Restricted Equivalence Research Articles

The effectiveness of aggregation functions used in fuzzy local contrast constructions

In this paper, we explore the concept of local contrast of a fuzzy relation, which can be perceived as a measure for distinguishing the degrees of membership of elements within a defined region of an image. We introduce four distinct methods for constructing fuzzy local contrast: one uses a similarity measure, the second relies on the aggregation of similarity, the third is based on the aggregation of restricted equivalence, and the fourth utilizes the notion of equivalence. We further divide the constructions using similarity measures into two categories based on the two known definitions of similarity: distance-based similarity and aggregation function-based similarity. These construction methods also incorporate fuzzy implications and negations. Aggregation functions, which can be manipulated to enhance the effectiveness of the constructed fuzzy local contrast, play a significant role in most of our proposed constructions. For each construction method, several examples of fuzzy local contrasts are provided. The usefulness of the new fuzzy local contrasts is examined by applying them in image processing for salient region detection.

STRUCTURE-PRESERVING MODEL ORDER REDUCTION OF RANDOM PARAMETRIC LINEAR SYSTEMS VIA REGRESSION

We investigate model order reduction (MOR) of random parametric linear systems via the regression method. By sampling the random parameters contained in the coefficient matrices of the systems, the iterative rational Krylov algorithm (IRKA) is used to generate sample reduced models corresponding to the sample data.We assemble the resulting reduced models by interpolating the coefficient matrices of reduced sample models with the regression technique, where the generalized polynomial chaos (gPC) is adopted to characterize the random dependence coming from the original systems. Noting the invariance of the transfer function with respect to restricted equivalence transformations, the regression method is conducted based on the controllable canonical form of reduced sample models in such a way to improve the accuracy of reduced models greatly.We also provide a posteriori error bound for the projection reduction method in the stochastic setting. We showcase the efficiency of the proposed approach by two large-scale systems along with random parameters: a synthetic model and a mass-spring-damper system.

ωA-IvE methodology: Admissible interleaving entropy methods applied to video streaming traffic classification

This work focuses on the width-based interval fuzzy entropy notion, considering the interval data diameter as a measure of the lack of knowledge and uncertainty related to the precise membership degrees of elements in an interval-valued fuzzy set. The w-preserving notion relates the uncertainty from input to output in system information. Such a concept generates a new entropy by applying width-based average functions and admissible order to compare interval data and define width-based fuzzy connectives in data fuzzy computations. A new admissible total order is introduced, requesting just one injective and increasing function, illustrated by a Decimal Digit Interleaving (DDI) function. The proposal methods are based on the admissible interleaving order and related expressions for width-based interval entropy considering different conditions for composition among width-based interval fuzzy operators, as negations with equilibrium and average functions, also including idempotent aggregation function and restricted equivalence functions. Finally, we illustrate the application of the proposed methods for solving a video streaming traffic classification problem. The admissible interleaving entropy analysis is performed over the attributes modeling the FuzzyNetClass approach, a computational model for traffic classification related to video streaming, exploring the integration of inference systems based on interval-valued fuzzy logic and machine learning algorithms. The results by comparison with other width-based interval fuzzy entropy methods were promising and pointed to continuing study and research efforts.

Supervised penalty-based aggregation applied to motor-imagery based brain-computer-interface

In this paper we propose a new version of penalty-based aggregation functions, the Multi Cost Aggregation choosing functions (MCAs), in which the function to minimize is constructed using a convex combination of two relaxed versions of restricted equivalence and dissimilarity functions instead of a penalty function. We additionally suggest two different alternatives to train a MCA in a supervised classification task in order to adapt the aggregation to each vector of inputs. We apply the proposed MCA in a Motor Imagery-based Brain–Computer Interface (MI-BCI) system to improve its decision making phase. We also evaluate the classical aggregation with our new aggregation procedure in two publicly available datasets. We obtain an accuracy of 82.31% for a left vs. right hand in the Clinical BCI challenge (CBCIC) dataset, and a performance of 62.43% for the four-class case in the BCI Competition IV 2a dataset compared to a 82.15% and 60.56% using the arithmetic mean. Finally, we have also tested the goodness of our proposal against other MI-BCI systems, obtaining better results than those using other decision making schemes and Deep Learning on the same datasets.

From Restricted Equivalence Functions on $L^{n}$ to Similarity Measures Between Fuzzy Multisets

Restricted equivalence functions are well-known functions to compare two numbers in the interval between 0 and 1. Despite the numerous works studying the properties of restricted equivalence functions and their multiple applications as support for different similarity measures, an extension of these functions to an n-dimensional space is absent from the literature. In this paper, we present a novel contribution to the restricted equivalence function theory, allowing to compare multivalued elements. Specifically, we extend the notion of restricted equivalence functions from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{n}$</tex-math></inline-formula> and present a new similarity construction on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{n}$</tex-math></inline-formula> . Our proposal is tested in the context of color image anisotropic diffusion as an example of one of its many applications.

An Axiomatic System Based on Ladd-Franklin's Antilogism

This paper sketches the antilogism of Christine Ladd-Franklin and historical advancement about antilogism, mainly constructs an axiomatic system Atl based on first-order logic with equality and the wholly-exclusion and not-wholly-exclusion relations abstracted from the algebra of Ladd-Franklin, with soundness and completeness of Atl proved, providing a simple and convenient tool on syllogistic reasoning. Atl depicts the empty class and the whole class differently from normal set theories, e.g. ZFC, revealing another perspective on sets and set theories. Two series of Dotterer and the theorem proved by Russinoff are re-characterized in antilogistic formulas corresponding to the square of opposition and twenty-four moods of syllogism. Then the restricted equivalence between two forms of antilogism and minimal inconsistency is built up, implying the internal connection between antilogisms and minimal inconsistency and leaving two related conjectures not proved. Besides, this paper provides a new explanation for the reason why contrariety, subcontrariety, and subalternation and the nine special moods require the non-empty assumption.

Normal [formula omitted] functions-based fuzzy information measures with applications in MADM and pattern recognition

Fuzzy and non-standard fuzzy information measures act upon vague information entailed in linguistic variables or uncertain numerical data following some goal for which they discover the “best alternative” or “correct pattern”. This paper introduces the concept of normalKN functions instrumental to generate a class of fuzzy knowledge measures. In addition to the axiomatic approach, we investigate various general methods to obtain normalKN functions from the notions of auto-morphisms, restricted equivalence functions, and restricted dissimilarity functions. Further, aggregatingnormal KNfunctions, we propose three knowledge measures of a fuzzy set and exploit their significance in practical applications. This concerns Multiple Attribute Decision Making (MADM) problems, the issue of appropriate weight computation, and the understanding of structured linguistic variables. Additionally, we consider the real case study for the ranking of universities. A novel methodology to transform a given fuzzy knowledge measure into a family of similarity measures of fuzzy sets is also suggested. Furthermore, we discuss an application of the proposed similarity measures in the context of pattern recognition and the MADM problem using artificial and real data sets. The comparative study of the proposed measures shows their advantage over existing ones because of specific theoretical and practical aspects.

Interval-valued fuzzy discernibility pair approach for attribute reduction in incomplete interval-valued information systems

Interval-valued information systems provide rich semantic interpretation and greater flexibility compared to real-valued information systems. Meanwhile, incomplete interval-valued information systems containing missing values exist widely in reality. One of the key issues in handling incomplete interval-valued information systems is attribute reduction. Unfortunately, there are few studies on attribute reduction for incomplete interval-valued information systems. Moreover, most of the existing studies use fuzzy similarity relations to measure the similarity of objects. Due to the complexity of incomplete interval-valued information systems, traditional fuzzy similarity relations cannot adequately represent the information contained within incomplete interval-valued information systems. To address this issue, this study proposes interval-valued fuzzy min-max similarity relations for incomplete interval-valued information systems, along with two attribute reduction algorithms based on interval-valued fuzzy discernibility pairs model for incomplete interval-valued information systems. Firstly, this study defines interval-valued fuzzy min-max similarity relations using distance-based extended restricted equivalence functions. Secondly, this study presents the interval-valued fuzzy discernibility pairs and discerning ability measure for attribute sets. Furthermore, an attribute reduction algorithm is proposed for incomplete interval-valued information systems. Thirdly, taking into account decision attributes, the relative interval-valued fuzzy discernibility pairs and an attribute reduction algorithm for incomplete interval-valued decision information systems are proposed. Finally, the effectiveness of the proposed method is compared to four representative attribute reduction algorithms in experiments. The experimental results illustrate that our method is effective and has advantages over the compared algorithms.

On Presentations of Semigroup of Transformations Restricted by an Equivalence

For a non-empty set X denote the full transformation semigroup of X by T(X). Let \(\sigma\) be an equivalence relation on X and E(X, \(\sigma\)) denotes the semigroup (under composition) of all \(\alpha\) : X \(\mapsto\) X, such that \(\sigma\) \(\subseteq\) ker(\(\alpha\) ). Semigroup of transformations with restricted equivalence occur when we take all transformations whose kernel is contained in some fixed equivalence, E(X, \(\sigma\)). First, we found that E(X, \(\sigma\)) is a disjoint union copies of two generating sets. Next, we discuss the presentations, acts, subacts, direct products and bilateral semidirect product of the semigroup of transformation with restricted equivalence E(X, \(\sigma\)) and its application.

Extension of Restricted Equivalence Functions and Similarity Measures for Type-2 Fuzzy Sets

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

Generalized nonconservative gravitational field equations from Herglotz action principle

We formulate a nonconservative gravitational theory based on the Herglotz variational principle in a tensorial covariant form. The model presented here may be seen as an improvement of the theory proposed in [ et al., Phys. Rev. D 95, 101501 (2017)], whose resulting field equations are meaningful just in particular coordinate systems. The new theory we report in this work is free from such a restriction. In comparison to the standard general relativity, both theories based on the Herglotz principle contain an extra vector field, the Herglotz field, but the new theory is formulated by taking advantage of the restricted equivalence between Lagrangian functions in the scope of the Herglotz action principle. The more restricted class of equivalent Lagrangian functions, in comparison with the Hamilton variational principle, is the key point to finding a Lagrangian that generates a new alternative gravitational theory in a covariant form. Once the equations that govern the dynamics of the gravitational field are obtained, a few simple cosmological models are investigated. It is found that the Herglotz vector field reduces to a single function that, under certain conditions, plays the role of the cosmological constant in general relativity, turning unnecessary the introduction of dark energy to explain the accelerated expansion of the Universe. The linearized version of the theory is also investigated and it is verified that the theory shows a dissipative character regarding the propagation of gravitational perturbations. From observational data, in both scenarios, the magnitude of the Herglotz field is estimated.

Loanable funds versus money creation in banking: a benchmark result

We establish a benchmark result for the relationship between the loanable-funds and the money-creation approach to banking. In particular, we show that both processes yield the same allocations when there is no uncertainty. In such cases, using the much simpler loanable-funds approach as a shortcut does not imply any loss of generality. When there is aggregate risk with complete contracts and complete markets, we indicate that a restricted equivalence result holds.

Exploring fuzzy set consensus analysis in IoT resource ranking

The complexity of the procedures for discovering, classifying, and selecting suitable resources to meet customer demands is related to the growing resource offers connected to the Internet. This proposal takes into account the uncertainties in the specification and processing of customer preferences, via consensual analysis. In this work we study the relationship between restricted equivalence functions, more generally, consensus measures, and the possibility of building the latter using the former. Thus, consensus measures of fuzzy values and consensus measures on fuzzy sets are both defined by aggregations, such as the arithmetic mean and the exponential mean. Based on the interval-valued fuzzy logic we consider inaccuracies related to the measurements beyond the uncertainties, modeling the imprecision of expertise in classifying a set of resources in the IoT based on the IT2FL-EXEHDA-RR Model. Several results arise from these methods. Firstly, the methodology L[0,1]-FCM, which is performed to measure how similar are the corresponding fuzzy values of a fuzzy set on [0,1]. In particular, it is applied to the superior and the inferior limits of each interval-valued membership function, modeling linguistic variables related to the attributes of the IT2FL-EXEHDA-RR Model. And the second one, based on LFχ-FSCM, providing the consensus analysis among a family of fuzzy sets. In the reported case study, this is applied to obtain the consensus measure between corresponding upper and lower bounds of each interval-valued function. This research also investigates the conditions under which we can build convex sum based on LFχ-FSCM, including the analysis of their main properties.

Strong negations and restricted equivalence functions revisited: An analytical and topological approach

Throughout this paper, our main idea is to analyze the concepts of a strong negation and a restricted equivalence function, that appear in a natural way when dealing with theory and applications of fuzzy sets and fuzzy logic. Here we will use an analytical and topological approach, showing how to construct them in an easy way. In particular, we will also analyze some classical functional equation related to those key concepts.

A cohesion-driven consensus reaching process for large scale group decision making under a hesitant fuzzy linguistic term sets environment

Large-scale group decision-making (LSGDM) under uncertainty modelled by comparative linguistic expressions based on a hesitant fuzzy linguistic term set (HFLTS) has recently attracted the interest of many researchers and research, due to the necessity of its function in LSGDM, and the challenges it faces such as the managing of the scalability problem, uncertainty of experts’ opinions and dealing with polarized conflicting opinions. To smooth out such discrepancies and obtain agreed solutions Consensus Reaching Processes (CRPs) for LSGDM have been applied, in which experts are grouped into sub-groups according to the closeness of their opinions to deal with scalability. However, most CRPs for LSGDM are driven by a majority rule, in which larger sub-groups, where there might be internal disagreements, lead the consensus. In such processes, the internal disagreements can produce unsatisfactory solutions. Consequently, the majority view should be complemented by additional mechanisms that also measure the strength of the sub-groups’ opinions. A good measurement of such strength is the cohesion among the sub-group members. Therefore, in this paper, a new cohesion measure for HFLTS based on restricted equivalence functions for measuring the experts’ sub-group cohesiveness is introduced to drive the consensus process together the majority and thus reduce the impact of internal disagreements risen in majority driven CRPs. It is then integrated in a new cohesion-driven CRP approach based on LSGDM to deal with comparative linguistic expressions based on HFLTS. An experimental analysis on different large scale scenarios will show the performance and importance of cohesion in consensus based LSGDM.

Intuitionistic fuzzy approach for enhancement of low contrast mammogram images

AbstractMammogram image enhancement is very much necessary in diagnosing breast cancer or tumor at an early stage. Nonuniform illumination and low contrast images are commonly encountered in mammogram images. Conventional enhancement algorithms produce either some artifacts or cannot highlight minute details present in the images, particularly when dealing with mammogram images. In this article, we propose a new mammogram image enhancement scheme using Atanassov's intuitionistic fuzzy set (IFS) theory. IFS considers two uncertainties—membership and nonmembership degree apart from membership degree as in fuzzy set theory. As mammogram images are low contrast images and many of the image definitions are vague/unclear, so IFS theory may be suitable for better image enhancement. Initially, the image is transformed to an intuitionistic fuzzy image using a novel intuitionistic fuzzy generator. Hesitation degree is computed and using the hesitation degree, two membership levels are computed to form an interval type 2 fuzzy set. These two membership functions are then combined using Zadeh's fuzzy t‐conorm to form a new membership function. Threshold of interval type 2 fuzzy image is obtained using restricted equivalence function. Using the threshold, modified fuzzy hyperbolization is carried out. Real data experiments demonstrate that the proposed algorithm has better performance on contrast and visual quality of the images both quantitatively and qualitatively when compared with different existing methods.

Moderate deviation and restricted equivalence functions for measuring similarity between data

In this work we study the relation between moderate deviation functions, restricted dissimilarity functions and restricted equivalence functions. We use moderate deviation functions in order to measure the similarity or dissimilarity between a given set of data. We show an application of moderate deviate functions used in the same way as penalty functions to make a final decision from a score matrix in a classification problem.

Similarity between interval-valued fuzzy sets taking into account the width of the intervals and admissible orders

In this work we study a new class of similarity measures between interval-valued fuzzy sets. The novelty of our approach lays, firstly, on the fact that we develop all the notions with respect to total orders of intervals; and secondly, on that we consider the width of intervals so that the uncertainty of the output is strongly related to the uncertainty of the input. For constructing the new interval-valued similarity, interval valued aggregation functions and interval-valued restricted equivalence functions which take into account the width of the intervals are needed, so we firstly study these functions, both in line with the two above stated features. Finally, we provide an illustrative example which makes use of an interval-valued similarity measure in stereo image matching and we show that the results obtained with the proposed interval-valued similarity measures improve numerically (according to the most widely used measures in the literature) the results obtained with interval valued similarity measures which do not consider the width of the intervals.

Application of two different methods for extending lattice-valued restricted equivalence functions used for constructing similarity measures on L-fuzzy sets

Based on previous investigations, we have proposed two different methods to extend lattice-valued fuzzy connectives (t-norms, t-conorms, negations and implications) and other related operators, considering a generalized notion of sublattices. Taking into account the results obtained and seeking to analyze the behavior of both extension methods in face of fuzzy operators related to image processing, we have applied these methods so as to extend restricted equivalence functions, restricted dissimilarity functions and Ee, N-normal functions. We also generalize the concepts of similarity measure, distance measure and entropy measure for L-fuzzy sets constructing them via restricted equivalence functions, restricted dissimilarity functions and Ee, N-normal functions.

Mammogram Enhancement Using Intuitionistic Fuzzy Sets.

Conventional mammogram enhancement methods use transform-domain filtering, which possibly produce some artifacts or not well highlight all local details in images. This paper presents a new enhancement method based on intuitionistic fuzzy sets. The presented algorithm initially separates a mammogram via a global threshold and then fuzzifies the image utilizing the intuitionistic fuzzy membership function that adopts restricted equivalence functions. After that, the presented scheme hyperbolizes membership degrees of foreground and background areas, defuzzifies the fuzzy plane, and achieves a filtered image via normalization. Finally, an enhanced mammogram is obtained by fusing the original image with filtered one. These implementations can be processed in parallel. This algorithm can improve the contrast and visual quality of regions of interest. Real data experiments demonstrate that our method has better performance regarding the improvement of contrast and visual quality of abnormalities in mammograms (such as masses and/or microcalcifications), compared with classical baseline methods. This algorithm has potential for understanding and determining abnormalities.