facilitated great progress in both theory and practical applications of fuzzy measures. As well as simplifying a number of calculations used for interpretation, the Möbius representation of certain families like k -additive fuzzy measures significantly reduces the number of defining parameters. For some other classes, like k -interactive fuzzy measures, we can use the less well-known co-Möbius representation, however this is not as intuitive or straightforward when it comes to interpreting its values. In this contribution we propose a modification to this calculation that we call the complement co-Möbius representation, which leads to more natural expressions, useful simplifications and insights. We provide the conversion formulas between each of the representations as well as some examples, highlighting k -interactive and plausibility measures in particular.

The random generation of fuzzy measures under complex linear constraints holds significance in various fields, including optimization solutions, machine learning, decision making, and property investigation. However, most existing random generation methods primarily focus on addressing the monotonicity and normalization conditions inherent in the construction of fuzzy measures, rather than the linear constraints that are crucial for representing special families of fuzzy measures and additional preference information. In this paper, we present two categories of methods to address the generation of linearly constrained fuzzy measures using linear programming models. These methods enable a comprehensive exploration and coverage of the entire feasible convex domain. The first category involves randomly selecting a subset and assigning measure values within the allowable range under given linear constraints. The second category utilizes convex combinations of constrained extreme fuzzy measures and vertex fuzzy measures. Then we employ some indices of fuzzy measures, objective functions, and distances to domain boundaries to evaluate the coverage performance of these methods across the entire feasible domain. We further provide enhancement techniques to improve the coverage ratios. Finally, we discuss and demonstrate potential applications of these generation methods in practical scenarios.

When dealing with a fuzzy measure on$n$elements, verifying satisfaction of the monotonicity conditions typically requires performing$n2^{n-1}$comparisons on measure values, while checking the convexity conditions involves$\binom{n}{2} 2^{n-2}$comparisons among marginal contributions. The exponential computation required for these checks in fuzzy measure optimization models often leads heuristic algorithms into numerous challenging situations. In this contribution, we propose efficient comparison algorithms based on sorting methods, linear extensions of fuzzy measures, and partial orders on set pairs of marginal contributions. With the aid of these algorithms, the computational complexity is substantially reduced to a linear level on average. Our numerical experiments confirm the significant benefit when it comes to scenarios with large values of$n$, (e.g.,$n>10$), allowing us to apply these methods to problems that were previously intractable.

Agglomerative hierarchical clustering based on Ordered Weighted Averaging (OWA) operators not only generalises the single, complete, and average linkages, but also includes intercluster distances based on a few nearest or farthest neighbours, trimmed and winsorised means of pairwise point similarities, amongst many others. We explore the relationships between the famous Lance–Williams update formula and the extended OWA-based linkages with weights generated via infinite coefficient sequences. Furthermore, we provide some conditions for the weight generators to guarantee the resulting dendrograms to be free from unaesthetic inversions.

The useful balance of flexibility and robustness provided by fuzzy integrals for classification and regression tasks has been well established in the fuzzy research community. With the rise of artificial intelligence and neural networks, we have witnessed great advances in data analysis and decision-making, however the issue of explainability has become more prominent in recent times. It is in this regard that the theory of fuzzy integrals and other sophisticated aggregation frameworks have a lot to offer. In this contribution, a neural network architecture is proposed based on the inclusion-exclusion integral, which is defined with respect to a fuzzy measure and a triangular norm. After presenting the model and learning methodology, we provide some applications of the network to real datasets. The key benefit of this approach is in the ability to derive interpretations using the Shapley value and interaction indices, read directly from the model. However, we also highlight the predictive performance, which is comparable with state-of-the-art machine learning techniques. Our study hence contributes to the aims of explainable artificial intelligence, whereby model flexibility and accuracy is achieved without sacrificing interpretability.

Measures of diversity, spread and inequality can be important indicators in domains as diverse as ecology, economics and health. One of the key characteristics of such indices is the Pigou–Dalton (P-D) principle, also known as the principle of progressive transfers, whereby proportional redistribution from larger to smaller arguments should increase (or decrease, depending on the context) the overall measure of social or economic welfare, diversity and so on. Previous studies have identified the ordered weighted averaging operators as being appropriate for welfare measurement, subject to conditions on the weighting vectors. We propose the Choquet integral, defined with respect to a capacity or fuzzy measure, as a candidate for defining nonsymmetric measures of welfare. This allows for importance and interaction to be modelled between inputs while still satisfying the P-D principle. We extend the buoyancy concept to fuzzy measures and characterise the resulting classes of buoyant and antibuoyant fuzzy measures. We then turn to the problem of optimisation of the Choquet integral subject to linear constraints, which in the case of antibuoyant fuzzy measures permits an efficient linear programming solution.

The problem of learning symmetric capacities (or fuzzy measures) from data is investigated toward applications in data analysis and prediction as well as decision making. Theoretical results regarding the solution minimizing the mean absolute error are exploited to develop an exact branch-refine-and-bound-type algorithm for fitting Sugeno integrals (weighted lattice polynomial functions, max-min operators) with respect to symmetric capacities. The proposed method turns out to be particularly suitable for acting on ordinal data. In addition to providing a model that can be used for the general data regression task, the results can be used, among others, to calibrate generalized h-indices to bibliometric data.

We propose a version of ordered weighted averaging (OWA) operators, which are robust against inputs with outliers. Outliers may heavily bias the outputs of the standard OWA. The penalty-based method proposed here comprises both outlier detection and reallocation of weights of the OWA. At the first stage, the outliers are identified based on a robust criterion that can accommodate up to half the inputs being outliers, but at the same time not removing the inputs unnecessarily. Three numerical algorithms for calculating the optimal value of this criterion are proposed. At the second stage, the OWA weights are recalculated for a subset of clean data while preserving the overall character of the weighting vector. The method is numerically tested on simulated data and exemplified on aggregating a large number of online ratings where the outliers represent biased, missing, or erroneous evaluations.

With the rise of Massive Open Online Courses (MOOCs), online peer marking is an attractive contemporary tool for educational assessment. However its widespread use faces serious challenges, most significantly in the perceived and actual reliability of assessment grades, which can be affected by the ability of peers to mark accurately and the potential for collusion and bias. There exist a number of aggregation approaches for alleviating the impact of biased scores, usually involving either the down-weighting or removal of outliers. Here we investigate the use of the least trimmed squares (LTS) and Huber mean for the aggregation step, comparing their performance to weighting of markers based on divergence from other peers' marks. We design an experimental setup to generate scores and test a number of conditions. Overall we find that for a feasible number of peer markers, when the student pool comprises a significant number of `biased' markers, outlier removal techniques are likely to result in a number of very unfair assessments, while more standard approaches will have more grades unfairly influenced but to a lesser extent.

We consider the problem of aggregating a large number of online ratings where there may be outliers, representing biased, missing or erroneous evaluations. The penalty-based method proposed comprises both outlier detection and reallocation of weights and we focus on models dependent on the relative order of inputs, i.e. based on OWA operators, however we also define the model for weighted means.

Economic inequality measures are employed as a key component in various socio-demographic indices to capture the disparity between the wealthy and poor. Since their inception, they have also been used as a basis for modelling spread and disparity in other contexts. While recent research has identified that a number of classical inequality and welfare functions can be considered in the framework of OWA operators, here we propose a framework of penalty-based aggregation functions and their associated penalties as measures of inequality.

In group decision-making problems it is common to elicit preferences from human experts in the form of pairwise preference relations. When this is extended to a fuzzy setting, entries in the pairwise preference matrix are interpreted to denote strength of preference, however once logical properties such as consistency and transitivity are enforced, the resulting preference relation requires almost as much information as providing raw scores or a complete order over the alternatives. Here we instead interpret fuzzy degrees of preference to only apply where the preference over two alternatives is genuinely fuzzy and then suggest an aggregation procedure that minimizes a generalized Kemeny distance to the nearest complete or partial order. By focusing on the fuzzy partial order, the method is less affected by differences in the natural scale over which an expert expresses their preference, and can also limit the influence of extreme scores.

Rather than denoting fuzzy membership with a single value, orthopairs such as Atanassov's intuitionistic membership and non-membership pairs allow the incorporation of uncertainty, as well as positive and negative aspects when providing evaluations in fuzzy decision making problems. Such representations, along with interval-valued fuzzy values and the recently introduced Pythagorean membership grades, present particular challenges when it comes to defining orders and constructing aggregation functions that behave consistently when summarizing evaluations over multiple criteria or experts. In this paper we consider the aggregation of pairwise preferences denoted by membership and non-membership pairs. We look at how mappings from the space of Atanassov orthopairs to more general classes of fuzzy orthopairs can be used to help define averaging aggregation functions in these new settings. In particular, we focus on how the notion of `averaging' should be treated in the case of Yager's Pythagorean membership grades and how to ensure that such functions produce outputs consistent with the case of ordinary fuzzy membership degrees.

In the context of group decision making with fuzzy preferences, consensus measures are employed to provide feedback and help guide automatic or semi-automatic decision reaching processes. These measures attempt to capture the intuitive notion of how much inputs, individuals or groups agree with one another. Meanwhile, in ecological studies there has been an ongoing research effort to define measures of community evenness based on how evenly the proportional abundances of species are distributed. The question hence arises as to whether there can be any cross-fertilization from developments in these fields given their intuitive similarity. Here we investigate some of the models used in ecology toward their potential use in measuring consensus. We found that although many consensus characteristics are exhibited by evenness indices, lack of reciprocity and a tendency towards a minimum when a single input is non-zero would make them undesirable for inputs expressed on an interval scale. On the other hand, we note that some of the general frameworks could still be useful for other types of inputs like ranking profiles and that in the opposite direction consensus measures have the potential to provide new insights in ecology.

The mean defined by Bonferroni in 1950 (known by the same name) averages all non-identical product pairs of the inputs. Its generalizations to date have been able to capture unique behavior that may be desired in some decision-making contexts such as the ability to model mandatory requirements. In this paper, we propose a composition that averages conjunctions between the respective means of a designated subset-size partition. We investigate the behavior of such a function and note the relationship within a given family as the subset size is changed. We found that the proposed function is able to more intuitively handle multiple mandatory requirements or mandatory input sets.

In the face of mass amounts of information and the need for transparent and fair decision processes, aggregation functions are essential for summarizing data and providing overall evaluations. Although families such as weighted means and medians have been well studied, there are still applications for which no existing aggregation functions can capture the decision makers' preferences. Furthermore, extensions of aggregation functions to lattices are often needed to model operations on L-fuzzy sets, interval-valued and intuitionistic fuzzy sets. In such cases, the aggregation properties need to be considered in light of the lattice structure, as otherwise counterintuitive or unreliable behavior may result. The Bonferroni mean has recently received attention in the fuzzy sets and decision making community as it is able to model useful notions such as mandatory requirements. Here, we consider its associated penalty function to extend the generalized Bonferroni mean to lattices. We show that different notions of dissimilarity on lattices can lead to alternative expressions.

Atanassov's intuitionistic fuzzy sets (AIFS) and interval valued fuzzy sets (IVFS) are two generalizations of a fuzzy set, which are equivalent mathematically although different semantically. We analyze the median aggregation operator for AIFS and IVFS. Different mathematical theories have lead to different definitions of the median operator. We look at the median from various perspectives: as an instance of the intuitionistic ordered weighted averaging operator, as a Fermat point in a plane, as a minimizer of input disagreement, and as an operation on distributive lattices. We underline several connections between these approaches and summarize essential properties of the median in different representations.

Anonymous communication has become a hot research topic in order to meet the increasing demand for web privacy protection. However, there are few such systems which can provide high level anonymity for web browsing. The reason is the current dominant dummy packet padding method for anonymization against traffic analysis attacks. This method inherits huge delay and bandwidth waste, which inhibits its use for web browsing. In this paper, we propose a predicted packet padding strategy to replace the dummy packet padding method for anonymous web browsing systems. The proposed strategy mitigates delay and bandwidth waste significantly on average. We formulated the traffic analysis attack and defense problem, and defined a metric, cost coefficient of anonymization (CCA), to measure the performance of anonymization. We thoroughly analyzed the problem with the characteristics of web browsing and concluded that the proposed strategy is better than the current dummy packet padding strategy in theory. We have conducted extensive experiments on two real world data sets, and the results confirmed the advantage of the proposed method.

We consider an application of fuzzy measures to the problem of metric learning in semisupervised clustering. We investigate the necessary and sufficient conditions on the underlying fuzzy measure that make the discrete Choquet integral suitable for defining a metric. As a byproduct, we can obtain the analogous conditions for the ordered-weighted-averaging (OWA) operators, which constitute a special case. We then generalize these results for power-based Choquet and OWA operators. We show that this metric-learning problem can be formulated as a linear-programming problem and specify the required sets of linear constraints. We present the results of numerical experiments on artificial- and real-world datasets, which illustrate the potential, usefulness, and limitations of this construction.

k-nearest neighbors (kNN) is a popular method for function approximation and classification. One drawback of this method is that the nearest neighbors can be all located on one side of the point in question x. An alternative natural neighbors method is expensive for more than three variables. In this paper we propose the use of the discrete Choquet integral for combining the values of the nearest neighbors so that redundant information is canceled out. We design a fuzzy measure based on location of the nearest neighbors, which favors neighbors located all around x.

We discuss the problem of texture recognition based on the grey level co-occurrence matrix (GLCM). We performed a number of numerical experiments to establish whether the accuracy of classification is optimal when GLCM entries are aggregated into standard metrics like contrast, dissimilarity, homogeneity, entropy, etc., and compared these metrics to several alternative aggregation methods.We conclude that k nearest neighbors classification based on raw GLCM entries typically works better than classification based on the standard metrics for noiseless data, that metrics based on principal component analysis inprove classification, and that a simple change from the arithmetic to quadratic mean in calculating the standard metrics also improves classification.