We present four new efficient algorithms for random generation of fuzzy measures based on random linear extensions of Boolean lattices. We verify experimentally the uniformity of the resulting distributions and design various informative criteria of uniformity. The algorithms involve Karzanov and Khachiyan random walks but with much shorter mixing times to achieve uniformity. We calculated some important statistical quantities, such as the centroid of the polytope of fuzzy measures, and found patterns for their behaviour in higher dimensions. By matching these patterns we developed random generation algorithms with much higher efficiency. We confirmed empirically the uniformity of the samples.

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.

Random generation of fuzzy measures is an important computational task in applied problems related to fuzzy integrals such as the Choquet, Sugeno or Shilkret integrals. In general, a desirable property is the uniformity over the set of fuzzy measures. However, testing this property is not an easy task. In this paper, properties of uniform random fuzzy measures are derived. Special attention is being paid to the families of balanced fuzzy measures, belief measures and possibilities measures. Then, based of such properties, statistical tests for the uniformity of random fuzzy measures are developed. Finally, the uniformity of the most used algorithms is tested using the proposed methods.

In a data-driven knowledge-based forecasting system, knowledge can be extracted from historical data by machine learning methods, and represented in the knowledge base. In the inference engine of such a system, the current inputs are used to forecast the likely outputs. The unpredictability of one (or more) inputs in the forecasting horizon, gives rise to one source of uncertainty in the reasoning process, named evidence uncertainty, which is the main focus of this article. To deal with this sort of uncertainty numerical scenarios are generated from historical data to substitute the uncertain input parameter using two approaches: Uniform sampling, and a classical nonuniform sampling acceptance–rejection method. Each input scenario has particular output, and at the end these outputs are aggregated with the discrete Choquet integral to account for outputs' dependencies. Fuzzy measures, also called capacities or nonadditive measures, assign the importance weight not to just each scenario but also to groups of scenarios. A fuzzy measure for the Choquet integral is learned from the historical data. The proposed reasoning approach is evaluated in a long-term relative electricity load forecasting, and treats the uncertainty which arises from unpredictability of the daily temperature in the long run. The results show the superiority of the proposed Choquet integral-based approach with respect to the$k$-interactive,$k$-additive, and$k$-tolerant fuzzy measures compared to traditional aggregators: the weighted arithmetic mean, and the ordered weighted averaging.

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.

Our study focuses on using fuzzy measures and integrals in optimization with interacting decision variables. We optimize piecewise linear objectives through the use of discrete Choquet integral, while s.t. a set of linear constraints. These optimization models are relevant in scenarios where decision variables interact in complex ways, with potential synergies or redundancies among them. Fuzzy measures, which are typically nonadditive and nonmodular set functions, provide a solid mathematical foundation for modeling such interactions. By extending traditional linear objectives, the discrete Choquet integral is a versatile tool for addressing these optimization problems. We decompose fuzzy measures with two opposite kinds of interactions, into submodular and supermodular parts, which accordingly transfer into the convex and concave components of optimized objectives. We customize the difference of convex algorithm for our specific objective functions that involve positive and negative interactions, and investigate its numerical scalability.

Random generation of fuzzy measures can be viewed as assigning$2^{n}$ordered random values from the unit interval to the linear extension of power-sets of inputs. Several recently proposed methods for constructing linear extensions have high computational cost. We propose a numerically efficient approach that directly obtains the linear extensions from convex combinations of 0-1 fuzzy measures, reducing the computational complexity to$O(n2^{n})$. The resulting algorithm is very short but is effective in generating a broad range of fuzzy measures as demonstrated by experiments. The full C++ code is presented.

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 collusion attack combines multiple multimedia files into one new file to erase the user identity information. The traditional anti-collusion methods (which aim to trace the traitors) can defend the collusion attack, but they cannot well defend some hybrid collusion attacks (e.g., a collusion attack combined with desynchronization attacks). To address this issue, we propose a frequency spectrum modification process (FSMP) to defend the collusion attack by significantly downgrading the perceptual quality of the colluded file. The severe perceptual quality degradation can demotivate the attackers from launching the collusion attack. Because FSMP is orthogonal to the existing traitor-trace-based methods, it can be combined with the existing methods to provide a double-layer protection against different attacks. In FSMP, after several signal processing procedures (e.g., uneven framing and smoothing), multiple signals (called FSMP signals) can be generated from the host signal. Launching collusion attack using the generated FSMP signals would lead to the energy disturbance and attenuation effect (EDAE) over the colluded signals. Due to the EDAE, FSMP can significantly degrade the perceptual quality of the colluded audio file, thereby thwarting the collusion attack. In addition, FSMP can well defend different hybrid collusion attacks. Theoretical analysis and experimental results confirm the validity of the proposed method.

We address the problem of optimising piece-wise linear objectives based on the discrete Choquet integral, subject to multiple linear constraints. Such problems arise when aggregating interacting decision variables, which can show positive (synergies) or negative (redundancies) interactions. Non-additivity of fuzzy measures offers a sound mathematical framework to model various interactions, and the Choquet integral is a suitable generalisation of the traditional linear objectives. We deal with fuzzy measures which offer both kinds of interactions, and we decompose them into sub- and supermodular parts. This leads to optimising objectives represented as the sum of convex and concave parts. We employ the Difference of Convex (DC) functions algorithm instantiated and tailored for our specific kinds of objectives, and study its numerical scalability.

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.

Discrete fuzzy measures (capacities) are a powerful tool to model interacting inputs and aggregate them using Choquet, Sugeno, and other fuzzy integrals. Simulation studies and probabilistic optimization algorithms require efficient random sampling from the set of fuzzy measures. It is a challenging problem due to an extremely high complexity of the polytope of capacities. This article addresses random sampling from two special classes: supermodular and submodular fuzzy measures. A suitable marginal contributions representation converts the problem to sampling from an order polytope, which is dealt efficiently using the Markov chain random walk.

We address the problems of multiple criteria decision-making and multiobjective optimization in which the criteria are mutually dependent. A practical application of land combat vehicle selection and its potential extension to force design for Australian Defence is considered. The criteria dependencies are modeled and formalised using the theory of fuzzy measures. The main challenge is an exponentially large set of parameters that quantify all criteria interactions. We develop a number of strategies to 1) simplify the model construction and reduce the number of parameters; 2) elicit preferences from numerical simulations and from decision makers; and 3) translate these preferences into a fuzzy measure learning problem from unlabeled data. We formulate and test a mathematical programming problem to determine fuzzy measure parameters for a relatively large number of decision criteria.

Desynchronization is a very challenging type of attack in audio watermarking. The traditional singular value decomposition (SVD) based audio watermarking methods embed the watermark information by modifying the singular value of individual segment, which have little resistance against desynchronization attacks. In this paper, we propose a novel frequency singular value coefficient (FSVC) feature, which reflects the ratio between the singular values of two consecutive segments and is insensitive to desynchronization attacks, to carry the watermark bits. To our best knowledge, it is the first time that the ratio between singular values is employed for audio watermarking. In the proposed method, the discrete cosine transform (DCT) is performed on two consecutive segments of the host audio signal and SVD is applied to the DCT coefficients of the mid frequency band of each segment to extract the FSVC. Then the watermark bits are embedded by adjusting the values of the FSVC. The watermark embedding procedure is optimized to minimize the perceptual quality degradation and an error buffer is created to enhance the robustness. As a result, the proposed method can achieve a much higher embedding capacity than the existing methods tackling desynchronization attacks. The impact of desynchronization and common signal processing attacks on the proposed watermarking method is mathematically modeled, and the effectiveness of the proposed method against these attacks is theoretically and experimentally validated.

This paper addresses reliable and efficient calculation of the mode of a multivariate sample, which is a classical fusion function. In particular, we focus on the inputs given on the unit simplex, when aggregating elements of Atanassov intuitionistic fuzzy sets, interval-valued fuzzy sets and their extensions, as well as compositional data. We outline the use of a specially designed 2-additive fuzzy measures and the Choquet integral for the purposes of reducing computational complexity in higher dimensions. We present computational analysis and benchmark four different methods of density-based mode estimation.

Integrals defined with respect to fuzzy measures (capacities) are powerful tools in multicriteria decision making. Monotonicity is a basic property of capacity, which means that the marginal contribution of any single criterion to any subset of criteria is always nonnegative. In this paper, we present the capacity-based decision making theory in terms of marginal contributions, which provides an alternative perspective to this widely used decision making strategy. We construct the marginal contribution representations of the equivalent transformations of capacities, some particular capacities, three types of nonlinear integrals, and discuss the capacity identification methods. We also introduce some new concepts and representations, such as nonadditivity and nonmodularity indices, 0 to 1 variables-based linear constraints of k-maxitive capacity, a special representation of the Choquet integral and pan integral. We discuss constraints on marginal contributions which ensure supermodularity of capacities. Finally, an illustrative example is given to show the use of marginal contribution presentation in capacity-based decision making methods.

When applied to interval-valued data, averaging aggregation functions do not, in general, produce a good representative value of an interval data set. For real-valued scalar and vector data the mode has been shown to be both a robust averaging function, insensitive to noise and outliers, and to produce a good representative value of a larger set. This article introduces a method for computing the mode of sets of interval-valued data based on density estimation using the 2-additive Choquet integral over nearest neighbours on the unit simplex. This method is applied to compute the mode of daily temperature range data sets for Australian state capital cities. It is found that dissimilarity between the mode and the interval formed from averaging range end points independently increases significantly in months having highly skewed maximum or minimum temperature distributions.

The discrete Sugeno integral is an aggregation function particularly suited to aggregation of ordinal inputs. It accounts for inputs interactions, such as redundancy and complementarity, and its learning from empirical data is a challenging optimisation problem. The methods of ordinal regression involve an expensive objective function, whose complexity is quadratic in the number of data. We formulate ordinal regression using a much less expensive objective computed in linear time by the pool-adjacent-violators algorithm. We investigate the learning problem numerically and show the superiority of the new algorithm.

The probabilistic simultaneous interaction index has been widely adopted to measure the interaction among the decision criteria. However, this type of indices sometimes fails to reflect the kind of interaction associated with the nonadditivity of a fuzzy measure (capacity). For example, any simultaneous interaction index of the universal set of criteria w.r.t. a strictly superadditive capacity is not always positive. The main reason is that the simultaneous interaction index generalizes the notion of value by replacing the marginal contribution of a single criterion with the marginal simultaneous interaction of criteria subset. In this paper, we reform the generalization process and replace the marginal contribution with the marginal bipartition interaction, which can better reflect the kind of interaction associated with the nonadditivity, for example, superadditivity, subadditivity, strict-superadditivity, or strict-subadditivity. We construct a family of probabilistic bipartition interaction indices of subsets of criteria and study its properties. We discuss the issue of capacity identification based on the bipartition interaction index and demonstrate that the new type of interaction index can be adopted as a feasible alternative to describing the interaction phenomenon among the decision criteria.

Nonadditive robust ordinal regression (NAROR) is a widely adopted approach to analyze and reveal the dominance relationships among all decision alternatives based on nonadditive measures, called capacities. In this paper, we first investigate some advantages of the nonadditivity index as an explicit interaction index, as compared with the traditional probabilistic simultaneous interaction indices, and show that nonadditivity index can serve as an equivalent representation of a capacity. Then we enhance the NAROR method by using nonadditivity index as well as multiple-goal linear programming, where the former is used to replace the traditional interaction index to more naturally represent the decision maker's preferences, and the latter aims to replace the 0 to 1 mixed integer programming to enhance the ability to detect and adjust contradictory and redundant preference information. The updated NAROR's steps are constructed and discussed in detail and illustrated with a practical example.

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.

This paper advocates the use of weighted ordered weighted averaging (WOWA) functions in decision-making processes, where the alternatives are not directly comparable. In particular, WOWA allows one to compare the strongest points of each alternative, also weighted by the importance of each criterion. Four different approaches to applying weights in OWA functions are reviewed. Torra's method based on interpolating regular increasing monotone quantifier and the pruned n-ary tree are compared to the WOWA obtained from recently proposed implicit averaging. Computationally efficient algorithms are outlined. The use of WOWA is illustrated in several examples.

We propose a new generalization of the discrete Choquet integral based on an arbitrary bivariate symmetric averaging function (mean). So far only the means with a natural multivariate extension were used for this purpose. In this paper, we use a general method based on a pruned binary tree to extend symmetric means with no obvious multivariate form, such as the logarithmic, identric, Heronian, Lagrangean, and Cauchy means. The generalized Choquet integral is built by computing the extensions of the bivariate means of the ordered inputs, and includes some existing extensions as special cases. Our construction is illustrated with multiple examples.

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.

This article investigates image filtering and smoothing from the perspective of a recent generalisation of the notion of aggregation functions in fuzzy systems, called pre-aggregation functions. Mixture functions describing a broad class of robust spatial-tonal filters and smoothers are derived using penalty-based methods. Several existing filters are re-derived using this approach and several novel filters are proposed, which are able to better handle filtering in contexts where the pixel to be filtered is itself an outlier in the local neighbourhood. The proposed class of Robust Bilateral Filters formalises and generalises a recent result of Chaudhury, who noted that using a filtered version of an image to compute tonal weights for a Bilateral Filter gave more robust denoising. Filter performance is validated using standard test images and quantified using peak signal-to-noise ratio and visual similarity, finding novel filters that exceed the performance of the standard Bilateral Filter.

We propose weighted aggregation algorithms for creating general idempotent weighted aggregators of n variables derived from related symmetric idempotent aggregators of two variables. This computational method, together with interpolative aggregation, can be used for the development of general idempotent logic aggregators that satisfy a variety of conditions necessary for building decision models in the area of weighted compensative logic.

A multilayer watermarking system is a system that is able to embed watermarks to a host media signal repeatedly in an overlaying manner, without incurring troubles in extracting the watermarks in each layer. In this paper, we present a novel patchwork-based audio watermarking algorithm that can embed and extract watermark bits successfully in such a multilayer framework. In the proposed method, a new watermark embedding algorithm is designed to ensure that the embedded watermarks in a certain layer do not affect the detection of watermarks in other layers. Adding multiple layers of watermark bits inevitably reduces the perceptual quality. However, to minimize the perceptual quality degradation in multilayer watermarking, the audio fragments for watermark embedding are selected from a set of specially arranged discrete cosine transform coefficients of the host audio signal. Watermark embedding is achieved by modifying the mean values of selected sample fragments. With the use of an embedding error buffer, the proposed system can withstand a wide range of common attacks. To maintain the balance between the perceptual quality and robustness, watermark embedding strength is adjusted according to the specific layer used. The proposed multilayer scheme ensures the independence of the processing in different layers. The effectiveness of the proposed system is demonstrated and verified by extensive simulation results.

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.

Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions. It is also a limiting property that results in many important nonmonotonic averaging functions being excluded from the theoretical framework. This work proposes a definition for weakly monotonic averaging functions, studies some properties of this class of functions, and proves that several families of important nonmonotonic means are actually weakly monotonic averaging functions. Specifically, we provide sufficient conditions for weak monotonicity of the Lehmer mean and generalized mixture operators. We establish weak monotonicity of several robust estimators of location and conditions for weak monotonicity of a large class of penalty-based aggregation functions. These results permit a proof of the weak monotonicity of the class of spatial-tonal filters that include important members such as the bilateral filter and anisotropic diffusion. Our concept of weak monotonicity provides a sound theoretical and practical basis by which (monotonic) aggregation functions and nonmonotonic averaging functions can be related within the same framework, allowing us to bridge the gap between these previously disparate areas of research.

Cropping and random bending are two common attacks in image watermarking. In this paper we propose a novel image-watermarking method to deal with these attacks, as well as other common attacks. In the embedding process, we first preprocess the host image by a Gaussian low-pass filter. Then, a secret key is used to randomly select a number of gray levels and the histogram of the filtered image with respect to these selected gray levels is constructed. After that, a histogram-shape-related index is introduced to choose the pixel groups with the highest number of pixels and a safe band is built between the chosen and nonchosen pixel groups. A watermark-embedding scheme is proposed to insert watermarks into the chosen pixel groups. The usage of the histogram-shape-related index and safe band results in good robustness. Moreover, a novel high-frequency component modification mechanism is also utilized in the embedding scheme to further improve robustness. At the decoding end, based on the available secret key, the watermarked pixel groups are identified and watermarks are extracted from them. The effectiveness of the proposed image-watermarking method is demonstrated by simulation examples.

Certain tasks in image processing require the preservation of fine image details, while applying a broad operation to the image, such as image reduction, filtering, or smoothing. In such cases, the objects of interest are typically represented by small, spatially cohesive clusters of pixels which are to be preserved or removed, depending on the requirements. When images are corrupted by the noise or contain intensity variations generated by imaging sensors, identification of these clusters within the intensity space is problematic as they are corrupted by outliers. This paper presents a novel approach to accounting for spatial organization of the pixels and to measuring the compactness of pixel clusters based on the construction of fuzzy measures with specific properties: monotonicity with respect to the cluster size; invariance with respect to translation, reflection, and rotation; and discrimination between pixel sets of fixed cardinality with different spatial arrangements. We present construction methods based on Sugeno-type fuzzy measures, minimum spanning trees, and fuzzy measure decomposition. We demonstrate their application to generating fuzzy measures on real and artificial images.

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.

Pixel-scale fine details are often lost during image processing tasks such as image reduction and filtering. Block or region based algorithms typically rely on averaging functions to implement the required operation and traditional function choices struggle to preserve small, spatially cohesive clusters of pixels which may be corrupted by noise. This article proposes the construction of fuzzy measures of cluster compactness to account for the spatial organisation of pixels. We present two construction methods (minimum spannning trees and fuzzy measure decomposition) to generate measures with specific properties: monotonicity with respect to cluster size; invariance with respect to translation, reflection and rotation; and, discrimination between pixel sets of fixed cardinality with different spatial arrangements. We apply these measures within a non-monotonic mode-like averaging function used for image reduction and we show that this new function preserves pixel-scale structures better than existing monotonie averages.

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.

In image processing, particularly in image reduction, averaging aggregation functions play an important role. In this work we study the aggregation of color values (RGB) and we present an image reduction algorithm for RGB color images. For this purpose, we define and study aggregation functions and penalty functions in product lattices. We show how the arithmetic mean and the median can be obtained by minimizing specific penalty functions. Moreover, we study other penalty functions and we show that, in general, aggregation functions on product lattices do not coincide with the cartesian product of the corresponding aggregation functions. Finally, we make an experimental study where we test our reduction algorithm and we analyze the stability of the penalty functions in images affected by noise.

The explosion of the Web 2:0 platforms, with massive volume of user generated data, has presented many new opportunities as well as challenges for organizations in understanding consumer's behavior to support for business planning process. Feature based sentiment mining has been an emerging area in providing tools for automated opinion discovery and summarization to help business managers with achieving such goals. However, the current feature based sentiment mining systems were only able to provide some forms of sentiments summary with respect to product features, but impossible to provide insight into the decision making process of consumers. In this paper, we will present a relatively new decision support method based on Choquet Integral aggregation function, Shapley value and Interaction Index which is able to address such requirements of business managers. Using a study case of Hotel industry, we will demonstrate how this technique can be applied to effectively model the user's preference of (hotel) features. The presented method has potential to extend the practical capability of sentiment mining area, while, research findings and analysis are useful in helping business managers to define new target customers and to plan more effective marketing strategies.

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.

We investigate the problem of averaging values on lattices and, in particular, on discrete product lattices. This problem arises in image processing when several color values given in RGB, HSL, or another coding scheme need to be combined. We show how the arithmetic mean and the median can be constructed by minimizing appropriate penalties, and we discuss which of them coincide with the Cartesian product of the standard mean and the median. We apply these functions in image processing. We present three algorithms for color image reduction based on minimizing penalty functions on discrete product lattices.

Sleep stage identification is the first step in modern sleep disorder diagnostics process. K-complex is an indicator for the sleep stage 2. However, due to the ambiguity of the translation of the medical standards into a computer-based procedure, reliability of automated K-complex detection from the EEG wave is still far from expectation. More specifically, there are some significant barriers to the research of automatic K-complex detection. First, there is no adequate description of K-complex that makes it difficult to develop automatic detection algorithm. Second, human experts only provided the label for whether a whole EEG segment contains K-complex or not, rather than individual labels for each subsegment. These barriers render most pattern recognition algorithms inapplicable in detecting K-complex. In this paper, we attempt to address these two challenges, by designing a new feature extraction method that can transform visual features of the EEG wave with any length into mathematical representation and proposing a hybrid-synergic machine learning method to build a K-complex classifier. The tenfold cross-validation results indicate that both the accuracy and the precision of this proposed model are at least as good as a human expert in K-complex detection.

We advance the theory of aggregation operators and introduce non-monotone aggregation methods based on minimization of a penalty for inputs disagreements. The application in mind is processing data sets which may contain noisy values. Our aim is to filter out noise while at the same time preserve signs of unusual values. We review various methods of robust estimators of location, and then introduce a new estimator based on penalty minimisation.

In this work we present a new image thresholding algorithm for the segmentation of MRI brain images into two classes: gray matter and white matter. The proposed algorithm is based on the concept of incomparability proposed by Fodor and Roubens for fuzzy preference relations. We test our algorithm for local and global segmentation of brain images. We proof that global segmentation performs better results than local segmentation and improves the results obtained by other thresholding algorithm.

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.

Stereo matching problem attempts to find corresponding locations between pairs of displaced images of the same scene. Correspondence estimation between pixels suffers from occlusions, noise, and bias. This paper introduces a novel approach to represent images by means of interval-valued fuzzy sets. These sets allow one to overcome the uncertainty due to the aforementioned problems. The aim is to take advantage of the new representation to develop a stereo matching algorithm. The interval-valued fuzzification process for images that is proposed here is based on image segmentation. Interval-valued fuzzy similarities are introduced to compare windows whose pixels are represented by intervals. To make use of color information, the similarities of the RGB channels were aggregated using the luminance formula. The experimental analysis makes a comparison with other methods. The new representation that is proposed together with the new similarity measure show a better overall behavior, providing more accurate correspondences, mainly near depth discontinuities and for images with a large amount of color.

In this paper, a simple and effective image-magnification algorithm based on intervals is proposed. A low-resolution image is magnified to form a high-resolution image using a block-expanding method. Our proposed method associates each pixel with an interval obtained by a weighted aggregation of the pixels in its neighborhood. From the interval and with a linear K(α) operator, we obtain the magnified image. Experimental results show that our algorithm provides a magnified image with better quality (peak signal-to-noise ratio) than several existing methods.

In this work we propose an image reduction algorithm based on weak local reduction operators. We use several averaging functions to build these operators and we analyze their properties. We present experimental results where we apply the algorithm and weak local reduction operators in procedures of reduction, and later, reconstruction of images. We analyze these results over natural images and noisy images.

We consider an application of fuzzy logic connectives to statistical regression. We replace the standard least squares, least absolute deviation, and maximum likelihood criteria with an ordered weighted averaging (OWA) function of the residuals. Depending on the choice of the weights, we obtain the standard regression problems, high-breakdown robust methods (least median, least trimmed squares, and trimmed likelihood methods), as well as new formulations. We present various approaches to numerical solution of such regression problems. OWA-based regression is particularly useful in the presence of outliers, and we illustrate the performance of the new methods on several instances of linear regression problems with multiple outliers.

This paper examines the practical construction ofk-Lipschitz triangular norms and conorms from empirical data. We apply a characterization of such functions based onk-convex additive generators and translatek-convexity of piecewise linear strictly decreasing functions into a simple set of linear inequalities on their coefficients. This is the basis of a simple linear spline-fitting algorithm, which guaranteesk-Lipschitz property of the resulting triangular norms and conorms.

We discuss the problem of learning fuzzy measures from empirical data. Values of the discrete Choquet integral are fitted to the data in the least absolute deviation sense. This problem is solved by linear programming techniques. We consider the cases when the data are given on the numerical and interval scales. An open source programming library which facilitates calculations involving fuzzy measures and their learning from data is presented.

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.

This paper describes an approach to pointwise construction of general aggregation operators, based on monotone Lipschitz approximation. The aggregation operators are constructed from a set of desired values at certain points, or from empirically collected data. It establishes tight upper and lower bounds on Lipschitz aggregation operators with a number of different properties, as well as the optimal aggregation operator, consistent with the given values. We consider conjunctive, disjunctive and idempotent n-ary aggregation operators; p-stable aggregation operators; various choices of the neutral element and annihilator; diagonal, opposite diagonal and marginal sections; bipolar and double aggregation operators. In all cases we provide either explicit formulas or deterministic numerical procedures to determine the bounds. The findings of this paper are useful for construction of aggregation operators with specified properties, especially using interpolation schemata.

This paper examines disjunctive aggregation operators used in various recommender systems. A specific requirement in these systems is the property of noble reinforcement: allowing a collection of high-valued arguments to reinforce each other while avoiding reinforcement of low-valued arguments. We present a new construction of Lipschitz-continuous aggregation operators with noble reinforcement property and its refinements.

The fourteen papers in this special section are devoted to aggregation operators with respect to knowledge based systems.

This paper treats the problem of fitting general aggregation operators with unfixed number of arguments to empirical data. We discuss methods applicable to associative operators (t-norms, t-conorms, uninorms and nullnorms), means and Choquet integral based operators with respect to a universal fuzzy measure. Special attention is paid to k-order additive symmetric fuzzy measures.

This article discusses a range of regression techniques specifically tailored to building aggregation operators from empirical data. These techniques identify optimal parameters of aggregation operators from various classes (triangular norms, uninorms, copulas, ordered weighted aggregation (OWA), generalized means, and compensatory and general aggregation operators), while allowing one to preserve specific properties such as commutativity or associativity. © 2003 Wiley Periodicals, Inc.

The need for monotone approximation of scattered data often arises in many problems of regression, when the monotonicity is semantically important. One such domain is fuzzy set theory, where membership functions and aggregation operators are order preserving. Least squares polynomial splines provide great flexbility when modeling non-linear functions, but may fail to be monotone. Linear restrictions on spline coefficients provide necessary and sufficient conditions for spline monotonicity. The basis for splines is selected in such a way that these restrictions take an especially simple form. The resulting non-negative least squares problem can be solved by a variety of standard proven techniques. Additional interpolation requirements can also be imposed in the same framework. The method is applied to fuzzy systems, where membership functions and aggregation operators are constructed from empirical data.

This paper investigates the problem of obtaining the weights of the ordered weighted aggregation (OWA) operators from observations. The problem is formulated as a restricted least squares and uniform approximation problems. We take full advantage of the linearity of the problem. In the former case, a well known technique of non-negative least squares is used. In a case of uniform approximation, we employ a recently developed cutting angle method of global optimisation. Both presented methods give results superior to earlier approaches, and do not require complicated nonlinear constructions. Additional restrictions, such as degree of orness of the operator, can be easily introduced.

This paper discusses various extensions of the classical within-group sum of squared errors functional, routinely used as the clustering criterion. Fuzzy c-means algorithm is extended to the case when clusters have irregular shapes, by representing the clusters with more than one prototype. The resulting minimization problem is non-convex and non-smooth. A recently developed cutting angle method of global optimization is applied to this difficult problem.

Fuzzy logic provides a mathematical formalism for a unified treatment of vagueness and imprecision that are ever present in decision support and expert systems in many areas. The choice of aggregation operators is crucial to the behavior of the system that is intended to mimic human decision making. The paper discusses how aggregation operators can be selected and adjusted to fit empirical data: a series of test cases. Both parametric and nonparametric regression are considered and compared. A practical application of the proposed methods to electronic implementation of clinical guidelines is presented.