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As an important technology in artificial intelligence, Granular Computing has emerged as a new multi-disciplinary paradigm and received much attention in recent years. Information granules forming an abstract and efficient characterization of large volumes of numeric data have been considered as the fundamental constructs of Granular Computing. By generating centroids (prototypes) and partition matrix, fuzzy clustering is a commonly encountered way of information granulation. As a reverse process of granulation, degranulation involves data reconstruction completed on a basis of the granular representatives (decoding information granules into numeric data). Previous studies have shown that there is a relationship between the reconstruction error and the performance of the granulation process. Typically, the lower the degranulation error is, the better performance of granulation process becomes. However, the existing methods of degranulation usually cannot restore the original numeric data, which is one of the important reasons behind the occurrence of the reconstruction error. To enhance the quality of reconstruction (degranulation), in this study, we develop an augmented scheme through modifying the partition matrix. By proposing the augmented scheme, we elaborate on a novel collection of granulation-degranulation mechanisms. In the constructed approach, the prototypes can be expressed as the product of the dataset matrix and the partition matrix. Then, in the degranulation process, the reconstructed numeric data can be decomposed into the product of the partition matrix and the matrix of prototypes. By modifying the partition matrix, the new partition matrix is constructed through a series of matrix operations. We offer a thorough analysis of the developed scheme. The experimental results are in agreement with the underlying conceptual framework. The results obtained on both synthetic and publicly available datasets are reported to show the enhancement of the data reconstruction performance thanks to the proposed method. It is pointed out that by using the proposed approach in some cases the reconstruction errors can be reduced close to zero by using the proposed approach. [ABSTRACT FROM AUTHOR]

Fuzzy rule-based models are widely employed to tackle regression problems due to their simplicity and comprehensibility. Numeric functions (e.g., linear ones) are generally utilized to represent the conclusion part of a rule, but this simple module cannot well describe the behavior of data located within local area defined by the premise part. In this study, a novel reinforced fuzzy clustering-based rule model (RFCRM) is presented to address this issue. First, information granulation (Fuzzy C-Means) approach is used to reveal the structure in the data and divide the input space into local regions, as well as form the premise parts. Second, the augmented Random Vector Functional Link Networks (RVFLNs) are employed as the conclusion parts to strengthen the description and representation of the behavior of data positioned within the local regions. Weighted Least Squares Error estimation with regularization is exploited to estimate the coefficients of the connection weights of RVFLN. Also, an exponential weight method benefited from the weight function theory encountered in harmonic analysis is introduced to improve the penalty strategy of regularization in coefficient estimation. More specifically, an exponential weighted ℓ 2 regularization (EW ℓ 2) strategy equipped with the exponential functions as the penalty terms is proposed to realize the targeted penalty on coefficients. The merit of EW ℓ 2 over ordinary ℓ 2 is that different types of polynomial terms can be identified and penalized separately in coefficient estimation. This strategy not only prevents the decline of generalization ability, but also effectively enhances the prediction potential of the model. By combining weighted LSE based on EW ℓ 2 with FCM partition and the augmented RVFLN leads to better prediction accuracy as well as lower complexity (viz., number of rules). Finally, detailed experimental studies are conducted to demonstrate the effectiveness of the proposed RFCRM. [ABSTRACT FROM AUTHOR]

This paper introduces some recent approaches to the fuzzy fractional Abel integral equations with respect to another function, which is called the fuzzy fractional Abel k −integral equations. The problem proposed here allows for the interpolation of different types of fuzzy classical fractional Abel integral equations, and the solvability of each type of integral equation is also discussed. Additionally, a new approach to the linear fuzzy fractional differential equations in the space of generalized fuzzy functions is introduced. Unlike previous approaches with gH-differentiability, this approach avoids making any prior assumptions about the solution's monotonic behavior during the solving process. It also overcomes the problem of multiple solutions for fuzzy fractional differential equations. To demonstrate the benefits of this approach, two initial value problems of linear fuzzy differential equations are compared using two different concepts of fractional derivatives: the Caputo fractional gH-differentiability and the Caputo fractional derivative in the generalized fuzzy space. Furthermore, some applications are provided in a viscoelastic material model and a population growth with harvesting model. [ABSTRACT FROM AUTHOR]

In this paper, we present a method of extending t-norms and t-conorms to a given closed subinterval [ a , b ] in [ − ∞ , + ∞ ] , which preserves the basic properties of these fuzzy connectives and their one-to-one correspondence with operations in a I-semigroup. Subsequently, the generalized De Morgan triples, weighted generalized interval t-norms and t-conorms are provided to construct the aggregation function for extended basic uncertain information. Eventually, we apply the proposed aggregation function to establish the basic uncertain linguistic information-based aggregation operator. [ABSTRACT FROM AUTHOR]

Bisimulation techniques have been recently introduced to fuzzy transition systems to compare their behavior. When studying bisimulation, one of the most popular methods is relational lifting. In this paper, we first realize the relational lifting through a function S ˜. The new relational lifting method relaxes the condition of lifting relation from using weight functions to function S ˜ defined by residua (implications) in complete residuated lattices. We first discuss some properties of the lifting operation, and then define an α -bisimulation under S ˜ , where α ∈ [ 0 , 1 ] , for nondeterministic fuzzy transition systems (NFTSs). The new bisimulation can handle the case that two states are equivalent intuitively but are simply treated as distinguishable by some behavioral measures, and so it is much more natural and robust. Then we provide some properties of α -bisimulation under S ˜. The parallel composition operator of NFTSs is addressed and we show that α -bisimilarity under S ˜ is nonexpansive with respect to the operator, which makes compositional verification possible. In addition, we give a fixed-point characterization of α -bisimulation under S ˜ and compute α -bisimilarity under S ˜. Finally, we introduce a real-valued logic to characterize α -bisimilarity under S ˜. [ABSTRACT FROM AUTHOR]

The generalized extended Bonferroni mean (GEBM) is a powerful tool for modeling the complex process of aggregating information, whether it is homogeneously or heterogeneously connected, within a composite aggregation structure. It maintains several favorable characteristics and effectively captures the diverse and interconnected nature of expert opinions or criteria, which is commonly observed in various decision-making contexts. This research expands upon the existing GEBM framework by applying it to the specific domains of q-rung orthopair fuzzy sets (q -ROFSs) and extended q -rung orthopair fuzzy sets (E q -ROFSs). Furthermore, it examines the transformation processes among different variants of GEBMs. To facilitate the development of generalized aggregation functions, the de Morgan triplets for q -ROFSs and E q -ROFSs are established. By introducing an isomorphism, the transformation relationship between the aggregation functions for q -ROFSs and E q -ROFSs is analyzed. Based on this foundation, the Bonferroni mean de Morgan triplet-based GEBMs for q -ROFSs and E q -ROFSs are proposed, and the keeping-order relations for these proposed GEBMs are discussed. Finally, several special cases of the GEBMs for q -ROFSs and E q -ROFSs are obtained, and several relevant theorems are verified. [ABSTRACT FROM AUTHOR]

Recently, the polynomial-based radial basis function neural networks (P-RBFNNs) have been successfully applied to regression tasks. However, the redundant and non-linear partitioned data easily interfere with accurate partitioning of clusters completed in P-RBFNNs, affecting the regression performance of this existing model. Because the squared error is used as the cost function of the learning method, P-RBFNNs are sensitive to noise interference. In order to cope with these problems, this study introduces a double iterative learning-based polynomial based-RBFNNs (DP-RBFNNs) modeling that focuses on the formation of architectures to improve the accuracy of regression performance as well as enhance the robustness through double iterative learning as follows: a) support vector-based Gaussian kernel fuzzy c-means (SV-GKFCM) as a kind of the support vector-based kernel fuzzy clustering are designed to determine connections (weights) between the input and hidden layers of the proposed model. SV-GKFCM helps effectively reduce the number of redundant data to re-modify the partitioning of clusters in the DP-RBFNNs. In addition, the cluster centers can be accurately updated from the non-linear partitioned data with the aid of Gaussian kernel distance in SV-GKFCM; b) least absolute shrinkage deviations (LASD) as a robust estimation are designed to update connection weights between the hidden and output layers. The SAE (sum of absolute error) function in the LASD method is used as a cost function to reduce the noise interference in the procedure of weight estimation as well as enhance the robustness of the DP-RBFNNs. The superiority of the proposed model is demonstrated through the experimental results. [ABSTRACT FROM AUTHOR]

This study is concerned with the concept of Ensemble Fuzzy Radial Basis Function Neural Networks (EFRBFNN) endowed with clustering techniques, polynomial-based LSE/WLSE learning as well as their optimization implemented by Multi-Objective Particle Swarm Optimization (MOPSO). The network architecture of the proposed EFRBFNN classifier is considered three types of clustering techniques and four forms of regression polynomials. The proposed classifier can not only fully capture the local distribution (feature) information contained in the data, but also choose effective network architecture for improving classification performance and compactness. Three classes of clustering techniques (such as iterative self-organizing data analysis techniques algorithm (ISODATA), affinity propagation (AP), and extended K-means) are considered to effectively produce membership degrees based on feature information extraction suitable to fit data characteristics. Least Square Error Estimation (LSE) and Weighted Least Square Error Estimation (WLSE)-based learning with four types of regression polynomials are considered to estimate to coefficients of polynomials. MOPSO is exploited for choosing the effective architecture among different clustering techniques, polynomials, and polynomial-based learning with multi-objective parametric optimization. MOPSO helps achieve a sound compromise between the preferred classification performance and the compactness realized with the aid of four objective functions such as a) classification accuracy (CA), b) complexity of clustering, c) complexity of polynomial and d) sum of squared coefficients (SSC). The improvement and effectiveness of the proposed network architecture are quantified on a basis of the comprehensive experimental results and also a comparative analysis is offered to demonstrate the superiority of the proposed classifier. [ABSTRACT FROM AUTHOR]

Enhancing the ability to make informed decisions stands as a significant challenge in modern IT. Specifically, there is a growing need to improve the efficiency of classification algorithms. When faced with multiple results derived from various methods, one can select the most probable decision using a robust aggregation operator. A common class of algorithms employed for this purpose is based on extensions of the Choquet integral (CI). In this study, we introduce and extensively analyze a novel aggregation operator concept founded on the generalization of the CI. This approach leverages quadrature formulae to calculate Choquet integral values, but with a unique modification involving a smoothing operation. This refinement results in more precise values, preserving the essential characteristics of the Choquet integral. These refined values can be effectively applied in the aggregation of classifiers and, more broadly, in information fusion processes. A series of numerical experiments demonstrates the efficiency of our approach. Furthermore, we thoroughly discuss and provide mathematical proofs for the properties of the newly constructed operators. • We propose novel and efficient generalization of Choquet integral. • The generalization is based on the differences between quadratures applied to calculate a numerical function value. • The operation leads to the smoothness process. • Numerical experiments' results demonstrate the efficiency of the proposal. • Anomaly detection and multi-class problems are analyzed in the experimental series. [ABSTRACT FROM AUTHOR]

Fuzzy c -means (FCM) clustering is a clustering method based on fuzzy theory. This method shows good adaptability by assigning membership values to each sample to represent the degree of membership of the sample to each cluster. However, when dealing with fuzzy boundary data, FCM also generates uncertainty and randomness, which in turn affects the accuracy of clustering results and the number of iterations required for algorithm convergence. In order to solve this problem, fuzzy rough set, as a method of dealing with uncertain data, provides a more accurate and strict description method for the processing of boundary data. Considering this advantage, this paper proposes a new fuzzy rough c -means (FRCM) clustering algorithm to improve the performance and iteration efficiency of FCM. Specifically, in this paper, the similarity based on the Gaussian kernel and the membership information of the object for each cluster are firstly used to construct a fuzzy rough set model to describe the fuzzy roughness between the object and the cluster center, which is used to more accurately represent the relationship between the object and the cluster. Secondly, based on the fuzzy rough model, the fuzzy rough degree of the object is calculated, which is used to describe the approximation degree of the object to the center of each cluster. This fuzzy expression can better handle the fuzzy boundary problem between the object and the cluster center, thereby improving the clustering results and enhancing interpretability. Finally, based on fuzzy rough degree information, the FRCM algorithm is designed. The experimental results show that our proposed method has better performance compared to other comparative clustering methods on both synthetic and real datasets. Specifically, compared to FCM, this algorithm exhibits higher iteration efficiency. [ABSTRACT FROM AUTHOR]

This article aims to introduce and investigate the analytical fuzzy solution of the incommensurate non-homogeneous system of fuzzy linear fractional differential equations (INS-FLFDEs) using trivariate Mittag-Leffler functions. Entries of the coefficient matrix of the given system are treated as real numbers, initial-values are triangular fuzzy numbers (TFNs), and the forcing function is a fuzzy set (or a bunch) of real function. We extract the potential solution in the form of a fuzzy bunch of real functions (FBoRFs) rather than the solution of fuzzy-valued functions. We formulate the fuzzy initial value problem as a set of classical initial value problems by taking the forcing function from the class of FBoRFs and the initial value from the collection of TFNs (as a special case). The solution of this system is in the form of a trivariate Mittag-Leffler function. We interpret this solution as an element of the fuzzy solution set and assign the minimum value of membership that takes from the forcing function and the initial value in the fuzzy set. The originality of the proposed technique is that the uncertainty is smaller compared to the uncertainty extracted from other techniques. In addition, generalized derivatives increase the order and dimension of the system. Therefore, the proposed technique is better in terms of complexity because it reduces the order and dimension of the system. Finally, to grasp the proposed technique, we solve the electrical network and multiple mass-spring systems as applications and analyze their graphs to visualize and support theoretical results. [ABSTRACT FROM AUTHOR]

This study focuses on two critical design aspects of a progressive fuzzy polynomial neural network (PFPNN): the influence of the gated recurrent unit (GRU) structure and the implementation of fitness-based candidate neuron selection (FCNS) through two probabilistic strategies. The primary objectives are to enhance modeling accuracy and to reduce the computational load associated with nonlinear regression tasks. Compared with the existing fuzzy rule-based modeling architecture, the proposed dynamic model consists of the GRU structure and the hybrid fuzzy polynomial architecture. In the initial two layers of the PFPNN, we introduce three types of polynomial and fuzzy rules into the GRU neurons (GNs) and fuzzy polynomial neurons (FPNs), which can effectively reveal potential complex relationships in the data space. The synergy of the FCNS strategies and the ℓ 2 regularization learning method is to design a progressive regression model adept at melding the GRU structure with a self-organizing architecture. The proposed GRU structure and polynomial-based neurons significantly improve the modeling accuracy for time-series datasets. The rational utilization of FCNS strategies can reinforce the network structure and discover the potential performance of neurons of the network. Furthermore, the inclusion of ℓ 2 norm regularization provides additional stability to the proposed model and mitigates the overfitting issue commonly encountered in many existing learning methods. We validated the proposed neural networks using six time-series, four machine learning, and two real-world datasets. The PFPNN outperformed other models in the comparison studies in 83.3% of the datasets, emphasizing its superiority in terms of developing a stable deep structure from diverse candidate neurons and reducing computational overhead. [ABSTRACT FROM AUTHOR]