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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]

Three-way approximations of fuzzy sets are an important scheme of granular computing, by abstracting a fuzzy set to its discrete three option-alternatives which adhere to human cognitive behaviors and reduce the computational burden. The key point of such three-way approximations of fuzzy sets is how to choose a suitable design leading to their realization. Undesired three-way approximations might be produced if the selected mechanism is unsuitable to data distribution. In this study, the principles for constructing three-way approximations of fuzzy sets are summarized. The following taxonomy of these principles is provided, namely (i) uncertainty balance-based principle, (ii) prototype-based principle, and (iii) model-based invoking the tradeoff between classification error and the number of data that have to be classified. A number of detailed optimization models are discussed in detail. To evaluate the performance of different construction principles, a general unsupervised learning framework based on three-way approximations of fuzzy sets is exhibited. Some synthetic data sets along with sixteen data sets from UCI repository are involved for experiments. Friedman testing followed by Holm-Bonferroni testing are exploited to test the performance significance of the proposed criteria, which can provide insights and deliver guidance when choosing a principle for constructing three-way approximations of fuzzy sets in the real-world scenarios. The research methods in this paper can also be extended to supervised and semi-supervised learning areas. [ABSTRACT FROM AUTHOR]

Fuzzy clustering is regarded as an unsupervised learning process that constitutes a prerequisite for many other data mining techniques. Deciding how to classify data efficiently and accurately has been one of the topics pursued by many researchers. We anticipate that the classification performance of the clustering is strongly dependent on the boundary data (viz. data located at the boundaries of the clusters). The boundary data hold some levels of uncertainties and as such contain more information than others. Usually the greater the uncertainty, the more information contained in such data. To improve the quality of clustering, this study develops an augmented scheme of fuzzy clustering, in which a novel weighted data-based fuzzy clustering is proposed. In the introduced scheme, a dataset is composed of boundary data and non-boundary data. The partition matrix is used to determine the boundary data and the non-boundary data to be next considered in the clustering process. Then, we assign different weights to each datum to construct the weighted data. During this process, we make the weights for the boundary data and the non-boundary data different, which makes the contributions of the boundary data and the non-boundary data to the prototypes being reduced and enhanced, respectively. Furthermore, we build a weighting function to determine the weights of the data. The weighted data are used to optimize the prototypes. With the optimized prototypes, the partition matrix can be refined, which ultimately makes the boundaries of the clusters optimized. Finally, the classification performance of fuzzy clustering is enhanced. We offer a thorough analysis of the developed scheme. Comprehensive experimental studies involving synthetic and publicly available datasets are reported to demonstrate the performance of the proposed approach. [ABSTRACT FROM AUTHOR]

This study proposes a novel methodology for the extraction of a hierarchical Takagi-Sugeno fuzzy rule-based architecture from data. This architecture reduces the number and complexity of involved fuzzy rules, and, in many cases, improves the predictive performance of the model. The proposed hierarchical architecture takes the form of a cascading topology in which the predicted result computed at the previous layer is considered in the output part of the fuzzy rules. We propose a well-defined general methodology for the extraction of this hierarchical topology from data and discuss strategies for feature selection and choosing the number of rules at each level. The performance of the proposed methodology is demonstrated through extensive experiments, including case studies outlining specific behaviors and parameterizations, and comparative experiments showing the performance of the proposed architecture compared to a standard flat fuzzy rule-based system. [ABSTRACT FROM AUTHOR]

Abstract Fuzzy rule-based models have attracted attention because of their modular architectures, well-developed design methodologies and practices as well as interpretability aspects. Methods exploiting factors of randomness offer significant efficiency and implementation simplicity that are essential in numerous application areas. In this study, we propose an original development of fuzzy rule-based models established with the aid of concepts of randomization algorithms. Several design strategies involving different random prototypes generation and basis functions approximation are studied. We investigate performance aspects of randomized rule-base and look at the performance versus the key components of the models such as the number of rules and the use of the randomized algorithms in the development. Furthermore, a comparative study is offered to quantify the efficiency of randomized algorithms. Experimental studies are reported for a series of publicly available data sets to illustrate the effectiveness of the proposed method and discuss its main features. [ABSTRACT FROM AUTHOR]

We present a spatiotemporal clustering method, namely SEFCM, which is a generalization of the extended fuzzy C-Means (EFCM) method for detecting hotspots in spatial analysis. Each pattern is formed by three features: the geographical coordinates and the period in which a certain event is occurred. This method is applied to a spatial dataset (formed by earthquake epicenters occurred in Southern Italy since 2001 till to 2014) for prediction of the hotspots obtained in a given year. Comparisons of the prediction results are also made with the ones obtained by applying the known ST-DBSCAN algorithm. [ABSTRACT FROM AUTHOR]

Under the assumption of rational economics, the consistency of judgments is one of the important issues in multiple criteria decision making (MCDM) methods. We propose three axiomatic properties of the consistent judgments in relative measurements being considered from the perspective of strict logic and rationality: ( A 1) the reciprocal property of pairwise comparisons, ( A 2) the invariance of consistency with respect to permutations of alternatives, and ( A 3) the robustness of the ranking of alternatives. These properties are further applied to analyze the consistency of the judgments expressed by positive real numbers and triangular fuzzy numbers, respectively. The inconsistency of comparison matrices encountered in the Analytic Hierarchy Process (AHP) can be considered to be the case of weakening the axiomatic property ( A 3). Fuzzifying the judgments is due to the weakening of axiomatic property ( A 1). A method of weakening the axiomatic properties for triangular fuzzy reciprocal matrices is proposed and a new concept of approximation-consistency is put forward to distinguish from the typical consistency. By considering the permutations of alternatives, an illustrative example is presented to show the use of the proposed procedures for solving the decision making problems with triangular fuzzy reciprocal preference relations. [ABSTRACT FROM AUTHOR]

Owning to their abilities to reveal structural relationships in data, fuzzy clustering plays a pivotal role in fuzzy modeling, pattern recognition, and data analysis. As supporting an unsupervised mode of learning, fuzzy clustering, brings about unique opportunities to build a structural backbone of numerous constructs in the areas identified above. A follow-up phase is required when the structural findings developed in the form of fuzzy clusters need some refinements, usually when the clustering results are used afterwards in the model being developed in a supervised mode. Following this general line of thought, in this study we propose a novel approach to optimize the clustering and classification performance of the Fuzzy C-Means (FCM) algorithm. Proceeding with a collection of clusters (information granules) produced by the FCM, we carry out the refinements of the results (in order to improve the representation or classification capabilities of fuzzy clusters) by adjusting a location of the prototypes so that a certain performance index becomes optimized. At this phase, the optimization is carried out in a supervised mode with the aid of Differential Evolution (DE). We propose five different strategies to adjust a location of the prototypes. Experimental studies completed on synthetic data and publicly available real-world data quantify the improvement of the representation and classification abilities of the clustering method. [ABSTRACT FROM AUTHOR]

Rules in fuzzy rule-based models convey essential knowledge about the system under discussion. As such, they capture the essence of relationships occurring among input and output variables. While the quality of such fuzzy models is predominantly related with the accuracy and eventual interpretability of rules (although to a limited extent), the quality of rules being regarded as generic pieces of knowledge has not been studied so far. In this study, we formulate and investigate this problem by looking at the quality of rules, including aspects of (a) stability, (b) generalizability, and (c) conflict. We identify a concept of rule multiplicity, conflict, and study an emergence of rule generalization. A number of pertinent performance indices are developed, and their usage is presented through a series of experimental studies. [ABSTRACT FROM AUTHOR]

Processing realized in fuzzy rule-based models is associated with matching individual rules with the existing input data and aggregation of the levels of matching and conclusions of the rules. Rules are processed as individual entities. In this study, we introduce an augmentation of fuzzy models by facilitating interaction among the rules leading to more flexible type membership functions of fuzzy sets forming conditions of the rules (thus resulting in substantially advanced topology of the partition of the input space). Different ways of realizing interaction among the rules are studied. In the sequel, we develop a granular fuzzy model implied by the rule-based model showing how granular parameters of the original rule-based model enhance its quality expressed in terms of coverage of experimental data. The two evaluation criteria of the constructed granular model, namely coverage and specificity are studied. Experimental results are reported for a series of publicly available data. [ABSTRACT FROM AUTHOR]

This note offers some personal views on the two pioneers of fuzzy sets, late Professors Hideo Tanaka and Kiyoji Asai. The intent is to share some personal memories about these remarkable researchers and humans, highlight their long-lasting research accomplishments and stress a visible impact on the fuzzy set community. The note elaborates on new and promising research avenues initiated by fuzzy regression and identifies future developments of these models emerging within the realm of Granular Computing and giving rise to a plethora of granular fuzzy models and higher-order and higher-type granular constructs. [ABSTRACT FROM AUTHOR]

Abstract: In this study, we present a comprehensive comparative analysis of kernel-based fuzzy clustering and fuzzy clustering. Kernel based clustering has emerged as an interesting and quite visible alternative in fuzzy clustering, however, the effectiveness of this extension vis-à-vis some generic methods of fuzzy clustering has neither been discussed in a complete manner nor the performance of clustering quantified through a convincing comparative analysis. Our focal objective is to understand the performance gains and the importance of parameter selection for kernelized fuzzy clustering. Generic Fuzzy C-Means (FCM) and Gustafson–Kessel (GK) FCM are compared with two typical generalizations of kernel-based fuzzy clustering: one with prototypes located in the feature space (KFCM-F) and the other where the prototypes are distributed in the kernel space (KFCM-K). Both generalizations are studied when dealing with the Gaussian kernel while KFCM-K is also studied with the polynomial kernel. Two criteria are used in evaluating the performance of the clustering method and the resulting clusters, namely classification rate and reconstruction error. Through carefully selected experiments involving synthetic and Machine Learning repository (http://archive.ics.uci.edu/beta/) data sets, we demonstrate that the kernel-based FCM algorithms produce a marginal improvement over standard FCM and GK for most of the analyzed data sets. It has been observed that the kernel-based FCM algorithms are in a number of cases highly sensitive to the selection of specific values of the kernel parameters. [Copyright &y& Elsevier]

Abstract: Ordinal feature values are totally ordered labels that can be considered as fuzzy sets. The formulation of proper fuzzy sets for ordinal labels is important for the systems that deal with the objects of mixed feature types. When a proper ordinal–numerical mapping for an ordinal feature of interest is given, proper fuzzy sets for the labels of the ordinal feature can be easily formulated. In this paper, we propose an adaptive method to learn proper ordinal–numerical mappings for ordinal features of interest from a given objects of mixed features including the ordinal features. The method starts with uniform ordinal–numerical mappings, and performs two steps iteratively. The first step computes a fuzzy partition over the given object set with the ordinal–numerical mappings. The second step learns new ordinal–numerical mappings from the new fuzzy partition in the way that the new mappings make the similarity between two ordinal labels be similar to the average similarity between the objects having the two labels, respectively. Through the alternate repetition of the two steps, both of the ordinal–numerical mappings and the clustering quality become gradually improved. The validity of the proposed method is strongly supported through the experiments with a modified fuzzy C-means clustering algorithm in which the proposed method is implemented. [Copyright &y& Elsevier]

Abstract: Frequency-based cluster prototypes have been used to cluster categorical objects, based on the simple matching dissimilarity measure. This paper introduces a new generalization called fuzzy p-mode prototype, of frequency-based prototypes. A fuzzy p-mode cluster prototype at a categorical feature is expressed as a list of p labels that have larger frequencies than others in the cluster. This paper also presents a new generalization of the fuzzy C-means clustering algorithm for the objects of mixed features. In the general fuzzy C-means clustering algorithm, any dissimilarity measures at the categorical feature level are assumed, not like other clustering algorithms that use the simple matching dissimilarity. The convergence of the general fuzzy C-means clustering algorithm under the optimization framework is proved. It is also explained through experiments over real object sets that the size of fuzzy p-mode prototypes and the fuzzification coefficients affect clustering performance. [Copyright &y& Elsevier]

Abstract: The ultimate challenges of system modeling concern designing accurate yet highly transparent and user-centric models. We have witnessed a plethora of neurofuzzy architectures which are aimed at addressing these two highly conflicting requirements. This study is concerned with the design and the development of transparent logic networks realized with the aid of fuzzy neurons and fuzzy unineurons. The construction of networks of this form requires a formation of efficient interfaces that constitute a conceptually appealing bridge between the model and the real-world experimental environment in which the model is to be used. In general, the interfaces are constructed by invoking some form of granulation of information; and binary (Boolean) discretization, in particular. We introduce a new discretization environment that is realized by means of particle swarm optimization (PSO) and data clustering implemented by the K-Means algorithm. The underlying structure of the network is optimized by invoking a combination of the PSO and the mechanisms of conventional gradient-based learning. We discuss various optimization strategies by considering Boolean as well as fuzzy data coming as the result of discretization of original experimental data and then involving several learning strategies. We elaborate on the interpretation aspects of the network and show how those could be strengthened through efficient pruning. We also show how the interpreted network leads to a simpler and more accurate logic description of the experimental data. A number of experimental studies are included. [Copyright &y& Elsevier]

Abstract: Statistical learning theory based on real-valued random samples has been regarded as one of the influential developments for small samples statistical estimation and learning. The key theorem of learning theory and the bounds on the rate of convergence of learning process are the most important theoretical fundamentals of the statistical learning theory. In this paper, we discuss a statistical learning theory based on fuzzy complex random samples. Firstly, the definition of fuzzy complex numbers is introduced and the fuzzy complex random variables along with their numeric characteristic are investigated. Secondly, we carry out further research focused on a special type of fuzzy complex number, namely rectangular fuzzy complex number and establish some properties and develop important theorems. We also prove the strong law of large numbers based on fuzzy complex random variables. Thirdly, the definitions of the fuzzy complex expected risk functional, the fuzzy complex empirical risk functional, the fuzzy complex empirical risk minimization principle and the consistency are provided and discussed. Finally, the key theorem of learning theory and the bounds on the rate of convergence of learning process based on fuzzy complex random samples are discussed. [Copyright &y& Elsevier]

Abstract: Many classification problems involve features whose specificity demand some form of feature space transformation (preprocessing) coupled with post-processing consensus analysis in order to accomplish a successful discrimination between different classes. In this study, we present a new methodology, which systematically addresses these design classification issues. At the preprocessing phase we offer a new approach of stochastic feature selection. This type of feature selection, collates quadratically transformed feature subsets for presentation to a collection of respective classifiers. In the sequel, independent classification outcomes are aggregated through fuzzy integration. The motivation behind the proposed methodology is twofold. Often, only a subset of features possesses discriminatory power while the remainder has a tendency to confound the effectiveness of the underlying classifier. Quite commonly, classification based on some consensus of classification outcomes coming from a set of classifiers operating upon different feature subsets becomes more accurate than the classification results produced by any individual classifier. To illustrate this design methodology, we discuss a classification problem coming from software engineering. Here we are concerned with a dataset comprosed of features describing a collection of qualitative attributes of a software system. The experiments demonstrate that the aggregated classification results using fuzzy integration are superior to the predictions from the respective best single classifiers. [Copyright &y& Elsevier]

Abstract: In this study, we introduce the concept of collaborative fuzzy clustering—a conceptual and algorithmic machinery for the collective discovery of a common structure (relationships) within a finite family of data residing at individual data sites. There are two fundamental features of the proposed optimization environment. First, given existing constraints which prevent individual sites from exchanging detailed numeric data, any communication has to be realized at the level of information granules. The specificity of these granules impacts the effectiveness of ensuing collaborative activities. Second, the fuzzy clustering realized at the level of the individual data site has to constructively consider the findings communicated by other sites and act upon them while running the optimization confined to the particular data site. Adhering to these two general guidelines, we develop a comprehensive optimization scheme and discuss its two-phase character in which the communication phase of the granular findings intertwines with the local optimization being realized at the level of the individual site and exploits the evidence collected from other sites. The proposed augmented form of the objective function is essential in the navigation of the overall optimization that has to be completed on a basis of the data and available information granules. The intensity of collaboration is optimized by choosing a suitable tradeoff between the two components of the objective function. The objective function based clustering used here concerns the well-known Fuzzy C-Means (FCM) algorithm. Experimental studies presented include some synthetic data, selected data sets coming from the machine learning repository and the weather data coming from Environment Canada. [Copyright &y& Elsevier]

In this paper, we propose an iterative algorithm for multiple regression with fuzzy variables. While using the standard least-squares criterion as a performance index, we pose the regression problem as a gradient-descent optimisation. The separation of the evaluation of the gradient and the update of the regression variables makes it possible to avoid undue complication of analytical formulae for multiple regression with fuzzy data. The origins of fuzzy input data are traced back to the fundamental concept of information granulation and an example FCM-based granulation method is proposed and illustrated by some numerical examples. The proposed multiple regression algorithm is applied to one-, three-and nine-dimensional synthetic data sets as well as the 13-dimensional Boston Housing dataset from the machine learning repository. The algorithm''s performance is illustrated by the corresponding plots of convergence of regression parameters and the values of the prediction error of the resulting regression model. General comments on the numerical complexity of the proposed algorithm are also provided. [Copyright &y& Elsevier]

Abstract: In this paper, we study the minimization of lattice-valued finite automata with membership values in a distributive lattice and its relationships to the decomposition of lattice-valued regular languages. First, we establish the equivalence of (nondeterministic) lattice finite automata (LA) and deterministic lattice finite automata (DLA). Furthermore, we provide some characterization of lattice-valued regular languages and regular operations on family of lattice-valued regular languages. In the sequel, we introduce some notions that help clarify the concept of minimal DLAs and present an effective algorithm to obtain a minimal DLA from a given LA. Using the construction of minimal DLA, we introduce some simple classes of lattice-valued regular languages such as L-unitary and L-prefix ones. We demonstrate that any lattice-valued regular language can be decomposed as disjoint joins of such kinds of simple languages. [Copyright &y& Elsevier]

Abstract: Fuzzy neurons and fuzzy neural networks (FNN) are constructs of computational intelligence that come with significant learning abilities and inherent transparency (interpretability). Consequently, they exhibit high potential to provide strong mechanisms for building intelligent systems that must operate in rapidly changing environments. However, to fully exploit the potential of FNN structures, efficient parallel-processing implementations are highly desired. In this study, our objective is to investigate this avenue and identify various critical hardware design issues as we propose a versatile neurofuzzy platform with a topology strongly influenced by theories of fuzzy modelling. With a novel hybrid-learning scheme involving structural and parametric optimization, we demonstrate how fuzzy neural networks are well suited in forming the adaptive logic-processing core of this platform, supporting intelligent information processing. The core emulates aspects of human thought, dealing with information at a granular level and using logic-oriented approximate reasoning to solve a problem. It is able to learn and approximate real-world concepts, building a knowledge base that may be interpreted and modified by the user. Drawing upon this knowledge, a high-speed parallel implementation has potential for performing inference of many simultaneous concepts in real-time, realizing cognition as it perceives the current state of its environment. [Copyright &y& Elsevier]

Abstract: In this study, we are concerned with the design of granular modeling being originally proposed by Pedrycz and Vasilakos. The enhancement of the development process comes in the form of the boosting mechanism applied to the generic model. In comparison with the original topology of the model studied so far, we augment it by a bias term and investigate its role in the overall architecture. Second, treating the granular model as a weak learner, we discuss the underlying mechanisms of boosting which in this setting has to be refined so that it handles the continuous case. From the design standpoint, we are interested in studying the following issues: (a) effectiveness of boosting when applied in this modeling framework along with its numeric quantification, and (b) impact of information granularity at which the granular models are developed on the improvements offered by boosting procedure itself. Numeric experiments help quantify the performance of the boosted granular models and gain a detailed view at the efficiency of the boosting strategy vis-à-vis different design scenarios. [Copyright &y& Elsevier]

Abstract: Semi-supervised (or partial) fuzzy clustering plays an important and unique role in discovering hidden structure in data realized in presence of a certain quite limited fraction of labeled patterns. The objective of this study is to investigate and quantify the effect of various distance functions (distances) on the performance of the clustering mechanisms. The underlying goal of endowing the clustering algorithms with a higher level of flexibility is done via the use of various distances. The enhancement of this character is evaluated by means of a comprehensive assessment of quality of clusters, their ensuing discrimination abilities and the accuracy of clusters themselves. In addition to the standard Euclidean distance being commonly exploited in fuzzy clustering, three more versatile and adaptive distance measures are considered such as its weighted version, a full adaptive distance, and a kernel-based distance. Using Fuzzy C-Means (FCM) coming in its generic format, we show its semi-supervised enhancements, derive detailed formulas and analyze their effectiveness. The improvements of semi-supervised clustering are empirically evaluated and numerically quantified with the use of several Machine Learning data sets. [Copyright &y& Elsevier]

Abstract: We study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice-ordered monoid . For these -fuzzy finite automata (-FFA, for short), we provide necessary and sufficient conditions for the extendability of the state-transition function. It is shown that nondeterministic -FFA (N-FFA, for short) are more powerful than deterministic -FFA (D-FFA, for short). Then, we give necessary and sufficient conditions for the simulation of an N-FFA by an equivalent D-FFA. Next, we turn to the closure properties of languages defined by -FFAs: we establish closure under the regular operations and provide conditions for closure under intersection and reversal, in particular we show that the family of fuzzy languages accepted by D-FFAs is not closed under Kleene closure operation, and the family of fuzzy languages accepted by N-FFAs is not closed under complement operation. Furthermore, we introduce the notions of -fuzzy regular expressions and give the Kleene theorem for N-FFAs. The description of D-FFAs by -fuzzy regular expressions is also given. Finally, we investigate the level structures of -FFAs. Our results provide some insight as to what extend properties of -FFAs and their languages depend on the algebraic properties of . [Copyright &y& Elsevier]

Abstract: This study is concerned with the genetic development of logic-based fuzzy models (logic networks). The models are constructed with the aid of AND and OR fuzzy neurons. These neurons are logic-driven processing units realizing s–t and t–s composition operations of fuzzy sets and fuzzy connections with “t” and “s” being triangular norms and co-norms. The fundamental architecture of the network comprises a hidden layer formed by a collection of AND neurons that is followed by an output layer consisting of OR neurons. We show that this structure of the network directly translates into a collection of “if-then” statements. The learning procedure in such networks consists of two main development phases. First, a blueprint of the network is constructed genetically (through the use of a genetic algorithm) with the underlying intent of capturing the structural (logical) essence of the data and encapsulating it into the network. At this development phase the resulting network is kept binary (Boolean) with anticipation that the binary connections are a sound model of the blueprint of the logic nature of the data. The second phase of the design of the network is aimed at it refinement in which we move from the binary connections and optimize them to reflect the details of the data. There are two main optimization mechanisms applied in the above sequence: the structural binary development uses a genetic algorithm (GA) while a parametric refinement of the network is realized through a gradient-based learning. This organization of the optimization process exhibits several evident advantages. The GA concentrates on a preliminary and rough binary optimization and in this way helps combat an eventual curse of dimensionality. This curse is inherent to high-dimensional problems to which gradient-based learning is quite vulnerable. Having the blueprint developed, further refinement being guided by gradient-based computing becomes more effective. An interesting optimization aspect addressed in the genetic optimization environment deals with a construction of optimal subspaces of the overall feature space: in highly dimensional spaces we can envision that only a small subset of features is essential and their reduction may help develop a compact network. We introduce a concept of network connectivity to quantify the aspect of feature reduction. A carefully presented numerical example serves as a detailed illustration of the development approach. [Copyright &y& Elsevier]

In this study, we introduce and study a proximity-based fuzzy clustering. As the name stipulates, in this mode of clustering, a structure “discovery” in the data is realized in an unsupervised manner and becomes augmented by a certain auxiliary supervision mechanism. The supervision mechanism introduced in this algorithm is realized via a number of proximity “hints” (constraints) that specify an extent to which some pairs of patterns are regarded similar or different. They are provided externally to the clustering algorithm and help in the navigation of the search through the set of patterns and this gives rise to a two-phase optimization process. Its first phase is the standard FCM while the second step is concerned with the gradient-driven minimization of the differences between the provided proximity values and those computed on a basis of the partition matrix computed at the first phase of the algorithm. The proximity type of auxiliary information is discussed in the context of Web mining where clusters of Web pages are built in presence of some proximity information provided by a user who assesses (assigns) these degrees on a basis of some personal preferences. Numeric studies involve experiments with several synthetic data and Web data (pages). [Copyright &y& Elsevier]

Experimental software data sets describing software projects in terms of their complexity and development time have been a subject of intensive modeling. A number of various modeling methodologies and modeling designs have been proposed including neural networks, fuzzy, and neurofuzzy models. In this study, we introduce a concept of Self-organizing neurofuzzy networks (SONFN), a hybrid modeling architecture combining neurofuzzy networks (NFN) and polynomial neural networks (PNN). The development of the SONFN dwells on the technologies of Computational Intelligence (CI), namely fuzzy sets, neural networks, and genetic algorithms. The architecture of the SONFN results from a synergistic usage of neurofuzzy networks (NFNs) and polynomial neural networks (PNNs). NFNs contribute to the formation of the premise part of the rule-based structure of the SONFN. The consequence part of the SONFN is designed using PNNs. We discuss two classes of SONFN architectures and propose comprehensive learning algorithms. The experimental results include well-known software data such as the NASA data set concerning software cost estimation and the one describing software modules of the Medical Imaging System (MIS). [Copyright &y& Elsevier]

Software quality is one of the most important practical features of software development. Project managers and developers look for methods and tools supporting software development processes and ensuring a required level of quality. To make such tools relevant, they should provide the designer/manager with some quantitative input useful for purposes of interpretation of the results. Knowledge provided by the tools leads to better understanding of the investigated phenomena. In this paper, we propose a comprehensive development methodology of logic-based models represented by fuzzy neural networks. A process of model development is performed in the stages of structural and parametric optimization. The structural optimization framework utilizes mechanisms of evolutionary computing, which become especially attractive in light of the structural optimization of the models. The parametric optimization is performed using the gradient-base method. The study comprises two detailed case studies dealing with existing software data. The first one deals with the quality assessment of software objects in an exploratory biomedical data analysis and visualization system. The second case is concerned with the model of software development effort discussed in the setting of some medical information system. [Copyright &y& Elsevier]

In this study, we introduce and investigate a class of neural architectures of polynomial neural networks (PNNs), develop a comprehensive design methodology and carry out a series of numeric experiments. PNN is a flexible neural architecture whose structure (topology) is developed through learning. In particular, the number of layers of the PNN is not fixed in advance but is generated on the fly. In this sense, PNN is a self-organizing network. We distinguish between two kinds of SOPNN architectures, that is, (a) polynomial neuron (PN) based and (b) fuzzy polynomial neuron (FPN) based self-organizing networks. This taxonomy is based on the character of each neuron structure in the network. Each of them comes with two structures referred here to as basic and the modified topology. Moreover, for each topology of the SOPNN we identify two types that is a generic and advanced type. The essence of the design procedure of self-organizing polynomial neural networks (SOPNN) dwells on the group method of data handling (GMDH) (IEEE Trans. Systems Man and Cybernet. 12 (1971) 364). Each node of the PN based SOPNN exhibits a high level of flexibility and realizes a polynomial type of mapping (linear, quadratic, and cubic) between input and output variables. FPN based SOPNN dwells on the ideas of fuzzy rule-based computing and neural networks. Especially in FPN based SOPNN, the generic rules in the system assume the form “if A then y = P(x)” where A is a fuzzy relation in the condition space while P(x) is a polynomial forming a conclusion part of the rule. Each FPN (processing element) consists of a series of the nonlinear inference rules. The conclusion part of the rules, especially the regression polynomial uses several types of high-order polynomials such as linear, quadratic, modified quadratic, and cubic. As the premise part of the rules, both triangular and Gaussian-like membership functions are studied. Simulations involve a series of synthetic as well as experimental data used across various neurofuzzy systems. A detailed comparative analysis is included as well. [Copyright &y& Elsevier]

The study is concerned with a new temporal version of fuzzy Petri nets—fuzzy timed Petri nets (ftPNs). These nets are augmented by temporal fuzzy sets that allow for the representation of timing effect (e.g., aging of information). We show how the time factor can be added as an integral part of the models of transitions and places. The formalism of the extended net is introduced and studied in depth. A series of illustrative examples are presented with intent of developing a better qualitative insight into the temporal behavior of the nets. A detailed learning scheme for the net is also given. [Copyright &y& Elsevier]

This paper introduces an identification method for nonlinear models in the form of Fuzzy-Neural Networks (FNN). In this model, we use two forms of the fuzzy inference methods—a simplified and linear fuzzy inference, and exploit a standard Error Back Propagation learning algorithm. The FNN modeling and identification environment realizes parameter identification through a synergistic usage of clustering techniques, genetic optimization and a complex search method. We use a Hard C-Means (HCM) clustering algorithm to determine initial apexes of the membership functions of the information granules used in this fuzzy model. The parameters such as apexes of membership functions, learning rates, and momentum coefficients are then adjusted using hybrid algorithm. The proposed hybrid identification algorithm is carried out by combining both genetic optimization (genetic algorithm, GA) and the improved complex method. An aggregate objective function (performance index) with a weighting factor is introduced to achieve a sound balance between approximation and generalization of the model. According to the selection and adjustment of the weighting factor of this objective function, we reveal how to design a model with sound approximation and generalization abilities. The proposed model is experimented with using several time series data (gas furnace, sewage treatment process and NOx emission process data of gas turbine power plant). [Copyright &y& Elsevier]

A process of information granulation takes care of an enormous flood of numerical details that becomes summarized and hidden (encapsulated in the form of fuzzy sets) at the time of the design of a neural network. Information granules play an important role in the development of neural networks. First, they substantially reduce the amount of training as the designed network needs to deal with a significantly reduced and highly compressed number of data that falls far below the size of the original training set. The same granulation mechanism delivers some highly advantageous regularization properties. Second, information granules support the design of more transparent and easily interpretable neural networks. The necessary effect of information granulation is accomplished in the framework of fuzzy sets, especially via context-sensitive (conditional) fuzzy clustering. Subsequently, the resulting neural network becomes an architecture with nonnumeric connections. A thorough analysis of results of computing carried out in the setting of linguistic neurocomputing is also given. [Copyright &y& Elsevier]

Modeling of temporal functions has been studied extensively in last few decades. Temporal functions or time series are found practically in the gamut of applications, ranging from engineering analysis to financial and commercial transactions.Numerous time series models been proposed, and they can be found in the relevant time series literature. Deterministic and stochastic models have dominated much of the research in this area. Nonetheless, these models have their limitations, and particularly specific models solve specific problem domains. Therefore, there is no general mechanism that can address the issues of modeling of temporal functions.Granular information is now being used to address a few of the problems in the theory of time series. This paper proposes two models that use the concepts of regression and interleaved information granules—intervals and fuzzy sets to identify the structure existing in a given time series. In brevity, we are attempting to combine regression and fuzzy rules to model temporal systems. [ABSTRACT FROM AUTHOR]

This study focuses on fuzzy relational calculus viewed as a basis of data compression. Images are fuzzy relations. We investigate fuzzy relational equations as a basis of image compression. It is shown that both compression and decompression (reconstruction) phases are closely linked with the way in which fuzzy relational equations are developed and solved. The theoretical findings encountered in the theory of these equations are easily accommodated as the backbone of the relational compression. The character of the solutions to the equations makes them ideal for reconstruction purposes as they specify the extremal elements of the solution set and in such a way help establish some envelopes of the original images under compression. The flexibility of the conceptual and algorithmic framework arising there is also discussed. Numerical examples provide a suitable illustrative material emphasizing the main features of the compression mechanisms. [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]

This study is concerned with a design of a face recognition algorithm realized based on feature extraction with the aid of the 2-directional 2-dimensional linear discriminant analysis referred to as (2D) 2 LDA. The 2DLDA algorithm yields high accuracy, but comes with an unresolved problem of handling large feature matrices present in face recognition problems. The proposed (2D) 2 LDA algorithm is computationally more efficient and produces more powerful discrimination decision. The proposed P-RBF NNs is used as a recognition module. The architecture of this module consists of three functional components. The coefficients of the P-RBF NNs model are obtained by fuzzy inference method forming the inference part of fuzzy rules. The essential design parameters (including learning rate, momentum, fuzzification coefficient, and the feature selection mechanism) of the networks are optimized by means of differential evolution (DE). The experimental results reported for benchmark face datasets – the Yale, ORL database, and IC&CI dataset demonstrate the effectiveness and efficiency of (2D) 2 LDA algorithm compared with other pre-processing approaches such as LPP, 2D-PCA, 2D-LPP, SR, PCA, (2D) 2 PCA and fusion of PCA-LDA. The experimental results show that (2D) 2 LDA based P-RBF NNs achieves higher performance than being reported by other methods. [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]

In this paper, we propose a new passive auto focusing algorithm to acquire images with enhanced quality through a digital camera. The focus measure operations play a key role in a passive auto focusing method. The proposed focus measure operation is based on a localized version of image variance method. The localization of the image variance focus measure operation can be implemented with the use of fuzzy partition and Fuzzy Transform. In order to validate our approach, experimental comparative studies are conducted on test images with the use of different focus measure operators. The selected operators have been chosen from an extensive review of the state-of-the-art approaches. A comparative analysis demonstrates that the proposed algorithm is superior to the conventional methods. [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]

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]

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]

Consensus is defined as a cooperative process in which a group of decision makers develops and agrees to support a decision in the best interest of the whole. It is a questioning process, more than an affirming process, in which the group members usually modify their choices until a high level of agreement within the group is achieved. Given the importance of forming an accepted decision by the entire group, the consensus problem has attained a great attention as it is a major goal in group decision making. In this study, we propose the concept of the information granularity being regarded as an important and useful asset supporting the goal to reach consensus in group decision making. By using fuzzy preference relations to represent the opinions of the decision makers, we develop a concept of a granular fuzzy preference relation where each pairwise comparison is formed as a certain information granule (say, an interval, fuzzy set, rough set, and alike) instead of a single numeric value. As being more abstract, the granular format of the preference model offers the required flexibility to increase the level of agreement within the group using the fact that we select the most suitable numeric representative of the fuzzy preference relation. [ABSTRACT FROM AUTHOR]

Abstract: Fuzzy modeling of complex systems is a challenging task, which involves important problems of dimensionality reduction and calls for various ways of improving the accuracy of modeling. The IG-FRBFNN, a hybrid architecture of the IG-FIS (Fuzzy Inference System) and FRBFNN (Fuzzy Radial Basis Function Neural Networks), is proposed to address these problems. The paper is concerned with the analysis and design of IG-FRBFNNs and their optimization by means of the Hierarchical Fair Competition-based Parallel Genetic Algorithm (HFC-PGA). In the proposed network, the membership functions of the premise part of the fuzzy rules of the IG-based FRBFNN model directly rely on the computation of the relevant distance between data points and the use of four types of polynomials such as constant, linear, quadratic and modified quadratic are considered for the consequent part of fuzzy rules. Moreover, the weighted Least Square (WLS) learning is exploited to estimate the coefficients of the polynomial forming the conclusion part of the rules. Since the performance of the IG-RBFNN model is affected by some key design parameters, such as a specific subset of input variables, the fuzzification coefficient of the FCM, the number of rules, and the order of polynomial of the consequent part of fuzzy rules, it becomes beneficial to carry out both structural as well as parametric optimization of the network. In this study, the HFC-PGA is used as a comprehensive optimization vehicle. The performance of the proposed model is illustrated by means of several representative numerical examples. [Copyright &y& Elsevier]

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]

Abstract: In this study, we introduce a concept of a granular representation of numeric membership functions of fuzzy sets, which offers a synthetic and qualitative view at fuzzy sets and their ensuing processing. The notion of consistency of the granular representation is formed, which helps regard the problem as a certain optimization task. More specifically, the consistency is referred to a certain operation , which gives rise to the concept of -consistency. Likewise introduced is a concept of granular consistency with regard to a collection of several operations, Given the essential role played by logic operators in computing with fuzzy sets, detailed investigations include and- and or-consistency as well as (and, or)-consistency of granular representations of membership functions with the logic operators implemented in the form of various t-norms and t-conorms. The optimization framework supporting the realization of the -consistent optimization process is provided through particle swarm optimization. Further conceptual and representation issues impacted processing fuzzy sets are discussed as well. [Copyright &y& Elsevier]