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In this article, a reinforced iterative fuzzy radial basis function neural networks (IFRBFNN) is introduced as an augmented FRBFNN architecture generated through the iterative refinement process of weighted fuzzy C-means (WFCM) clustering and weighted least square error estimation. It is well known that the location of the clusters have an effect on the classification performance of the FRBFNN. The underlying idea behind this article is how to define the center points of clusters based on the data distribution analysis as well as the improvement of classification performance. In a nutshell, while the fuzzy C-means clustering to define the positions of radial basis functions (RBFs) based in the unsupervised learning manner is usually used in a FRBFNN, the parameters (cluster centers and their ensuing polynomial coefficients) refinement of FRBFNN through the proposed iterative method lead to superb classification performance by relocating the positions of RBFs over the input space implied by supervised learning. The idea of the proposed approach is to relocate the centers (prototypes) of the fuzzy clusters by using WFCM clustering algorithm and to re-estimate the coefficients by weighted least square estimation with the aid of the cross-entropy loss values of data so that the classification performance is improved. The weight related to each data is determined by some auxiliary information (i.e., the modified version of the cross-entropy loss value of each data). When it comes to the estimation of the coefficients of the consequent polynomial in IFRBFNN, they are estimated by the weighted least square error estimation technique, where the weights of the coefficients are defined based on the cross-entropy loss values. The auxiliary information such as the weights for relocation of the centers of RBFs and the weights for the re-estimation of the polynomial coefficients should be defined from the viewpoint of enhancement of the classification performance. The cross-entropy loss value of each data is able to meet the necessity of the required supervision signal. Several numerical experiments are provided to demonstrate the usefulness of the proposed design method for the FRBFNN for classification problems. [ABSTRACT FROM AUTHOR]

With the explosive growth in the density of acquired point cloud data, point cloud processing tasks will face tremendous challenges. LiDAR point cloud simplification is a key phase in addressing this issue, which effectively promotes the development of LiDAR technology in many engineering fields. In this study, an innovative point cloud simplification algorithm with the fuzzy encoding-decoding mechanism is proposed. In the developed scheme, an approach for curvature estimation is first designed on the basis of the k-neighbor searching and principal component analysis. Then, a collection of feature point sets is set up with the ordered curvatures. Subsequently, a Fuzzy C-Means clustering based encoding mechanism is employed to capture the level point cloud structures in depth and establish a reasonable and streamlined strategy for point clouds. Each feature point set and non-feature point set are encoded into a prototype matrix and a partition (membership) matrix. The membership degree of each feature point to its prototype becomes the basis for the simplification strategy. Finally, the simplification result of the point cloud is formed through merging the simplification results of all subsets. The method proposed in this study effectively preserves the point cloud features and ensures a uniform distribution of the simplified point cloud. A comparative analysis of the point cloud simplification is conducted. The experimental results demonstrate that the developed algorithm outperformed other point cloud simplification algorithms. • The curvature is used as a feature to screen out feature points, which can fully guarantee the preservation of local features. • The proposed method creatively introduces the fuzzy encoding-decoding mechanism into the sub-cluster division of the point cloud. • The simplification rate of the point cloud can be also flexibly controlled in the simplification process. • We carry out a comprehensive theoretical analysis and present a number of experiments to demonstrate the feasibility of the developed scheme. • This study opens a newway to simplify the LiDAR point cloud. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Interpretability of fuzzy rule-based models has always been of significant interest to the research community and the research in this area led to a number of far-reaching results. In this study, we briefly revisit the methodology and concepts of interpretability of Takagi–Sugeno (T-S) rule-based models and develop a conceptual framework involving several levels at which rules are interpreted. The layers at which interpretability is positioned are structured hierarchically by starting with the initial fuzzy set level (originating from the design of the rules), moving to information granules of finite support (where interval calculus is engaged) and finally ending up with symbols built at the higher level. As T-S rule-based models are endowed with local functions forming the conclusion parts of the rules, with the use of the principle of justifiable granularity, we develop a way of forming an interpretable conclusion in the form of information granule. To facilitate interpretability of conditions of the rules, multidimensional fuzzy sets (coming as a result of clustering) are decomposed into a Cartesian product of 1-D fuzzy sets and the quality of the resulting decomposition is evaluated. The quality of granular rules is assessed by analyzing the relationship between specificity of condition and conclusion information granules. The rules emerging at the level of symbols are further interpreted by engaging linguistic approximation, which helps approximate a collection of linguistic terms of subconditions producing a linguistic summarization in the form τ (inputs are A) consisting of a certain linguistic quantifier τ. The performance of summarization is provided in the form of ranking of the relevance of the rules. Experimental studies using publicly available data are completed and analyzed. [ABSTRACT FROM AUTHOR]

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]

Information granulation and degranulation play a fundamental role in granular computing (GrC). Given a collection of information granules (referred to as reference information granules), the essence of the granulation process (encoding) is to represent each data (either numeric or granular) in terms of these reference information granules. The degranulation process (decoding) that realizes the reconstruction of original data is associated with a certain level of reconstruction error. An important issue is how to reduce the reconstruction error such that the data could be reconstructed more accurately. In this study, the granulation process is realized by involving fuzzy clustering. A novel neural network is leveraged in the consecutive degranulation process, which could help significantly reduce the reconstruction error. We show that the proposed degranulation architecture exhibits improved capabilities in reconstructing original data in comparison with other methods. A series of experiments with the use of synthetic data and publicly available datasets coming from the machine-learning repository demonstrates the superiority of the proposed method over some existing alternatives. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the formation of type-2 information granules in a two-stage approach. We present a comprehensive algorithmic framework which gives rise to information granules of a higher type (type-2, to be specific) such that the key structure of the local granular data, their topologies, and their diversities become fully reflected and quantified. In contrast to traditional collaborative clustering where local structures (information granules) are obtained by running algorithms on the local datasets and communicating findings across sites, we propose a way of characterizing granular data (formed) by forming a suite of higher type information granules to reveal an overall structure of a collection of locally available datasets. Information granules built at the lower level on a basis of local sources of data are weighted by the number of data they represent while the information granules formed at the higher level of hierarchy are more abstract and general, thus facilitating a formation of a hierarchical description of data realized at different levels of detail. The construction of information granules is completed by resorting to fuzzy clustering algorithms (more specifically, the well-known Fuzzy C-Means). In the formation of information granules, we follow the fundamental principle of granular computing, viz., the principle of justifiable granularity. Experimental studies concerning selected publicly available machine-learning datasets are reported. [ABSTRACT FROM AUTHOR]

Fuzzy clustering techniques, especially fuzzy C-means (FCM) and its weighted variants, are typical partitive clustering models that are widely used for revealing possible hidden structures in data. Although they can quantitatively depict the overlapping areas with a partition matrix, their performances deteriorate when dealing with high-dimensional data because the distance computations may be negatively impacted by the irrelevant features, and then the concentration effect may arise. Moreover, they are sensitive to noisy environments. To tackle these obstacles, a robust jointly sparse fuzzy clustering method (RJSFC) is proposed in this study. The representative prototypes, sparse membership grades, and an orthogonal projection matrix are simultaneously learnt when optimizing RJSFC. The obtained low-dimensional embeddings can preserve the local neighborhood structure, and the clustering is conducted in the transformed lower dimensional space rather than the original space, which improves the capability of fuzzy clustering for dealing with high-dimensional scenarios. Furthermore, ${L_{2,1}}$ -norm is exploited as the basic metric for both loss and regularization parts in RJSFC, the robustness of the model and the interpretability of the extracted features are enhanced. The notions of fuzzy clustering, neighborhood structure preservation, and feature extraction are seamlessly integrated into a unified model. The limitation of the previous two-stage clustering framework when dealing with high-dimensional data entailing dimensionality reduction and clustering procedures separately can be effectively addressed. Extensive experimental results on various well-known datasets demonstrate the usefulness of RJSFC when comparing with some state-of-the-art methods. [ABSTRACT FROM AUTHOR]

While granular computing has experienced rapid growth in the past decades and some milestones have been reached, a comprehensive study of the representation capabilities delivered by numeric prototypes and granular prototypes produced by different techniques still calls for comprehensive research and a comparative analysis. Well-constructed information granules are reflective of the nature of the numeric evidence and serve as backbones of granular classifiers and granular models. The objective of this study is to review a number of clustering paradigms aimed at the construction of information granules, discuss the development of granular prototypes, and conduct a comprehensive evaluation of quality of numeric prototypes and their corresponding augmentations coming in the form of granular prototypes. We have been witnessing many studies devoted to the construction of information granules, but a comparative analysis of the quality of information granules constructed on a basis of prototypes produced by different clustering algorithms is still lacking. In this regard, the review of the clustering algorithms supporting the formation of information granules and the comprehensive comparative study of their usefulness in classification and modeling tasks offered in this study make sense. This will promote the usage of information granules in various future works, especially classification problem and system modeling. [ABSTRACT FROM AUTHOR]

In this article, we advocate that a knowledge tidbit residing in the output space could be helpful in improving the performance (accuracy) of the fuzzy rule-based model. It states that if two outputs are far apart from each other, it is advisable to place their corresponding inputs in different clusters when forming subspaces of the input space. Considering this knowledge guidance mechanism, we propose two different methods to partition the input space. In the first method, input data are first partitioned with the use of the standard clustering algorithm, say fuzzy C-means; here, a constructed partition matrix is reflective of the structure present in the input space. Then, the knowledge tidbit is used to adjust the entries of the original partition matrix in such a way that those input data whose corresponding output data are far apart from each other are assigned with low values of proximity. In the second method, we propose two strategies to modify the distance between input data and a prototype (cluster center) identified in the input space. The crux of this method is that if there are many input data (which, in virtue of the knowledge tidbit, are regarded as being far-apart from the input data of interest) around a certain prototype, the distance between the input data of interest and this prototype should be penalized. Thus, the membership of these input data to the prototype is reduced. The comprehensive experimental studies carried out on both synthetic and publicly available data are used to examine the usefulness of the proposed methods. [ABSTRACT FROM AUTHOR]

Rule-based models are applicable to model the behavior of complex and nonlinear systems. Due to limited experience and randomness involving constructing information granules, an insufficient credible rules division could reduce the model’s accuracy. This paper proposes a new rule-based modeling approach, which utilizes density-based spatial clustering of applications with noise (DBSCAN)-based information granules to construct the rules. First, bear in mind the advantages of density-based clustering, DBSCAN is proposed to generate data structures. Based on these data structures, two rule-based models are constructed: 1) models using DBSCAN clusters to construct granules and rules directly and 2) models generating subgranules in each DBSCAN cluster for rule formation. Experiments involving these two models are completed, and obtained results are compared with those generated with a traditional model involving fuzzy C-means-based granules. Numerical results show that the rule-based model, which builds rules from subgranules of DBSCAN structures, performs the best in analyzing system behaviors. [ABSTRACT FROM AUTHOR]

Information granules are fundamental, abstract, and easy-to-operate constructs supporting the human-centered handling way in granular computing (GrC). One of the basic properties of information granules comes with its hierarchy. Information granularity is the quantification expression of hierarchy of an information granule. Forming information granules with multigranularity to hierarchically describe the nature of data is an important task in GrC. In this article, a cone-shaped fuzzy set-based granular description method with multigranularity is proposed for multidimensional data. Its fundamental idea is to realize the synergy of quantification of information granularity and a hierarchical description of data by means of $\alpha$ –cuts of cone-shaped fuzzy sets. The proposed method first partition the entire data set into a series of data chunks. Then, some mutually nonoverlapping cone-shaped fuzzy sets are constructed by optimally determining their cores and support radii in terms of the coverage of $\alpha$ –cuts of these fuzzy sets to the data located in individual chunks and the corresponding specificity. Finally, by implementing $\alpha$ –cuts processing for these constructed cone-shaped fuzzy sets, a collection of families of hyper-spherical information granules used to hierarchically describe the structural characteristics of the data set are completely emerged. Besides, the quality of the resulting families of hyper-spherical information granules is evaluated along the granular perspective and the application perspective, respectively. A series of experimental studies concerning several synthetic and publicly available data sets are covered. The experimental results demonstrate the superiority of the proposed granular description method of data with multigranularity. [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]

As an augmentation of classic fuzzy models, granular fuzzy models (GFMs) have been applied to many fields being in rapport with experimental data, models, and users. However, most of the existing methods used to construct GFMs are based on the principle of optimal allocation of information granularity, which requires that a numeric model be provided in advance. In this paper, a straightforward and convincing modeling method is proposed to directly construct GFM on a basis of experimental data. The method first granulates the output space to form some interval information granules with distinct semantics and then uses them to partition the entire input space into a series of input subspaces. Subsequently, an initial GFM is emerged by using “If-Then” rules to relate with those interval information granules positioned in the output space and structures expressed in prototypes that are produced by clustering individual input subspaces. Further, the initial GFM is also refined by continuously migrating prototypes in individual input subspaces. The experimental studies using the synthetic dataset and several real-world datasets are reported. They offer a useful insight into the feasibility and effectiveness of the proposed modeling method and reveal the impact of parameters on the performance of the ensuing GFMs. An application example is also presented to exhibit the advantages of the resulting GFM. [ABSTRACT FROM AUTHOR]

The study is concerned with a description of large numeric data with the aid of building a limited collection of representative information granules with the objective of capturing the structure of the original data. The proposed development scheme consists of two steps. First, a clustering algorithm characterized by high flexibility of coping with the diverse geometry of data structure and efficient computational overhead is invoked. At the second step, a clustering algorithm applied to the clusters already formed during the first phase, yielding a collection of numeric prototypes is involved and the numeric prototypes produced there are then generalized into their granular prototypes. The quality of granular prototypes is quantified while their build-up is supported by the mechanisms of granular computing such as the principle of justifiable granularity. In this paper, the clustering algorithms of DBSCAN and fuzzy ${C}$ -means were used in successive phases of the processed approach. The experimental studies concerning synthetic data and publicly available data are covered and the performance of the developed approach is assessed along with a comparative analysis. [ABSTRACT FROM AUTHOR]

Big data with a large number of observations (samples) have posed genuine challenges for fuzzy clustering algorithms and fuzzy C-means (FCM), in particular. In this article, we propose an original algorithm referred to as a hyperplane division method to split the entire data set into disjoint subsets. By disjoint subsets, we mean that the data subspaces (parts of the entire data space), each of which is supported or spanned by the data points in the corresponding subset, do not overlap each other. The disjoint subsets turned out to be beneficial to the improvement of the quality of the clusters formed by the clustering algorithms. Moreover, considering that either a large number (say, thousands) or a small number (say, a few) of clusters may be pursued in the clustering task, we propose corresponding strategies (based on the hyperplane division method) to make clustering processes feasible, efficient, and effective. By validating the proposed strategies on both synthetic and publicly available data, we show their superiority (in terms of both efficiency and effectiveness) manifested in a visible way over the method of clustering the entire data and over some representative big data clustering methods. [ABSTRACT FROM AUTHOR]

We present a hierarchical model based on the extended fuzzy C-means (EFCM) clustering algorithm to develop a granular view of hotspots on a geographic map. The objective is to establish an overview of the spatial distribution of a phenomenon when the relevant data are partitioned into different datasets. The EFCM algorithm is applied to each dataset to detect local hotspots, represented as circles, on the map. The local hotspots constitute information granules at lower level of abstraction in the model. A weighted EFCM algorithm is then applied to a dataset formed by the centers of all the local hotspots to extract circular prototypes, defined as global hotspots, which constitute information granules at the higher level, and hence, they deliver a global overview of the spatial distribution of the phenomenon on the map. Two indices related to the essential criteria of the principle of justifiable granularity are used. The results demonstrate that the most justifiable overview is obtained by using the radius of the local hotspot as weight. Comparisons with a hierarchical model based on FCM algorithm show that our algorithm gives a better granular view of the phenomenon with respect to the latter. [ABSTRACT FROM AUTHOR]

Clustering offers a general methodology and comes with a remarkably rich conceptual and algorithmic framework for data analysis and data interpretation. As one of the most representative algorithms of fuzzy clustering, fuzzy C-means (FCM) is a widely used objective function-based clustering method exploited in various applications. In this study, a virtual-based fuzzy clustering algorithm is proposed to improve the classification performance coming as a result of using fuzzy clustering. This improvement is achieved by forming a virtual space based on the original data space. First, we construct a piecewise linear transformation function to modify the similarity matrix of the original data and build the so-called virtual similarity matrix (VSM). Considering the VSM, the effect of closeness becomes amplified; in other words, high similarity values (say, larger than α which is a cutoff value of the large and small similarity in this paper) present in the original similarity matrix are made higher, whereas lower similarity levels (say, smaller than α) are further reduced. In addition, data with high similarity (say, larger than a certain threshold value) observed in the original space will overlap (the attributes of the samples are exactly the same) significantly in the virtual space; the overlapping samples can be treated as one sample. This modification makes possible easier to identify clusters. Second, we build a relationship matrix between the original dataset and the determined similarity values and present two closed-form solutions to the problem of building the relationship matrix. Subsequently, a virtual space of the original data space is derived through the modified similarity matrix and the introduced relationship matrix. We offer a thorough analysis behind the developed clustering algorithm. The experimental results are in agreement with the underlying conceptual basis. Furthermore, the resulting classification performance is significantly improved compared with the results produced by the FCM and the kernel-based fuzzy C-means. [ABSTRACT FROM AUTHOR]

At present, the development of most of the granular fuzzy models depends upon some well-established numeric ones. In this study, a layered approach used to directly construct granular fuzzy models based on multidimensional numeric data is presented by engaging design methodology of granular computing. The crux of the approach involves a construction of interval information granules in the output space and the corresponding hyperbox information granules in the input space. A method of constructing these information granules and the hyperbox-based granular fuzzy model formed around them is studied in detail. Two different schemes to decode the formed hyperbox-based granular fuzzy model are also presented. Furthermore, a measure of a composite quality of the formed hyperbox-based granular fuzzy model is proposed along with the concept of coverage and specificity of resulting information granules. A number of experimental studies are reported, which offer a useful insight into the effectiveness of the presented approach, as well as reveal the impact of critical parameters on the performance of the established models. [ABSTRACT FROM AUTHOR]

Fuzzy clustering has emerged as one of the fundamental conceptual and algorithmic frameworks supporting the development of information granules. Generic fuzzy clustering such as fuzzy C-means (FCM) has been utilized in a broad range of applications. However, the constructs resulting from fuzzy clustering, namely a partition matrix and prototypes, are numeric and as such are not capable of fully capturing the essence of the overall data. In this study, we propose an alternative augmented way of building information granules by generating hypercube-like information granules. A collection of hypercubes is referred to as a family of ϵ-information granules. This family is constructed around numeric prototypes generated through a modified version of the FCM algorithm whose running time is linear with respect to the number of clusters. By admitting a certain level of information granularity $(\varepsilon)$ , a collection of hypercubes is formed around the prototypes. The quality of information granules realized in this way is assessed by involving them in the granulation—degranulation process as well as determining a value of the coverage criterion. The level of information granularity and the number of the granular prototypes in the family of $\varepsilon $ -information granules form an important design asset directly impacting the obtained coverage level of the data. The computational facet of the approach is stressed. It has been demonstrated that the granular enhancements of the description of data come with a very limited computing overhead. Experimental studies involve synthetic data as well as data coming from the UCI Machine Learning repository. The granular reconstruction capabilities delivered by the family of $\varepsilon $ -information granules are discussed. [ABSTRACT FROM AUTHOR]

Granular computing (GrC) has emerged as a unified conceptual and processing framework. Information granules are fundamental constructs that permeate concepts and models of GrC. This paper is concerned with a design of a collection of meaningful, easily interpretable ellipsoidal information granules with the use of the principle of justifiable granularity by taking into consideration reconstruction abilities of the designed information granules. The principle of justifiable granularity supports designing of information granules based on numeric or granular evidence, and aims to achieve a compromise between justifiability and specificity of the information granules to be constructed. A two-stage development strategy behind the construction of justifiable information granules is considered. First, a collection of numeric prototypes is determined with the use of fuzzy clustering. Second, the lengths of the semi-axes of ellipsoidal information granules to be formed around such prototypes are optimized. Two optimization criteria are introduced and studied. Experimental studies involving synthetic data set and data sets coming from the machine learning repository are reported. [ABSTRACT FROM PUBLISHER]

Data science is a research field concerned with processes and systems that extract knowledge from massive amounts of data. In some situations, however, data shortage renders existing data-driven methods difficult or even impossible to apply. Transfer learning has recently emerged as a way of exploiting previously acquired knowledge to solve new yet similar problems much more quickly and effectively. In contrast to classical data-driven machine learning methods, transfer learning methods exploit the knowledge accumulated from data in auxiliary domains to facilitate predictive modeling in the current domain. A significant number of transfer learning methods that address classification tasks have been proposed, but studies on transfer learning in the case of regression problems are still scarce. This study focuses on using transfer learning techniques to handle regression problems in a domain that has insufficient training data. We propose an original fuzzy regression transfer learning method, based on fuzzy rules, to address the problem of estimating the value of the target for regression. A Takagi–Sugeno fuzzy regression model is developed to transfer knowledge from a source domain to a target domain. Experimental results using synthetic data and real-world datasets demonstrate that the proposed fuzzy regression transfer learning method significantly improves the performance of existing models when tackling regression problems in the target domain. [ABSTRACT FROM AUTHOR]

Fuzzy models are regarded as numeric constructs and as such are optimized and evaluated at the numeric level. In this study, we depart from this commonly accepted position and propose a granular evaluation of fuzzy models and present an augmentation of fuzzy models by forming information granules around numeric values of the parameters and constructions of the models. The concepts and algorithms of granular fuzzy models are discussed in the setting of Takagi–Sugeno rule-based architectures. We show how different protocols of forming and allocating information granules lead to the improvement of the granular performance of the models. Different from the standard numeric performance measure of fuzzy models coming in the form of the root mean squared error index, two performance measures are introduced that are pertinent to granular constructs, namely coverage and specificity. Furthermore, we propose a global indicator implied by these two measures, called an area under the curve, being computed for the characteristics of the granular model expressed in the coverage-specificity coordinates. A series of experimental studies is reported, which offers a comprehensive overview of the introduced performance measure criteria as well as the underlying realization of the granular fuzzy models. [ABSTRACT FROM PUBLISHER]

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]

Associative and bidirectional associative memories are examples of associative structures studied intensively in the literature. The underlying idea is to realize associative mapping so that the recall processes (onedirectional and bidirectional ones) are realized with minimal recall errors. Associative and fuzzy associative memories have been studied in numerous areas yielding efficient applications for image recall and enhancements and fuzzy controllers, which can be regarded as one-directional associative memories. In this study, we revisit and augment the concept of associative memories by offering some new design insights where the corresponding mappings are realized on the basis of a related collection of landmarks (prototypes) over which an associative mapping becomes spanned. In light of the bidirectional character of mappings, we have developed an augmentation of the existing fuzzy clustering (fuzzy c-means, FCM) in the form of a socalled collaborative fuzzy clustering. Here, an interaction in the formation of prototypes is optimized so that the bidirectional recall errors can be minimized. Furthermore, we generalized the mapping into its granular version in which numeric prototypes that are formed through the clustering process are made granular so that the quality of the recall can be quantified. We propose several scenarios in which the allocation of information granularity is aimed at the optimization of the characteristics of recalled results (information granules) that are quantified in terms of coverage and specificity. We also introduce various architectural augmentations of the associative structures. [ABSTRACT FROM AUTHOR]

Fuzzy rough sets (FRSs) are considered to be a powerful model for analyzing uncertainty in data. This model encapsulates two types of uncertainty: 1) fuzziness coming from the vagueness in human concept formation and 2) roughness rooted in the granulation coming with human cognition. The rough set theory has been widely applied to feature selection, attribute reduction, and classification. However, it is reported that the classical FRS model is sensitive to noisy information. To address this problem, several robust models have been developed in recent years. Nevertheless, these models do not consider a statistical distribution of data, which is an important type of uncertainty. Data distribution serves as crucial information for designing an optimal classification or regression model. Thus, we propose a data-distribution-aware FRS model that considers distribution information and incorporates it in computing lower and upper fuzzy approximations. The proposed model considers not only the similarity between samples, but also the probability density of classes. In order to demonstrate the effectiveness of the proposed model, we design a new sample evaluation index for prototype-based classification based on the model, and a prototype selection algorithm is developed using this index. Furthermore, a robust classification algorithm is constructed with prototype covering and nearest neighbor classification. Experimental results confirm the robustness and effectiveness of the proposed model. [ABSTRACT FROM PUBLISHER]

Incomplete data clustering is often encountered in practice. Here the treatment of missing attribute value and the optimization procedure of clustering are the important factors impacting the clustering performance. In this study, a missing attribute value becomes an information granule and is represented as a certain interval. To avoid intervals determined by different cluster information, we propose a congeneric nearest-neighbor rule-based architecture of the preclassification result, which can improve the effectiveness of estimation of missing attribute interval. Furthermore, a global fuzzy clustering approach using particle swarm optimization assisted by the Fuzzy C-Means is proposed. A novel encoding scheme where particles are composed of the cluster prototypes and the missing attribute values is considered in the optimization procedure. The proposed approach improves the accuracy of clustering results, moreover, the missing attribute imputation can be implemented at the same time. The experimental results of several UCI data sets show the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

In this study, we discuss a design of fuzzy associative structures (memories) realized within the framework of fuzzy clustering. Associative memories are inherently direction-free structures (and the recall of objects can be realized for any variable or a subset of variables). Fuzzy clustering being direction-free comes here as a sound design alternative. Two recall proposals are studied: one involves prototypes (being the key descriptors of the structure of the data) and their activation in the presence of partially available data to be recalled whereas the second proposal involves fuzzy correlation matrices and in principle exhibits some resemblance with a standard correlation associative memories. In the setting of associative memories, Fuzzy C-Means (FCM) is studied. The recall error is discussed with regard to the essential parameters of the FCM (the number of clusters and the fuzzification coefficient). Furthermore we discuss an optimization of the distance function used in the clustering algorithm realized with regard to the recall error. Experimental results are provided along with a comparative study involving correlation-based associative memories. [ABSTRACT FROM AUTHOR]

In this study, we introduce a concept of hierarchical granular clustering and establish its algorithmic framework. We show that the proposed model naturally gives rise to information granules that are both of higher order and higher type, offering a compelling justification behind their emergence. In a concise way, we can capture the overall architecture of information granules as a hierarchy exhibiting conceptual layers of increasing abstraction: numeric data → information granules → information granules of type-2, order-2 → … information granules of higher type/order. The elevated type of information granules is reflective of the visible hierarchical facet of processing and the inherent diversity of the individual locally revealed structures in data. While the concept and the methodology deliver some general settings, the detailed algorithmic aspects are discussed in detail when using fuzzy clustering realized by means of fuzzy c-means. Furthermore, for illustrative purposes, we mainly focus on interval-valued fuzzy sets and granular interval fuzzy sets arising at the higher level of the hierarchy. Higher type fuzzy sets are formed with the help of the principle of justifiable granularity. The conceptually sound hierarchy is established in a general way, which makes it equally applicable to various formalisms of representation of information granules. Experiments are reported for synthetic and publicly available datasets. [ABSTRACT FROM AUTHOR]

The study is devoted to the clustering of granular data and an evaluation of the results of such clustering. A comprehensive and systematic approach is developed, which is composed of three fundamental phases: 1) representation of granular data; 2) clustering carried out in the representation space of information granules; and 3) evaluation of quality of clusters following the reconstruction criterion. The reconstruction criterion formed originally for numeric data and leading to an idea of granular prototypes is revisited. We show here an emergence of granular information of higher type, which are used to implement granular interval prototypes. We discuss a way of forming granular data in the context of representation of time series and present clustering of granular time series. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

In this study, we propose a cluster-oriented development of fuzzy models. An overall design process is focused on an efficient usage of fuzzy clustering, Fuzzy C-Means (FCM), in particular, to form information granules—clusters that are used in the construction of the fuzzy model. Fuzzy models are regarded as mappings from information granules expressed in the input and output spaces. This position motivates us to look at the development of the models through the perspective of the construction and efficient usage of information granules. This study directly associates fuzzy clustering with fuzzy modeling both in terms of conceptual and algorithmic linkages. The augmented FCM method is formed predominantly for modeling purposes so that a balance between the structural content present in the input and output spaces is achieved and this way the performance of the resulting fuzzy model is optimized. It is shown that the cluster-oriented modeling gives rise to the Mamdani-like fuzzy rules and a zero-order Takagi–Sugeno model (under a certain decoding scheme). We identify an interesting and direct linkage between the developed fuzzy models and a fundamental idea of encoding–decoding (or granulation–degranulation) encountered in processing fuzzy sets and Granular Computing, in general. Furthermore, refinements of zero-order fuzzy models are investigated leading to first-order fuzzy models with linear functions standing in the conclusions of the rules. A series of experiments is reported where we used synthetic and real-world data using which an issue of generalization capabilities is elaborated in detail. [ABSTRACT FROM AUTHOR]

A class of clustering problems that is studied here is concerned with the development of a structure of a global nature given a collection of structures (clusters) constructed locally for data that are represented by several collections (blocks) of features. These blocks of features come with a well-defined semantics. For instance, in spatiotemporal data, a certain block of features concerns a spatial component of the data (say, x–y or x–y–z coordinates), while another one deals with the features that describe time series associated with the corresponding locations. The results of clustering that are being produced locally are reconciled by minimizing a distance between the proximity matrices that are formed at the higher conceptual level and induced by the individual partition matrices. The optimization problem is formulated and presented along with its iterative scheme. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

There are some variants of the widely used Fuzzy C-Means (FCM) algorithm that support clustering data distributed across different sites. Those methods have been studied under different names, like collaborative and parallel fuzzy clustering. In this study, we offer some augmentation of the two FCM-based clustering algorithms used to cluster distributed data by arriving at some constructive ways of determining essential parameters of the algorithms (including the number of clusters) and forming a set of systematically structured guidelines such as a selection of the specific algorithm depending on the nature of the data environment and the assumptions being made about the number of clusters. A thorough complexity analysis, including space, time, and communication aspects, is reported. A series of detailed numeric experiments is used to illustrate the main ideas discussed in the study. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract Rule-based modeling has been one of the key directions in fuzzy modeling since its inception. Nowadays it exhibits a number of conceptual development and algorithmic pursuits. Fuzzy clustering, especially Fuzzy C-Means (FCM), is a commonly used algorithmic tool to construct rules, in particular building fuzzy sets forming conditions of the rules. While being efficient with this regard and transforming data into a collection of fuzzy sets, one should note that the agenda of fuzzy clustering (and clustering, in general) does not fully align with the agenda of system modeling and because of this, it requires some attention and calls for further refinements. Clustering is a direction-free (relational) process and developed constructs (clusters) are optimized in light of the direction-free criterion (say, a commonly used objective function). In contrast, in fuzzy models the rules are direction-sensitive artifacts, which implies that the clusters themselves need to be reflective of this directionality requirement. This paper contributes to this direction of studies by bringing a collection of augmentations of the generic FCM algorithm along this line. There are three original enhancements considered in the study: (i) an accommodation of extreme (minimal and maximal) values encountered in the output variable, (ii) a reduction of spurious impact of rules being the result of variable overlap existing among fuzzy sets forming the condition parts of the rules, and (iii) a development of the core (granular) structure of rules and analysis of their features. The motivation behind these augmentations is presented followed by the detailed algorithms along with a series of illustrative examples. In the sequel, a number of numeric studies are conducted demonstrating in a quantitative manner the contributions delivered by the refinements. [ABSTRACT FROM AUTHOR]

Data clustering is aimed at discovering a structure in data. The revealed structure is usually represented in terms of prototypes and partition matrices. In some cases, the prototypes are simultaneously formed using data and features by running a co-clustering (bi-clustering) algorithm. Interval valued fuzzy clustering exhibits advantages when handling uncertainty. This study introduces a novel clustering technique by combining fuzzy co-clustering approach and interval-valued fuzzy sets in which two values of the fuzzifier of the fuzzy clustering algorithm are used to form the footprint of uncertainty (FOU). The study demonstrates the performance of the proposed method through a series of experiments completed for various datasets (including color segmentation, multi-spectral image classification, and document categorization). The experiments quantify the quality of results with the aid of validity indices and visual inspection. Some comparative analysis is also covered. [ABSTRACT FROM AUTHOR]