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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, 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]

Information granules can discover interpretable and meaningful relationships offering a full description for time series. This article presents an information granulation method with rectangular information granules and applies it to time-series similarity measurement. First, the fuzzy $c$ -means clustering algorithm transforms the time series and its first-order difference time series to data clusters. With the maximum volume of rectangular information granules viewed as the criterion, the optimal rectangular information granules are formed using the data cluster by the principle of justifiable granularity and the gravitational search algorithm. The time-series similarity is measured by calculating the similarity between the upper and lower bounds of the optimal rectangular information granules built from the time series. Finally, an experiment is performed on a public dataset to verify the feasibility of the proposed method. The result shows that the rectangular information granulation method can capture the change characteristics of time series. The similarity measurement method can effectively evaluate the similarity of the time series belonging to different classes. [ABSTRACT FROM AUTHOR]

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.

Financial time series are generally high-dimensional, nonstationary, and exhibit heteroscedasticity. To derive a suitable way to cluster financial time series, these characteristics have to be taken into consideration. With this aim, in this article, the financial time series is firstly modeled using generalized autoregressive conditional heteroscedasticity (GARCH) models, where the parameters of GARCH models can represent the dynamic feature of the volatility in each time series. Therefore, the following clustering is realized based on the GARCH model parameters, which can help reduce the dimensionality of the original time series at the same time. Then, to produce semantically sound clustering results, we granulate the parameters based on the axiomatic fuzzy set (AFS) theory and structure them into a collection of meaningful and semantically sound entities, i.e., AFS information granules. Furthermore, the hierarchical structure of AFS information granules is built to realize time series clustering under the framework of granular computing. In the proposed approach, the characteristics of financial time series is fully considered to proceed dimensionality reduction, and the semantic clustering results obtained for different numbers of clusters are guaranteed to be the most informative. In the experiments, an application for clustering the time series coming from Chinese Yuan exchange rates against international currencies is presented to demonstrate the performance of the proposed clustering method. The results of clustering of the proposed method are the same as those of the fuzzy C-means algorithm and the hierarchical clustering with ward linkage, where the clustering results produced by the AFS hierarchical clustering exhibit well-articulated semantics at each level of the hierarchy. [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 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]

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]

In this study, a reinforced Fuzzy Clustering-based Neural Network (FCNN) is introduced as an augmented FCNN architecture to address regression issues. It is widely recognized that regardless of the design method and rules employed by a fuzzy model, the determination of fuzzy sets remains a crucial aspect. FCNN and its improved variants utilize conventional fuzzy clustering algorithms to partition the feature space into fuzzy sets. However, this approach tends to disregard the distinctions inherent in data patterns. Although FCNN is a nonlinear model in relation to the input variables, it is a linear model with respect to the parameters that need to be estimated. Inspired by this, our method incorporates a pre-training phase where we utilize sample residuals from a linear regression algorithm to measure differences between data patterns. These differences are subsequently integrated into the fuzzy partition, yielding more refined fuzzy sets. To combat overfitting that can degrade the model's predictive capability, we introduce a convolutional L 2 regularization strategy that integrates the convolution operator from harmonic analysis into the construction of L 2 regularization. Compared to conventional L 2 regularization, this convolutional regularization strategy is more effective in improving the regularity of the design matrix, thereby reducing the variation between coefficients and enhancing the model's generalization ability. The efficacy of the presented method is substantiated by experimental studies conducted on both synthetic and real-world datasets. • A novel fuzzy clustering-based neural network is proposed as an enhanced FCNN architecture to address regression problems. • Residuals from linear regression, quantifying data pattern differences, are utilized in fuzzy partitions to create more rational fuzzy sets. • A convolutional L 2 regularization strategy, melding the convolutional operator with L 2 regularization, is introduced to mitigate overfitting, thereby improving model generalization • The classical L 2 regularization technique is revealed to be a particular instance of the proposed convolutional regularization strategy • A statistical analysis approach is employed to affirm the superior performance of the proposed neural network [ABSTRACT FROM AUTHOR]

Type-2 fuzzy sets provide an efficient vehicle for handling uncertainties of real-world problems, including noisy observations. Bringing type-2 fuzzy sets to clustering algorithms offers more flexibility to handle uncertainties associated with membership concepts caused by a noisy environment. However, the existing type-2 fuzzy clustering algorithms suffer from a time-consuming type-reduction process, which not only hampers the clustering performance but also increases the burden of understanding the clustering results. In order to alleviate the problem, this article introduces a set of typical characteristics of type-2 fuzzy sets and establishes a characteristic-based type-2 fuzzy clustering algorithm. Being different from the objective function used in the fuzzy C-means (FCM) algorithm that produces cluster centers and type-1 memberships, the objective function in the proposed algorithm contains additional characteristics of type-2 membership grades, namely, centers of gravity and cardinalities of the secondary fuzzy sets. The derived iterative formulas used for these parameters are much more efficient than the interval type-2 FCM algorithm. The experiments carried out in this study show that the proposed typical characteristic-based type-2 FCM algorithm has an ability of detecting noise as well as assigning suitable membership degrees to the individual data. [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]

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]

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

Aiming at the previously-proposed entropy-based differently implicational algorithm of fuzzy inference, this study analyzes its continuity. To begin with, for the FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) problems, the continuous as well as uniformly continuous properties of the entropy-based differently implicational algorithm are demonstrated for the Tchebyshev and Hamming metrics, in which the R-implications derived from left-continuous t-norms are employed. Furthermore, four numerical fuzzy inference examples are provided, and it is found that the entropy-based differently implicational algorithm can obtain more reasonable solution in contrast with the fuzzy entropy full implication algorithm. Finally, in the entropy-based differently implicational algorithm, we point out that the first fuzzy implication reects the effect of rule base, and that the second fuzzy implication embodies the inference mechanism. [ABSTRACT FROM AUTHOR]

Machine learning plays an important role in data classification and data-based prediction. In some real-world applications, however, the training data (coming from the source domain) and test data (from the target domain) come from different domains or time periods, and this may result in the different distributions of some features. Moreover, the values of the features and/or labels of the datasets might be nonnumeric and involve vague values. Traditional learning-based prediction and classification methods cannot handle these two issues. In this study, we propose a multistep fuzzy bridged refinement domain adaptation algorithm, which offers an effective way to deal with both issues. It utilizes a concept of similarity to modify the labels of the target instances that were initially predicted by a shift-unaware model. It then refines the labels using instances that are most similar to a given target instance. These instances are extracted from mixture domains composed of source and target domains. The proposed algorithm is built on a basis of some data and refines the labels, thus performing completely independently of the shift-unaware prediction model. The algorithm uses a fuzzy set-based approach to deal with the vague values of the features and labels. Four different datasets are used in the experiments to validate the proposed algorithm. The results, which are compared with those generated by the existing domain adaptation methods, demonstrate a significant improvement in prediction accuracy in both the above-mentioned datasets. [ABSTRACT FROM AUTHOR]

In this paper, we address problems of description and prediction of time series by developing architectures of granular time series. Granular time series are models of time series formed at the level of information granules expressed in the representation space and time. With regard to temporal granularity, time series is split into temporal windows leading in this way to the formation of temporal information granules. Information granules are also quantified and constructed over the space of amplitude and change of amplitude of the series collected over time windows. In the description of time series we involve clustering techniques and build information granules in the representation space (viz. the space of amplitude and change of amplitude) of the temporal data. Fuzzy relations forming the essence of the prediction model are optimized using particle swarm optimization. Experimental results are reported for a number of publicly available time series. [ABSTRACT FROM AUTHOR]

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]

This paper presents a new approach to modelling and control of hybrid systems with both continuous variables and discrete events. Applying the fuzzy set theory, a hierarchical fuzzy hybrid structure consisting of a fuzzy discrete event dynamic system and a continuous variable dynamic system is constructed, which not only captures the hybrid continuous/discrete dynamics but also handles the uncertainties in states and state transitions. The identification of continuous and discrete components is developed, and the hybrid control is then synthesised by fuzzy IF-THEN rules embedded in the fuzzy interface. An example of the optimisation of a production line in manufacturing shows the efficacy of the proposed approach. [ABSTRACT FROM AUTHOR]

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

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

Abstract: In this study, we are concerned with a concept of consensus-driven fuzzy clustering whose objective is to reconcile a structure developed for patterns in some data set with the structural findings already available for other related data sets (where these data sets are reflective of the same phenomenon which has led to the generation of the original patterns). The results of fuzzy clustering are provided in the form of prototypes and fuzzy partition matrices. Given this form of representation of granular results (clusters), we develop a suitable communication scheme using which consensus could be established in an effective manner. Here, we consider proximity matrices induced by the corresponding partition matrices. An overall optimization scheme is presented in detail along with a way of forming a pertinent criterion governing an intensity of collaboration between the data driven- and knowledge oriented hints guiding the process of consensus formation. Several illustrative numeric examples, using both synthetic data and the data coming from publicly available machine learning repositories are also included. [Copyright &y& Elsevier]

Established methods of Boolean minimization have previously unseen potential as an efficient and unrestricted means of fuzzy structure discovery, becoming particularly useful within a design methodology for the automatic development of fuzzy models. Traditionally used in digital systems design, logic minimization tools allow us to exploit the fundamental links between binary (two-valued) and fuzzy (multivalued) logic. In this paper, we show how logic optimization plays an integral role in a two-phase fuzzy model design process. Adaptive logic processing is realized as the discovered Boolean structures are augmented with fuzzy granules and then refined by adjusting connections of fuzzy neurons, helping to further capture the numeric details of the target systems behavior. Accurate and highly interpretable fuzzy models are the result of the entire development process. [ABSTRACT FROM AUTHOR]

Fuzzy clustering has emerged as a fundamental technique of information granulation. In this study, we introduce and discuss multivariable encoding and decoding mechanisms (referred altogether as a reconstruction problem) expressed in the language of fuzzy sets and fuzzy relations. The underlying performance index associated with the problem helps quantify a reconstruction error that arises when transforming a numeric datum through fuzzy sets (relations) and then reconstructing it into an original numeric format. The clustering platform considered in this study concerns the well-known algorithm of Fuzzy C-Means (FCM). The main design aspects deal with the relationships between the number of clusters versus the reconstruction properties and the resulting reconstruction error. The impact of the fuzzification coefficient on the reconstruction quality is investigated. This finding is of interest, given the fact that predominantly all applications involving FCM use the value of the fuzzification coefficient equal to 2. In light of the completed experiments, we demonstrate that this selection may not be experimentally. legitimate. We also carry out a comparative analysis of the reconstruction properties of the Boolean decoding that is induced by the fuzzy partition. Experimental investigations involve selected machine learning data. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and investigate a new category of fuzzy inference systems based on information granulation and genetic optimization used to system identification. We show the applications of such systems to identification of nonlinear systems. The formal framework of information granulation and resulting information granules themselves become an important design facet of the fuzzy models. By embracing fuzzy sets, the model is geared towards capturing essential relationship between information granules rather than concentrating on plain numeric data. Information granulation realized with the use of the commonly exploited C-Means clustering helps determine the initial values of the parameters of the fuzzy models. This in particular concerns such essential components of the rules as the initial apexes of the membership functions standing in the premise part of the fuzzy rules and the points of the polynomial functions standing in the consequence part. The initial apexes (center points) of the membership functions based on C-Means algorithm are tuned with the aid of the genetic algorithm (GA), while the tuned apexes are also used to adjust the points of the consequent polynomials (conclusions) of the rules. In particular, the initial apexes of the membership functions and the initial points of the consequent polynomials are adjusted and updated every time through successive evolution process. The overall design methodology involves a hybrid structural and parametric optimization. Genetic algorithms and C-Means clustering are used to optimize the model with respect to its structure and parameters. To determine the structure and estimate the values of the parameters of the fuzzy model we consider the successive tuning method with generation-based evolution by means of genetic algorithms. The model is evaluated with the use of numerical experimentation and its quality is compared with respect to some other fuzzy models already encountered in the literature. [Copyright &y& Elsevier]

Abstract: Vector quantization (VQ) is a fundamental and omnipresent mechanism of data compression with various conceptual underpinnings and diversified algorithmic realizations. The objective of this study is to investigate the concept of VQ in the setting of fuzzy sets by forming a coherent algorithmic framework referred to as a fuzzy VQ (FVQ). Given the nature of the framework of VQ in which fuzzy sets are involved, we may refer to the discussed processes of FVQ as a fuzzy granulation and fuzzy degranulation. In comparison to the winner-takes-all strategy encountered in VQ where a result of decoding typically arises as a single element of the codebook, in the FVQ we exploit an efficient usage of all components of the codebook (fuzzy sets) in the reconstruction of the original data. In this study, we present a complete development scheme of the FVQ and elaborate on its essential features. Its main design phases involve: (a) an encoding in which we encode data in terms of the elements of the given codebook; (b) a decoding during which we reconstruct the original data; and (c) a development of the codebook. The mechanisms of encoding and decoding are created as a result of some well-formed optimization tasks. The buildup of the codebook is completed through a mechanism of global optimization realized in the form of the particle swarm optimization (PSO). We offer a collection of experiments using synthetic data by focusing on and quantifying the role of fuzzy sets in VQ. While FVQ outperforms VQ (which seems to be an intuitively appealing finding), we also show that this improvement could be achieved through a careful optimization of the elements of the granulation scheme. It is also shown that without optimization of the FVQ scheme, the enhancements could not be possible or may become very much limited. A series of experiments involving synthetic data and data sets coming from the Machine Learning repository is included as well. [Copyright &y& Elsevier]

In this study, we introduce and discuss a category of genetically optimized fuzzy neural networks. As far as the underlying geometry of such networks is concerned, they are focused on revealing a hyperbox-based topology in numeric data. This class of the networks is developed around fuzzy tolerance neurons. Tolerance neurons form a generalized version of intervals (sets) arising in a form of fuzzy intervals. The architecture of the network reflects a hierarchy of geometric concepts typically exploited in data analysis: fuzzy intervals combined and-wise give rise to fuzzy hyperboxes and these in turn by being aggregated or-wise generate a summary of data as a collection of hyperboxes. We discuss a genetic form of optimization of the networks and provide an in-depth view into the geometry of the individual hyperboxes as well as the overall topology of the network. Numerical experiments deal with 2-D synthetic data. [Copyright &y& Elsevier]

Abstract: Decision trees are one among interesting and commonly encountered architectures used for learning, reasoning and organization of datasets. This study being positioned in the realm of decision trees is aimed at two main objectives. First, we propose a new algorithmic framework for building fuzzy sets (membership functions) and their logic operators based upon theoretical findings of the Axiomatic Fuzzy Set (logic) theory (AFS). Second, we cast the design processes of fuzzy decision trees in this framework. A number of illustrative examples are included. We demonstrate how the AFS setting results in the improvement of the performance of the resulting trees. The findings are contrasted with the outcomes produced by the decision trees studied by Janikow; in particular, we show the performance of different trees in the case of large number of fuzzy attributes. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new topology of fuzzy polynomial neural networks (FPNNs) that is based on a genetically optimized multilayer perceptron with fuzzy polynomial neurons (FPNs). The study offers a comprehensive design methodology involving mechanisms of genetic optimization, especially those exploiting genetic algorithms (GAs). Let us recall that the design of the ‘conventional’ FPNNs uses an extended group method of data handling (GMDH) and uses a fixed scheme of fuzzy inference (such as simplified, linear, and regression polynomial fuzzy inference) in each FPN of the network. It also considers a fixed number of input nodes (as being selected in advance by a network designer) at FPNs (or nodes) located in each layer. However such design process does not guarantee that the resulting FPNs will always result in an optimal networks architecture. Here, the development of the FPNN gives rise to a structurally optimized topology and comes with a substantial level of flexibility which becomes apparent when contrasted with the one we encounter in the conventional FPNNs. The design of each layer of the FPNN deals with its structural optimization involving a selection of preferred nodes (or FPNs) with specific local characteristics (such as the number of input variables, the order of the polynomial forming a consequent part of fuzzy rules and a collection of the specific subset of input variables) and addresses detailed aspects of parametric optimization. Along this line, two general optimization mechanisms are explored. The structural optimization is realized via GAs. In case of the parametric optimization we proceed with a standard least square method-based learning. Through the consecutive process of such structural and parametric optimization, an optimized and flexible fuzzy neural network becomes generated in a dynamic fashion. To evaluate the performance of the genetically optimized FPNN (gFPNN), we experimented with two time series data (gas furnace and chaotic time series) as well as some synthetic data. A comparative analysis reveals that the proposed FPNN exhibits higher accuracy and superb predictive capability in comparison to some previous models available in the literature. [ABSTRACT FROM AUTHOR]

Clustering with partial supervision finds its application in Situations where data is neither entirely nor accurately labeled. This paper discusses a semi-supervised clustering algorithm based on a modified version of the fuzzy C-Means (FCM) algorithm. The objective function of the proposed algorithm consists of two components. The first concerns traditional unsupervised clustering while the second tracks the relationship between classes (available labels) and the clusters generated by the first component. The balance between the two components is tuned by a scaling factor. Comprehensive experimental studies are presented. First, the discrimination of the proposed algorithm is discussed before its reformulation as a classifier is addressed. The induced classifier is evaluated on completely labeled data and validated by comparison against some fully supervised classifiers, namely support vector machines and neural networks. This classifier is then evaluated and compared against three semi-supervised algorithms in the context of learning from partly labeled data. In addition, the behavior of the algorithm is discussed and the relation between classes and clusters is investigated using a linear regression model. Finally, the complexity of the algorithm is briefly discussed. [ABSTRACT FROM AUTHOR]

The article introduces a concept and design of decision trees based on information granules, multivariable entities characterized by high homogeneity. The article shows how the tree is grown depending on some additional node expansion criteria such as cardinality at a given node and a level of structural dependencies of data existing there. A series of experiments is reported using both synthetic and machine learning data sets. The C-decision trees are classification constructs that are built on a basis of information granules, fuzzy clusters.

We introduce and study a new concept of fuzzy computing units. This construct is is aimed at coping with "negative" (inhibitory) information and accommodating it in the language of fuzzy sets. The essential concept developed in this study deals with computing units exploiting the concept of balanced fuzzy sets. We recall how the membership notion of fuzzy sets can be extended to the [-1,1] range giving rise to balanced fuzzy sets and then summarize properties of augmented (extended) logic operations for these constructs. We show that this idea is particularly appealing in neurocomputing as the "negative" information captured through balanced fuzzy sets exhibits a straightforward correspondence with inhibitory processing mechanisms encountered in neural networks. This gives rise to interesting properties of balanced computing units when compared with fuzzy and logic neurons developed on the basis of classical logic and classical fuzzy sets. Illustrative examples concerning topologies and properties and learning of balanced fuzzy computing units are included. A number of illustrative examples concerning topologies, properties and learning of balanced fuzzy fuzzy computing units are included. [ABSTRACT FROM AUTHOR]

This study is concerned with the Boolean satisfiability (SAT) problem and its solution in setting a hybrid computational intelligence environment of genetic and fuzzy computing. In this framework, fuzzy sets realize an embedding principle meaning that original two-valued (Boolean) functions under investigation are extended to their continuous counterparts resulting in the form of fuzzy (multivalued) functions. In the sequel, the SAT problem is reformulated for the fuzzy functions and solved using a genetic algorithm (GA). It is shown that a GA, especially its recursive version, is an efficient tool for handling multivariable SAT problems. Thorough experiments revealed that the recursive version of the GA can solve SAT problems with more than 1000 variables. [ABSTRACT FROM AUTHOR]

Defuzzification is an important operation in the theory of fuzzy sets. It transforms a fuzzy set information into a numeric data information. This operation along with the operation of fuzzification is critical to the design of fuzzy systems as both of these operations provide nexus between the fuzzy set domain and the real-valued scalar domain. We need the synergy of both of these domains to solve many of our ill-posed problems effectively. In this paper, we address the problem of defuzzification, we present merits and demerits of various defuzzification strategies that are used in the theory and practice, and in design and implementation of applications involving fuzzy theory, fuzzy control, and fuzzy rule base, and fuzzy inference-based systems. We also present in this paper a simple and yet a novel defuzzification mechanism. © 2001 John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

• A novel locality preserving based fuzzy C-means clustering method (LPFCM) is presented. • An orthogonally projected space, which preserves the locality of structural properties, can be generated in LPFCM. • The capability of FCM for handling high-dimensional data can be enhanced. • The ideas of fuzzy clustering, geometric structure preservation and feature extraction are seamlessly integrated. • Experimental results on some benchmark data sets show the effectiveness of LPFCM. Traditional partition-based clustering algorithms, hard or fuzzy version of C-means, could not deal with high-dimensional data sets effectively as redundant features may impact the computation of distances and local spatial structures among patterns are rarely considered. High dimensionality of space gives rise to so-called concentration effect that is detrimental. In this paper, a novel locality preserving based fuzzy C-means (LPFCM) clustering method and its optimization are presented. An orthogonally projected space, which preserves the locality of structural properties, can be generated in LPFCM, thus enhancing the capability of fuzzy C-means (FCM) for handling high-dimensional data. It is the first time to introduce projection techniques to the FCM optimization objective function directly, and the ideas of fuzzy clustering, geometric structure preservation and feature extraction are seamlessly integrated. LPFCM is also regarded as a unified model that combines two separate stages of spectral clustering. Experimental results on some benchmark data sets show the effectiveness of LPFCM in comparison with FCM and some state-of-the-art methods. [ABSTRACT FROM AUTHOR]

This study focuses on a granular representation of signals. The development process dwells upon the use of the principle of justifiable granularity encountered in Granular Computing and the least square error method. This process consists of two phases where the construction of granular representatives of a family of signals (temporal data) is realized by invoking the design at the local and global level. At the local design level involving individual elements of the universe of discourse (time moments), the principle of justifiable granularity is applied to construct (a vertical part) information granules. At the global level, the least square error method is invoked to develop the bounds (envelopes) of the information granules already formed at the local level. Experimental studies are reported for the granular representation of synthetic data and publicly available ECG signals. Furthermore we demonstrate that the proposed approach can be used to construct fuzzy sets of type-2. [ABSTRACT FROM AUTHOR]

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

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

Abstract: Various approaches to partitioning of high-dimensional input space have been studied with the intent of developing homogeneous clusters formed over input and output spaces of variables encountered in system modeling. In this study, we propose a new design methodology of a fuzzy model where the input space is partitioned by making use of some classification algorithm, especially, fuzzy K-nearest neighbors (K-NN) classifier being guided by some auxiliary information granules formed in the output space. This classifier being regarded in the context of this design as a supervision mechanism is used to capture the distribution of data over the output space. This type of supervision is beneficial when developing the structure in the input space. It is demonstrated that data involved in a partition constructed by the fuzzy K-NN method realized in the input space show a high level of homogeneity with regard to the data present in the output space. This enhances the performance of the fuzzy rule-based model whose premises in the rules involve partitions formed by the fuzzy K-NN. The design is illustrated with the aid of numeric examples that also provide a detailed insight into the performance of the fuzzy models and quantify several crucial design issues. [Copyright &y& Elsevier]

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

Fuzzy co-clustering algorithms are the effective techniques for multi-dimensional clustering in which all features are considered of equal importance (relevance). In fact, the features' importance could be different, even several of them could be considered redundant. The removal of the redundant features has formed the idea of feature-reduction in problems of the big data processing. In this paper, we propose a new unsupervised learning scheme by incorporating the feature-weighted entropy into the objective function of fuzzy co-clustering, called the Feature-Reduction Fuzzy Co-Clustering Algorithm (FRFCoC). First, a new objective function is formed on the basis of the original fuzzy co-clustering objective function which adds parameters representing the entropy weight of the different features. Next, a feature-reduction and clustering automatic schema are adjusted based on FCoC's original learning schema which calculates new parameters and conditions to eliminate irrelevant feature components. FRFCoC algorithm can be mathematically shown to converge after a finite number of iterations. The experiment results were conducted on some many-features data sets and hyperspectral images that have demonstrated the outstanding performance of FRFCoC algorithm compared with some previously proposed algorithms. [ABSTRACT FROM AUTHOR]