Search

Airway segmentation is crucial for the examination, diagnosis, and prognosis of lung diseases, while its manual delineation is unduly burdensome. To alleviate this time-consuming and potentially subjective manual procedure, researchers have proposed methods to automatically segment airways from computerized tomography (CT) images. However, some small-sized airway branches (e.g., bronchus and terminal bronchioles) significantly aggravate the difficulty of automatic segmentation by machine learning models. In particular, the variance of voxel values and the severe data imbalance in airway branches make the computational module prone to discontinuous and false-negative predictions, especially for cohorts with different lung diseases. The attention mechanism has shown the capacity to segment complex structures, while fuzzy logic can reduce the uncertainty in feature representations. Therefore, the integration of deep attention networks and fuzzy theory, given by the fuzzy attention layer, should be an escalated solution for better generalization and robustness. This article presents an efficient method for airway segmentation, comprising a novel fuzzy attention neural network (FANN) and a comprehensive loss function to enhance the spatial continuity of airway segmentation. The deep fuzzy set is formulated by a set of voxels in the feature map and a learnable Gaussian membership function. Different from the existing attention mechanism, the proposed channel-specific fuzzy attention addresses the issue of heterogeneous features in different channels. Furthermore, a novel evaluation metric is proposed to assess both the continuity and completeness of airway structures. The efficiency, generalization, and robustness of the proposed method have been proved by training on normal lung disease while testing on datasets of lung cancer, COVID-19, and pulmonary fibrosis.

A fuzzy cognitive map (FCM) is a graph-based knowledge representation model wherein the connections of the nodes (edges) represent casual relationships between the knowledge items associated with the nodes. This model has been applied to solve various modeling tasks including forecasting time series. In the original FCM-based forecasting model, causal relationships among concepts of the FCM remain unchanged. However, causal relationships may change in time. Therefore, we propose a new learning method for training an FCM resulting in an adaptive FCM which consists of several sub-FCMs. It can select different sub-FCMs at different moments. In an active processing scenario, in which we deal with a large-scale time series with new data being continuously generated, a forecasting model built on the old data should be updated when the new data arrive. Furthermore, retraining an FCM from scratch entails increasing computing overhead that will become a serious obstacle in many practical scenarios. To overcome the above-mentioned shortcomings, this study offers an original design setting in which the FCM is updated by knowledge-guidance learning mechanism for the first time. Compared with the existing classical forecasting models, the proposed model shows higher accuracy and efficiency. Its increased performance is demonstrated through a series of reported experimental studies.

In this article, we elaborate on a Kullback-Leibler (KL) divergence-based Fuzzy C -Means (FCM) algorithm by incorporating a tight wavelet frame transform and morphological reconstruction (MR). To make membership degrees of each image pixel closer to those of its neighbors, a KL divergence term on the partition matrix is introduced as a part of FCM, thus resulting in KL divergence-based FCM. To make the proposed FCM robust, a filtered term is augmented in its objective function, where MR is used for image filtering. Since tight wavelet frames provide redundant representations of images, the proposed FCM is performed in a feature space constructed by tight wavelet frame decomposition. To further improve its segmentation accuracy (SA), a segmented feature set is reconstructed by minimizing the inverse process of its objective function. Each reconstructed feature is reassigned to the closest prototype, thus modifying abnormal features produced in the reconstruction process. Moreover, a segmented image is reconstructed by using tight wavelet frame reconstruction. Finally, supporting experiments coping with synthetic, medical, and real-world images are reported. The experimental results exhibit that the proposed algorithm works well and comes with better segmentation performance than other peers. In a quantitative fashion, its average SA improvements over its peers are 4.06%, 3.94%, and 4.41%, respectively, when segmenting synthetic, medical, and real-world images. Moreover, the proposed algorithm requires less time than most of the FCM-related algorithms.

Rule-based fuzzy models play a dominant role in fuzzy modeling and come with extensive applications in the system modeling area. Due to the presence of system modeling error, it is impossible to construct a model that fits exactly the experimental evidence and, at the same time, exhibits high generalization capabilities. To alleviate these problems, in this study, we elaborate on a realization of granular outputs for rule-based fuzzy models with the aim of effectively quantifying the associated modeling errors. Through analyzing the characteristics of modeling errors, an error model is constructed to characterize deviations among the estimated outputs and the expected ones. The resulting granular model comes into play as an aggregation of the regression model and the error model. Information granularity plays a central role in the construction of granular outputs (intervals). The quality of the produced interval estimates is quantified in terms of the coverage and specificity criteria. The optimal allocation of information granularity is determined through a combined index involving these two criteria pertinent to the evaluation of interval outputs. A series of experimental studies is provided to demonstrate the effectiveness of the proposed approach and show its superiority over the traditional statistical-based method.

Fuzzy rule-based models (FRBMs) are sound constructs to describe complex systems. However, in reality, we may encounter situations, where the user or owner of a system only owns either the input or output data of that system (the other part could be owned by another user); and due to the consideration of data privacy, he/she could not obtain all the needed data to build the FRBMs. Since this type of situation has not been fully realized (noticed) and studied before, our objective is to come up with some strategy to address this challenge to meet the specific privacy consideration during the modeling process. In this study, the concept and algorithm of the collaborative fuzzy clustering (CFC) are applied to the identification of FRBMs, describing either multiple-input-single-output (MISO) or multiple-input-multiple-output (MIMO) systems. The collaboration between input and output spaces based on their structural information (conveyed in terms of the corresponding partition matrices) makes it possible to build FRBMs when input and output data could not be collected and used in unison. Surprisingly, on top of this primary pursuit, with the collaboration mechanism the input and output spaces of a system are endowed with an innovative way to comprehensively share, exchange, and utilize the structural information between each other, which results in their more relevant structures that guarantee better model performance compared with performance produced by some state-of-the-art modeling strategies. The effectiveness of the proposed approach is demonstrated by experiments on a series of synthetic and publicly available datasets.

Incomplete data are frequently encountered and bring difficulties when it comes to further processing. The concepts of granular computing (GrC) help deliver a higher level of abstraction to address this problem. Most of the existing data imputation and related modeling methods are of numeric nature and require prior numeric models to be provided. The underlying objective of this study is to introduce a novel and straightforward approach that uses information granules as a vehicle to effectively represent missing data and build granular fuzzy models directly from resulting hybrid granular and numeric data. The evaluation and optimization of this method are guided by the principle of justifiable granularity engaging the coverage and specificity criteria and carried out with the help of particle swarm optimization. We provide a collection of experimental studies using a synthetic dataset and several publicly available real-world datasets to demonstrate the feasibility and analyze the main features of this method.

Fuzzy-rough cognitive networks (FRCNs) are recurrent neural networks (RNNs) intended for structured classification purposes in which the problem is described by an explicit set of features. The advantage of this granular neural system relies on its transparency and simplicity while being competitive to state-of-the-art classifiers. Despite their relative empirical success in terms of prediction rates, there are limited studies on FRCNs' dynamic properties and how their building blocks contribute to the algorithm's performance. In this article, we theoretically study these issues and conclude that boundary and negative neurons always converge to a unique fixed-point attractor. Moreover, we demonstrate that negative neurons have no impact on the algorithm's performance and that the ranking of positive neurons is invariant. Moved by our theoretical findings, we propose two simpler fuzzy-rough classifiers that overcome the detected issues and maintain the competitive prediction rates of this classifier. Toward the end, we present a case study concerned with image classification, in which a convolutional neural network is coupled with one of the simpler models derived from the theoretical analysis of the FRCN model. The numerical simulations suggest that once the features have been extracted, our granular neural system performs as well as other RNNs.

A granular structure of intuitionistic fuzzy (IF) information presents simultaneously the similarity and diversity of samples. However, this structural representation has rarely displayed its technical capability in data mining and information processing due to the lack of suitable constructive methods and semantic interpretation for IF information with regard to real data. To pursue better performance of the IF-based technique in real-world data mining, in this article, we examine information granularity, information entropy of IF granular structures, and their applications to data reduction of IF information systems. First, several types of partial-order relations at different hierarchical levels are defined to reveal the granularity of IF granular structures. Second, the granularity invariance between different IF granular structures is characterized by using relational mappings. Third, Shannon's entropies are generalized to IF entropies and their relationships with the partial-order relations are addressed. Based on the theoretical analysis above, the significance of intuitionistic attributes using the information measures is then introduced and the information-preserving algorithm for data reduction of IF information systems is constructed. Finally, by inducing substantial IF relations from public datasets that take both the similarity/diversity between the samples from the same/different classes into account, a collection of numerical experiments is conducted to confirm the performance of the proposed technique.

To address the situation where the complete consistency is unnecessary, a stepwise optimization model-based method for testing the acceptably additive consistency (AAC) of hesitant fuzzy preference relations (HFPRs) is introduced. Then, an AAC concept for HFPRs is defined. Meanwhile, incomplete HFPRs (iHFPRs) are discussed and a series of optimization models to acquire complete HFPRs is constructed. If the consistency is unacceptable, an optimization model for revising unacceptably consistent HFPRs under the conditions of the AAC and maximizing the ordinal consistency (OC) is offered. Subsequently, a model for minimizing the number of adjusted variables is presented. Considering the weighting information and the consensus for group decision making (GDM), the weights of fuzzy preference relations (FPRs) obtained from each individual HFPR and the decision makers (DMs) are determined using the distance measure. With regard to the consensus, two models for reaching the consensus requirement and minimizing the amount of revised variables are separately constructed, which are both based on the analysis of maximizing the OC. Furthermore, the thresholds of the additive consistency and the consensus are studied using the Monte Carlo simulation method. A GDM algorithm with HFPRs is offered. Finally, an example and comparison are provided to show the efficiency of the new procedure.

Brace treatment is the most commonly used nonsurgical treatment for adolescents with idiopathic scoliosis. However, brace treatment is not always successful and the factors influencing its success are not completely clear. This makes treatment outcome difficult to predict. A computer model which can accurately predict treatment outcomes could potentially provide valuable treatment recommendations. This paper describes a fuzzy system that includes a prediction model and a decision support engine. The model was constructed using conditional fuzzy c-means clustering to discover patterns in retrospective patient data. The model's ability to predict treatment outcome was compared to the ability of eight Scoliosis experts. The model and experts each predicted treatment outcome retrospectively for 28 braced patients, and these predictions were compared to the actual outcomes. The model outperformed all but one expert individually and performed similarly to the experts as a group. These results suggest that the fuzzy model is capable of providing meaningful treatment recommendations. This study offers the first model for this application whose performance has been shown to be at or above the human expert level.

In the context of robust optimization with information granules for distributional parameters, this paper investigates a two-stage waste-to-energy feedstock flow planning problem with uncertain capacity expansion costs. The objective is to minimize the worst-case overall loss in a mean-risk criterion where the risk is measured by a conditional value-at-risk operator. As a salient feature, an integrated uncertainty is considered which consists of not only the uncertainty in distribution shapes of the uncertain variables, but also the manifold uncertainties of the mean parameters. To tackle the robust optimization under such integrated uncertainty, we first discuss a distributional robust two-stage feedstock flow planning model with precise mean parameters that handles the uncertainty in distribution shape, and the model can be equivalently transformed into a linear program (LP). Furthermore, the precise-mean-based robust model is extended into the case of multifaceted uncertainty for mean-parameters that are allowed to assume intervals, historical-data-based probabilistic estimates, and/or human-knowledge-centric fuzzy set estimates, under different circumstances. These multifaceted uncertain mean-parameters are uniformly represented by using information granules, and a granular robust optimization model is then developed which maximizes the robustness of the solution within a shortfall tolerance, and realizes a tradeoff between the solution conservativeness and robustness. It is showed that the granular robust model is equivalent to solving a series of LPs and can be efficiently handled by a nested binary search algorithm. Finally, the computational study illustrates the model performance, solution analysis, and underlines a much higher scalability of the developed robust model compared to the stochastic programming approach.

The study introduces a concept of mappings realized in presence of information granules and offers a design framework supporting the formation of such mappings. Information granules are conceptually meaningful entities formed on a basis of a large number of experimental input–output numeric data available for the construction of the model. We develop a conceptually and algorithmically sound way of forming information granules. Considering the directional nature of the mapping to be formed, this directionality aspect needs to be taken into account when developing information granules. The property of directionality implies that while the information granules in the input space could be constructed with a great deal of flexibility, the information granules formed in the output space have to inherently relate to those built in the input space. The input space is granulated by running a clustering algorithm; for illustrative purposes, the focus here is on fuzzy clustering realized with the aid of the fuzzy C-means algorithm. The information granules in the output space are constructed with the aid of the principle of justifiable granularity (being one of the underlying fundamental conceptual pursuits of Granular Computing). The construct exhibits two important features. First, the constructed information granules are formed in the presence of information granules already constructed in the input space (and this realization is reflective of the direction of the mapping from the input to the output space). Second, the principle of justifiable granularity does not confine the realization of information granules to a single formalism such as fuzzy sets but helps form the granules expressed any required formalism of information granulation. The quality of the granular mapping (viz. the mapping realized for the information granules formed in the input and output spaces) is expressed in terms of the coverage criterion (articulating how well the experimental data are “covered” by information granules produced by the granular mapping for any input experimental data). Some parametric studies are reported by quantifying the performance of the granular mapping (expressed in terms of the coverage and specificity criteria) versus the values of a certain parameters utilized in the construction of output information granules through the principle of justifiable granularity. The plots of coverage–specificity dependency help determine a knee point and reach a sound compromise between these two conflicting requirements imposed on the quality of the granular mapping. Furthermore, quantified is the quality of the mapping with regard to the number of information granules (implying a certain granularity of the mapping). A series of experiments is reported as well.

Fuzzy cognitive maps (FCMs) are convenient and widely used architectures for modeling dynamic systems, which are characterized by a great deal of flexibility and adaptability. Several recent works in this area concern strategies for the development of FCMs. Although a few fully automated algorithms to learn these models from data have been introduced, the resulting FCMs are structurally considerably different than those developed by human experts. In particular, maps that were learned from data are much denser (with the density over 90% versus about 40% density of maps developed by humans). The sparseness of the maps is associated with their interpretability: the smaller the number of connections is, the higher is the transparency of the map. To this end, a novel learning approach, sparse real-coded genetic algorithms (SRCGAs), to learn FCMs is proposed. The method utilizes a density parameter to guide the learning toward a formation of maps of a certain predefined density. Comparative tests carried out for both synthetic and real-world data demonstrate that, given a suitable density estimate, the SRCGA method significantly outperforms other state-of-the-art learning methods. When the density estimate is unknown, the new method can be used in an automated fashion using a default value, and it is still able to produce models whose performance exceeds or is equal to the performance of the models generated by other methods.

Clustering forms one of the most visible conceptual and algorithmic framework of developing information granules. In spite of the algorithm being used, the representation of information granules-clusters is predominantly numeric (coming in the form of prototypes, partition matrices, dendrograms, etc.). In this paper, we consider a concept of granular prototypes that generalizes the numeric representation of the clusters and, in this way, helps capture more details about the data structure. By invoking the granulation-degranulation scheme, we design granular prototypes being reflective of the structure of data to a higher extent than the representation that is provided by their numeric counterparts (prototypes). The design is formulated as an optimization problem, which is guided by the coverage criterion, meaning that we maximize the number of data for which their granular realization includes the original data. The granularity of the prototypes themselves is treated as an important design asset; hence, its allocation to the individual prototypes is optimized so that the coverage criterion becomes maximized. With this regard, several schemes of optimal allocation of information granularity are investigated, where interval-valued prototypes are formed around the already produced numeric representatives. Experimental studies are provided in which the design of granular prototypes of interval format is discussed and characterized.

Linguistic interface is a group of linguistic terms or fuzzy descriptions that describe variables in a system utilizing corresponding membership functions. Its transparency completely or partly decides the interpretability of fuzzy models. This paper proposes a GRadiEnt-descEnt-based Transparent lInguistic iNterface Generation (GREETING) approach to overcome the disadvantage of traditional linguistic interface generation methods where the consideration of the interpretability aspects of linguistic interface is limited. In GREETING, the widely used interpretability criteria of linguistic interface are considered and optimized. The numeric experiments on the data sets from University of California, Irvine (UCI) machine learning databases demonstrate the feasibility and superiority of the proposed GREETING method. The GREETING method is also applied to fuzzy decision tree generation. It is shown that GREETING generates better transparent fuzzy decision trees in terms of better classification rates and comparable tree sizes.

In this paper, we develop a general framework of a granular representation of ECG signals. The crux of the approach lies in the development and ongoing processing realized in the setting of information granules-fuzzy sets. They serve as basic conceptual and semantically meaningful entities using which we describe signals and build their models (such as various predictive schemes or classifiers). A comprehensive two-phase scheme of the design of the information granules is proposed and described. At the first phase, we discuss the temporal granulation through a series of temporal windows (granular windows) and an aggregation of the values of signal by means of fuzzy sets. To address this issue, offered is a detailed method of building a fuzzy set based on numeric data and a certain optimization criterion that strikes a balance between the highest experimental relevance of the fuzzy set supported by numeric data and its substantial specificity. At the next phase of the granular design, a collection of information granules is further summarized with the use of fuzzy clustering (Fuzzy C-Means). The resulting prototypes (centroids) formed by this grouping process serve as elements of the granular vocabulary. We discuss ways of using these vocabularies in the knowledge-based representation, modeling, and classification of ECG beats.

In this study, we introduce a novel clustering architecture, in which several subsets of patterns can be processed together with an objective of finding a common structure. The structure revealed at the global level is determined by exchanging prototypes of the subsets of data and by moving prototypes of the corresponding clusters toward each other. Thereby, the required communication links are established at the level of cluster prototypes and partition matrices, without hampering the security concerns. A detailed clustering algorithm is developed by integrating the advantages of both fuzzy sets and rough sets, and a measure of quantitative analysis of the experimental results is provided for synthetic and real-world data.

In this paper, we are concerned with the concept of fuzzy logic networks and logic-based data analysis realized within this framework. The networks under discussion are homogeneous architectures comprising of OR/AND neurons originally introduced by Hirota and Pedrycz. Being treated here as generic processing units, OR/AND neurons are neurofuzzy constructs that exhibit well-defined logic characteristics and are endowed with a high level of parametric flexibility and come with significant interpretation abilities. The composite logic nature of the logic neurons becomes instrumental in covering a broad spectrum of logic dependencies whose character spread in-between between those being captured by plain and and or logic descriptors (connectives). From the functional standpoint, the developed network realizes a logic approximation of multidimensional mappings between unit hypercubes, that is transformations from [0, 1]n to [0, 1]m. The way in which the structure of the network has been formed is highly modular and becomes reflective of a general concept of decomposition of logic expressions and Boolean functions (as being commonly encountered in two-valued logic). In essence, given a collection of input variables, selected is their subset and transformed into new composite variable, which in turn is used in the consecutive module of the network. These intermediate synthetic variables are the result of the successive problem (mapping) decomposition. The development of the network is realized through genetic optimization. This helps address important issues of structural optimization (where we are concerned with a selection of a subset of variables and their allocation within the network) and reaching a global minimum when carrying out an extensive parametric optimization (adjustments of the connections of the neurons). The paper offers a comprehensive and user-interactive design procedure including a simple pruning mechanism whose intention is to enhance the interpretability of the network while reducing its size. The experimental studies comprise of three parts. First, we demonstrate the performance of the network on Boolean data (that leads to some useful comparative observations considering a wealth of optimization tools available in two-valued logic and digital systems). Second, we discuss synthetic multivalued data that helps focus on the approximation abilities of the network. Finally, show the generation of logic expressions describing selected data sets coming from the machine learning repository.

In this paper, we develop an efficient fuzzy wavelet packet (WP) based feature extraction method for the classification of high-dimensional biomedical data such as magnetic resonance spectra. The key design phases involve: 1) a WP transformation mapping the original signals to many WP feature spaces and finding optimal WP decomposition for signal classification; 2) feature extraction based on the optimal WP decomposition; and 3) signal classification realized by a linear classifier. In contrast to the standard method of feature extraction used in WPs, guided by the criteria of signal compression or signal energy, our method is used to extract discriminatory features from the WP coefficients of the optimal decomposition. The extraction algorithm constructs fuzzy sets of features (via fuzzy clustering) to assess their discriminatory effectiveness. This paper includes a number of numerical experiments using magnetic resonance spectra. Classification results are compared with those obtained from common feature extraction methods in the WP domain.

The recent trend in the development of neurofuzzy systems has profoundly emphasized the importance of synergy between the fundamentals of fuzzy sets and neural networks. The resulting frameworks of the neurofuzzy systems took advantage of an array of learning mechanisms primarily originating within the theory of neurocomputing and the use of fuzzy models (predominantly rule-based systems) being well established in the realm of fuzzy sets. Ideally, one can anticipate that neurofuzzy systems should fully exploit the linkages between these two technologies while strongly preserving their evident identities (plasticity or learning abilities to be shared by the transparency and full interpretability of the resulting neurofuzzy constructs). Interestingly, this synergy still becomes a target yet to be satisfied. This study is an attempt to address the fundamental interpretability challenge of neurofuzzy systems. Our underlying conjecture is that the transparency of any neurofuzzy system links directly with the logic fabric of the system so the logic fundamentals of the underlying architecture become of primordial relevance. Having this in mind the development of neurofuzzy models hinges on a collection of logic driven processing units named here fuzzy (logic) neurons. These are conceptually simple logic-oriented elements that come with a well-defined semantics and plasticity. Owing to their diversity, such neurons form essential building blocks of the networks. The study revisits the existing categories of logic neurons, provides with their taxonomy, helps understand their functional features and sheds light on their behavior when being treated as computational components of any neurofuzzy architecture. The two main categories of aggregative and reference neurons are deeply rooted in the fundamental operations encountered in the technology of fuzzy sets (including logic operations, linguistic modifiers, and logic reference operations). The developed heterogeneous networks come with a well-defined semantics and high interpretability (which directly translates into the rule-based representation of the networks). As the network takes advantage of various logic neurons, this imposes an immediate requirement of structural optimization, which in this study is addressed by utilizing various mechanisms of genetic optimization (genetic algorithms). We discuss the development of the networks, elaborate on the interpretation aspects and include a number of illustrative numeric examples.

In this paper, we develop a method for recognizing face images by combining wavelet decomposition, Fisherface method, and fuzzy integral. The proposed approach is comprised of four main stages. The first stage uses the wavelet decomposition that helps extract intrinsic features of face images. As a result of this decomposition, we obtain four subimages (namely approximation, horizontal, vertical, and diagonal detailed images). The second stage of the approach concerns the application of the Fisherface method to these four decompositions. The choice of the Fisherface method in this setting is motivated by its insensitivity to large variation in light direction, face pose, and facial expression. The two last phases are concerned with the aggregation of the individual classifiers by means of the fuzzy integral. Both Sugeno and Choquet type of fuzzy integral are considered as the aggregation method. In the experiments we use n-fold cross-validation to assure high consistency of the produced classification outcomes. The experimental results obtained for the Chungbuk National University (CNU) and Yale University face databases reveal that the approach presented in this paper yields better classification performance in comparison to the results obtained by other classifiers.

In this study, we introduce and discuss a development of a highly interactive and user-friendly environment for an ECG signal analysis. The underlying neural architecture being a crux of this environment comes in the form of a self-organizing map. This map helps discover a structure in a set of ECG patterns and visualize a topology of the data. The role of the designer is to choose from some already visualized regions of the self-organizing map characterized by a significant level of data homogeneity and substantial difference from other regions. In the sequel, the regions are described by means of information granules-fuzzy sets that are essential in the characterization of the main relationships existing in the ECG data. The study introduces an original method of constructing membership functions that incorporates class membership as an important factor affecting changes in membership grades. The study includes a comprehensive descriptive modeling of highly dimensional ECG data.

In this article, we introduce a variant of the adaptive network-based fuzzy inference system (ANFIS). The proposed variant does not use backpropagation and grid partitioning, but the least-squares method with fractional Tikhonov regularization. The fractional regularization is a generalization of the standard regularization and is applied here to the learning process of the ANFIS scheme for the first time. This results in a simpler rule base, with a low number of rules, allowing to handle problems with many input variables with relatively low computational time while keeping high accuracy. We present new theoretical results on the fractional Tikhonov regularization. Such results are the basis for a formal discussion on how much the choice of a different architecture, resulting in a different matrix in the least-squares minimization, could affect the accuracy. We perform several numerical experiments on benchmark examples, first to assess the impact of the fractional regularization on the accuracy and then to compare our results against the most recent ones reported in the literature by other ANFIS-like or neuro-fuzzy systems. The numerical results show the good performance of the proposed approach. [ABSTRACT FROM AUTHOR]

Although the idea of information granulation has been shown to be a research craze in short-term time series forecasting, it is still urgent to develop a granular framework so that information granulation can characterize the trend distribution of data to the significant extent under a common concept of time. This article puts forward a novel granulation algorithm involving two-stage partitioning scheme so that information granule established there exhibits well-articulated semantics at the time level, while at the same time, it gives full consideration to the varying patterns of data. On this basis, a new association rule based on this type of information granules is presented. Unlike most fuzzy association rules, the proposed association rules can extract and derive the correlations between two collections of trend features corresponding to the past and future, which is in accord with human's reasoning. Keeping in mind that the prediction process should center on essential rules while freeing from the interference of irrelevant rules, which contributes to a reliable prediction result, thus, a rule selection algorithm is involved so as one makes sure the accuracy and interpretability of the forecasting results. The design of short-term forecasting model based on fuzzy inference system is implemented, where the concept of granulation eliminates the commonly used alternative, i.e., the recursive iterations of one-step prediction. Experimental results have verified the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

The notion of a rough set that was originally proposed by Pawlak underwent a number of extensions and generalizations. Dubois and Prade [4] introduced fuzzy rough sets that involve the use of rough sets and fuzzy sets within a single unified framework. Radzikowska and Kerre [5] proposed a broad family of fuzzy rough sets, referred to as ([phi], t)-fuzzy rough sets, which are determined by some implication operator (implicator) [phi] and a certain t-norm. In order to describe the linguistically represented concepts coming from data available in some information system, the concept of fuzzy rough sets are redefined and further studied in the setting of the Axiomatic Fuzzy Set (AFS) theory. Compared with the ([phi], t)-fuzzy rough sets, the advantages of AFS fuzzy rough sets are twofold. They can be directly applied to the data analysis present in any information system without resorting to the details concerning the choice of the implication [phi], t-norm, and a similarity relation S. Furthermore, such rough approximations of fuzzy concepts come with a well-defined semantics and therefore offer a sound interpretation. Some examples are included to illustrate the effectiveness of the proposed construct. It is demonstrated that the AFS fuzzy rough sets provide a far higher flexibility and effectiveness in comparison with rough sets and some of their generalizations. Index Terms--Rough sets, fuzzy rough sets, AFS algebras, AFS structures.

Multiagent systems are inherently associated with their distributivity, which enforces a great deal of communication mechanisms. To effectively arrive at meaningful solutions in a vast array of problem-solving tasks, it becomes imperative to establish a sound machinery of reconciling findings which might form partial solutions to an overall problem. In this paper, we focus on a broad category of problems of collaborative data analysis realized by a collection of agents having access to their individual data and exchanging findings through their collaboration activities. Such problems of data analysis arise in the context of building a global view at a certain phenomenon (process) by viewing it from different perspectives (and thus engaging various collections of attributes by various agents). Our goal is to develop some interaction between the agents so that they could form an overall perspective, where the knowledge available locally is shared and reconciled. The underlying format of knowledge built by the agents is that of information granules and fuzzy sets in particular. We develop a comprehensive optimization scheme and discuss its two-phase nature in which the communication phase of the granular findings intertwines with the local optimization being realized by the agents at the level of the individual datasite and exploits the evidence collected from other sites. We show how the mechanism of fuzzy granulation realized in the form of a well-known fuzzy c-means (FCM) clustering can be augmented to support collaborative activities required by the agents. For this purpose, we introduce augmented versions of the original objective function used in the FCM and derive algorithmic details. We also discuss an issue of optimizing the strength of collaborative linkages, so that the reconciled findings attain the highest level of consistency (agreement). The presented experimental studies include some synthetic data and selected data sets coming from the Machine Learning repository. Index Terms--Collaborative agents, data analysis, fuzzy clustering, information granules.

• The work presents a comprehensive review and discussion of logic-oriented fuzzy AND\OR network. • The architecture exhibits outstanding learning and interpretation capabilities. • Logic-oriented neurons are endowed with clearly delineated logic structure. • Interpretability is the main advantage of the neurons that make the model linguistically readable. • Knowledge is extracted in the form that is easily comprehended by humans. Data analysis and their thorough interpretation have posed a substantial challenge in the era of big data due to increasingly complex data structures and their sheer volumes. The black-box nature of neural networks may omit important information about why certain predictions have been made which makes it difficult to ground the reliability of a prediction despite tremendous successes of machine learning models. Therefore, the need for reliable decision-making processes stresses the significance of interpretable models that eliminate uncertainty, supporting explainability while maintaining high generalization capabilities. Logic-oriented fuzzy neural networks are capable to cope with a fundamental challenge of fuzzy system modeling. They strike a sound balance between accuracy and interpretability because of the underlying features of the network components and their logic-oriented characteristics. In this survey, we conduct a comprehensive review of logic-oriented fuzzy neural networks with a special attention being directed to AND\OR architecture. The architectures under review have shown promising results, as reported in the literature, especially when extracting useful knowledge through building experimentally justifiable models. Those models show balance between accuracy and interpretability because of the prefect integration between the merits of neural networks and fuzzy logic which has led to reliable decision-making processes. The survey discusses logic-oriented networks from different perspectives and mainly focuses on the augmentation of interpretation through vast array of learning abilities. This work is significantly important due to the lack to similar survey in the literature that discusses this particular architecture in depth. Finally, we stress that the architecture could offer a novel promising processing environment if they are integrated with other fuzzy tools which we have discussed thoroughly in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new topology of fuzzy-neural networks--fuzzy-set-based polynomial neural networks (FSPNNs). The two underlying design mechanisms of such networks involve genetic optimization and information granulation (IG). The resulting constructs come in the form of fuzzy polynomial neural networks with fuzzy-set-based polynomial neurons, regarded as their generic processing elements. First, we introduce a comprehensive design methodology using which we determine the optimal structure of the FSPNNs. This methodology hinges on the extended group method of data handling and fuzzy-set-based rules. It concerns the optimization of the FSPNN-related parameters such as the number of input variables, the order of the polynomial, the number of membership functions, and a collection of a specific subset of input variables realized through the mechanism of genetic optimization. Second, the fuzzy rules used in the networks exploit a notion of information granules defined over a system's variables and formed through the process of IG. This granulation is realized with the aid of the hard C-means clustering algorithm. The performance of the network is quantified through experimentation in which we use a number of modeling benchmarks already experimented with in the realm of fuzzy or neurofuzzy modeling. Index Terms--Fuzzy-set-based polynomial neural networks (FSPNNs), fuzzy-set-based polynomial neuron (FSPN), genetic algorithms (GAs), group method of data handling (GMDH), hard C-means (HCM) clustering, information granulation (IG).

In this paper, the fundamental idea of linguistic models introduced by Pedrycz and Vasilakos (1999) is followed and their comprehensive design framework is developed. The paradigm of linguistic modeling is concerned with constructing models that: 1) are user centric and 2) inherently dwell upon collections of highly interpretable and user-oriented entities such as information granules. The objective of this paper is to investigate and compare alternative design options, present an organization of the overall optimization process, and come up with a specification of several evaluation mechanisms of the performance of the models. The underlying design tool guiding the development of linguistic models revolves around the augmented version of fuzzy clustering known as a context-based or conditional fuzzy C-means (C-FCM). The design process comprises several main phases such as: 1) defining and further refining context fuzzy sets; 2) completing conditional fuzzy clustering; and 3) optimizing parameters (connections) linking information granules in the input and output spaces. An iterative process of forming information granules in the input and output spaces is discussed. Their membership functions are adjusted by the gradient-based learning guided by the minimization of some performance index. The paper comes with a comprehensive suite of experiments that lead to some design guidelines of the models. Furthermore, the performance of linguistic models is contrasted with that of other fuzzy models, especially radial basis function neural networks (RBFNNs) and related constructs that are based on concepts of fuzzy clustering. Index Terms--Context-based fuzzy clustering, fuzzy contexts, information granules, linguistic models, performance indexes, user-centric modeling.

In this paper, we are concerned with the granular representation of mappings (or experimental data) coming in the form R : R [right arrow] [0, 1] (for one-dimensional cases) and R : [R.sup.n] [right arrow] [0, 1] (for multivariable cases) with R being a set of real numbers. As the name implies, a granular mapping is defined over information granules and maps them into a collection of granules expressed in some output space. The design of the granular mapping is discussed in the case of set and fuzzy set-based granulation. The proposed development is regarded as a two-phase process that comprises: 1) a definition of an interaction between information granules and experimental evidence or existing numeric mapping and 2) the use of these measures of interaction in building an explicit expression for the granular mapping. We show how to develop information granules in case of multidimensional numeric data by resorting to fuzzy clustering (fuzzy C-means). Experimental results serve as an illustration of the proposed approach. Index Terms--Fuzzy clustering, fuzzy relational equations, granular computing, information granules, possibility and necessity measures, specificity of granular information.

This paper contributes to the conceptual and algorithmic framework of information granulation. We revisit the role of information granules that are relevant to several main classes of technical pursuits involving temporal and spatial granulation. A detailed algorithm of information granulation, regarded as an optimization problem reconciling two conflicting design criteria, namely, a specificity of information granules and their experimental relevance (coverage of numeric data), is provided in the paper. The resulting information granules are formalized in the language of set theory (interval analysis). The uniform treatment of data points and data intervals (sets) allows for a recursive application of the algorithm. We assess the quality of information granules through the application of fuzzy c-means (FCM) clustering algorithm. Numerical studies deal with two-dimensional (2-D) synthetic data and experimental traffic data. Index Terms--Complex systems, data mining-oriented time-series analysis, fuzzy sets, granular clustering, information granules and granulation, interval analysis, perception, time-series, traffic data.

Compensatory Fuzzy Logic is a transdisciplinary axiomatic theory, different from the Classical Norm and Conorm approach to improving interpretability by natural language. Archimedean Compensatory Fuzzy Logic (ACFL), introduced recently, uses different properties and interpretations of involved truth values. Membership functions involved are not studied explicitly in fuzzy theories, even though it is essential in solving problems. The definition of parameterized families of membership functions is not rare in fuzzy literature. However, according to our review, each of those families has the same shape except the recently introduced Continuous Linguistic Variables. That has been a limitation in the expressiveness of linguistic values. Besides, except for Dombi's theory, these functions are often not related to logical operators. This paper aims to use ACFL to overcome each of these drawbacks. We generalize some fuzzy concepts, only using the ACFL generator function. A Generalized Sigmoidal Function and a Generalized Linguistic Modifier are s-shaped functions generated by it. Those elements define a parameterized family containing different shape functions like an increasing sigmoidal, decreasing sigmoidal and convex function; we call it a Generalized Continuous Linguistic Variable. This paper improves ACFL by unifying it into single theory elements like logic generator functions, linguistic modifiers, membership functions, and linguistic variables. The improved ACFL is not just a Pluralist Logic that makes compatible the classical approach of Norm and Conorm with CFL theory, but a contextual pluralist logic able to select a logic that better expresses specific contextual knowledge. This theory is valuable in Knowledge Discovery; because it creates new searching elements that allow selecting the 'best logic´ for a particular dataset. We develop knowledge discovery cases for different databases to illustrate it and show its data sensitivity. [ABSTRACT FROM AUTHOR]

In this study, we elaborate on an important issue of membership function determination. The main point is that any membership estimation procedure should reconcile the semantics of a fuzzy set (regarded as an information granule arising at some level of information abstraction) with the experimental evidence conveyed by numeric data. This, in the sequel, calls for the development of the hybrid two-phase approach that starts from a rough specification of the support of the fuzzy set that is followed by detailed computations involving a specific type of membership function and an estimation of its parameters. The role of robust statistics in this setting is also raised. A number of experimental results are discussed. Index Terms--Energy and entropy measure of fuzziness, focal region, fuzzy sets, median, membership functions, robust estimator, triangular and parabolic membership functions.

In this study, we are concerned with a two-objective development of information granules completed on a basis of numeric data. The first goal of this design concerns revealing and representing a structure in a data set. As such, it is oriented toward coping with the underlying relational aspects of the experimental data. The second goal deals with the formation of a mapping between information granules constructed in two spaces (thus, it concentrates on the directional aspect of information granulation). The quality of the mapping is directly affected by the information granules over which it operates, so in essence, we are interested in the granules that not only reflect the data, but also contribute to the performance of such mapping. The optimization of information granules is realized through a collaboration occurring at the level of data and the mapping between the data sets. The operational facet of the problem is cast in the realm of fuzzy clustering. As the standard techniques of fuzzy clustering (including a well-known approach of FCM) are aimed exclusively at the first objective identified above, we augment them in order to accomplish sound mapping properties between the granules. This leads to a generalized version of the FCM (and any other clustering technique for this matter). We propose a generalized version of the objective function that includes an additional collaboration component to make the formed information granules in rapport with the mapping requirements (that comes with a directional component captured by the information granules). The additive form of the objective function with a modifiable component of collaborative activities makes it possible to express a suitable level of collaboration and avoid a phenomenon of potential competition in the case of incompatible structures and the associated mapping. The logic-based type of the mapping (that invokes the use of fuzzy relational equations) comes as a consequence of the logic framework of information granules. A complete optimization method is provided and illustrated with several numeral studies. Index Terms--Clustering, collaboration and competition, Fuzzy C-Means, fuzzy models, fuzzy relational equations, granular computing, granular modeling, models of collaborative computing.

In this article, we introduce novel reinforced hybrid fuzzy rule-based neural networks (RHFNNs). This article is concerned with the development of the design methodologies of hybrid fuzzy rule-based model for constructing the network structure and enhancing its predictive abilities through the combination of inhomogeneous neurons [i.e., clustering-based polynomial neurons (CPNs) and polynomial neurons (PNs)] and tournament selection. The key points of the proposed RHFNN are enumerated as follows: The first layer of the proposed network consists of CPNs. CPN can effectively reflect the complex nonlinear structure encountered in the data space, and refine (granulate) it with the help of the clustering algorithm. Two types of CPNs including hard C-means (HCM) clustering-based polynomial neuron (HCPN) and fuzzy C-means (FCM) clustering-based polynomial neuron (FCPN) are designed. According to the type of CPN used in the first layer, RHFNN can be categorized into two types, namely, RHFNN based on HCPN (HRHFNN) and RHFNN based on FCPN (FRHFNN). We use PNs to construct the second and consecutive layers. PN can identify and approximate the nonlinear relationship among system's inputs and outputs. A tournament-based performance selection (TPS) algorithm stemming from evolutionary computation is used for selection of neuron. TPS not only ensures that the candidate nodes have sufficient fitting ability but also enhances the individual diversity in the node set and provides the abilities to generate better prediction nodes. In addition, L2-norm regularization is considered to reduce the deviation between coefficients and ameliorate overfitting as well as boost generalization ability. The performance of RHFNN is discussed through a variety of publicly available machine learning datasets. From the experimental results, we conclude that RHFNN achieves the best prediction accuracy on 13 of 15 datasets; the statistical analysis also confirms the superiority of RHFNN. [ABSTRACT FROM AUTHOR]

This article is concerned with the development of fuzzy models realized with the aid of genetic programming (GP). The proposed architecture employs GP to form fuzzy logic expressions involving logic operators and information granules (fuzzy sets) located in the input space, used to predict information granules located in the output space. We propose an architecture realizing logic processing, with the structural optimization of the model accomplished by a multitree genetic programming and the parametric optimization completed by gradient-based learning. The granulation of information used in this architecture is developed using the Fuzzy C-means clustering algorithm. The novelty of this study is two-fold: 1) it comes with the flexibility of the logic-oriented structure of fuzzy models, and 2) our architecture is designed to handle high-dimensional data by alleviating the detrimental effect of distance concentration hampering the effectiveness of standard Takagi–Sugeno–Kang fuzzy rule-based models. The article is illustrated through some experiments that provide a detailed insight into the performance of the fuzzy models. A comprehensive comparative analysis is also covered. [ABSTRACT FROM AUTHOR]

Granular data streams (GDSs) are a class of high-level abstract multitime scale description of data streams. Prediction intervals (PIs) for GDSs that provide estimated values as well as their corresponding reliability play an important role for assisting on-site workers to perceive the nonstationary environment in real time. However, constructing reliable PIs for GDSs constitutes a significant challenge. To provide a solution to the problem, an interval type-2 (IT2) fuzzy granular neural network (FGNN) dynamic ensemble approach (IT2FGNNDEnsemble) is proposed in this article. To fully reflect the uncertainty of GDSs, an interval value learning algorithm based IT2FGNN is developed, which can automatically generate, prune, merge, and realize recall in a single-pass learning mode. In addition, an evolving dynamic ensemble method is presented by providing an adaptive structure that considers a tradeoff between coverage and width of PIs, which can dynamically generate and prune the element of an ensemble according to current data tendency. A number of synthetic and industrial data streams experimentally validate the performance of the proposed IT2FGNNDEnsemble by using the state-of-the-art comparative methods. It is demonstrated that the proposed approach exhibits a good performance on PIs for practical applications. [ABSTRACT FROM AUTHOR]

Neural networks (NNs) with least square error (LSE) estimation form a certain type of single hidden layer feed-forward NNs. In this class of networks, the input connections (weights) and the biases of hidden neurons are generated randomly and fixed after being generated. The output connections are estimated by the LSE method rather than the back-propagation method. The random generation of the input connection weights and the hidden biases results in the larger number of hidden neurons to assure the quality of classification performance. To reduce the number of neurons in the hidden layer while maintaining the classification performance, we apply a “divide and conquer” strategy in this article. In other words, we divide an overall input space into several subspaces by using information granulation technique (Fuzzy C-Means clustering algorithm) and determine the local decision boundaries among related subspaces. A decision boundary defined in the input space can be considered as being composed of several decision boundaries defined in subspaces that form the entire input space. For the decision boundaries defined in the subspaces, their nonlinearity becomes lower in comparison with the one being encountered when considering the entire input space. Through the weighted LSE estimation instead of using the LSE estimation method, the connections of several NNs can be estimated without interfering with each other. After estimating the weights, the decision boundaries defined in the related subspaces are merged to a single decision boundary by using fuzzy ensemble technique. Several machine learning datasets and one real world application dataset are used to evaluate and validate the proposed fuzzy ensemble classifier. Based on the experimental results, the proposed classifier shows better classification performance when compared with the performance of some selected classifiers. [ABSTRACT FROM AUTHOR]