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This study elaborates on a novel Hidden Markov Model (HMM)-based fuzzy model for time series prediction. Fuzzy rules (rule-based models) are employed to describe and quantify the relationship between the input and output time series, while the HMM is regarded as a vehicle for efficiently capturing the temporal behavior or changes of the multivariate time series which are not capable to capture through commonly encountered fuzzy rule-based models. Essentially, the proposed strategies control the contribution of different fuzzy rules so that the proposed model can well model the dynamic behavior of time series. Fuzzy C-Means clustering technique is an alternative way to construct fuzzy rules. Particle swarm optimization serves as a tool to optimize the parameters of the model (e.g., transition matrix and emission matrix). We construct and investigate the performance of the HMM-based fuzzy model by using a series of synthetic and publicly available multivariate time series. Experimental results demonstrate that the proposed model shows better performance than the fuzzy rule-based models used without the involvement of HMMs. [ABSTRACT FROM AUTHOR]

In fuzzy time series, a way of representing their original numeric data through a collection of fuzzy sets plays a pivotal role and impacts the prediction performance of the constructed forecasting models. An evident shortcoming of most existing models is that fuzzy sets (their membership functions) are developed in an intuitive manner so that not all aspects of the time series could be fully captured. In this study, using an idea of axiomatic fuzzy set clustering we take the distribution of data into account to position time series in the framework of fuzzy sets. The obtained clusters exhibit well-defined semantics. To produce numeric results of forecasting, we develop a method to determine the prototypes based on the corresponding fuzzy description of the clusters. The commonly used enrollment time series is applied to demonstrate how the proposed method works. The experimental results exploiting the Taiwan Stock Exchange Capitalization Weighted Stock Index demonstrate that the proposed method can effectively improve forecasting accuracy. Furthermore, the proposed approach is of a general form and as such can be easily integrated with various fuzzy time series models. [ABSTRACT FROM AUTHOR]

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

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

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]

In this study, we are concerned with the formation of interpretable descriptors of dependencies existing in experimental data and realized in the form of linguistic entities (information granules). We elaborate on a way of bridging numerically inclined fuzzy models (in which information granules are built on a basis of experimental data and described through numeric membership functions) and a qualitative way of system modeling (originating from symbol-based modeling). Proceeding with the principles of fuzzy modeling (especially, those residing with rule-based architectures), their numerical constructs of fuzzy sets-information granules are augmented with viable interpretation mechanisms abstracted from the detailed membership functions. In this regard, a linguistic view of the outcomes of fuzzy clustering (realized in terms of fuzzy C-means) are revisited and supplied with an interpretation at a higher level of abstraction. The buildup of the qualitative descriptors embraces two essential aspects of abstraction, namely (a) an abstraction of linguistic terms realized on the basis of membership functions (fuzzy sets) and (b) an abstraction of relationships completed on the basis of detailed functional dependencies present in the fuzzy model (say, the rules of the model). Two categories of problems are studied in detail along with their applications, namely a linguistic description of time series and a linguistic description of linearization tasks. Both of them are illustrated with a number of experimental studies. [ABSTRACT FROM AUTHOR]

In this study we propose a new approach to granular modeling of time series. In contrast to the existing fuzzy set-based models of time series, we engage information granules in time (granulation resulting in temporal segments). This method subsequently gives rise to information granules formed in the representation space of the series (in particular, the space of amplitude and space of changes of amplitude). Initially the time series is approximated as the sequence of granules forming a so-called granular time series (GTS). To develop a forecasting (prediction) model of the GTS, we cluster all information granules and regard the centers of the clusters obtained through fuzzy clustering as the concepts of the fuzzy cognitive map (FCM). We propose a matching mechanism to carry out description of the GTS and form the results as a vector of the concepts’ activations. In this way the GTS is represented as the sequence of vectors of the concepts’ activations, which is forecasted by the FCM. At the conceptual level, the forecasted granule is the FCM concept associated with the maximal degree of activation. At the numeric level, the predicted granule regarded as a fuzzy set is described in terms of its bounds and modal value. Experimental studies involving publicly available real-world data demonstrate the usefulness and satisfactory 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, a paradigm of fuzzy clustering is augmented by available domain knowledge expressed in the form of relational constraints built with the aid of a collection of fuzzy sets. These constraints are described as a collection of Cartesian products of fuzzy sets or their logic expressions are used to form an augmented data space and transform nonlinearly original data. Depending upon the nature of the constraints, discussed are two categories of resulting representations (clustering spaces), namely homogeneous spaces (in case when the transformations are fully expressed by means of the constraints) and heterogeneous spaces (when the resulting space is composed of some original variables present in the initially available data space and those being transformed and expressed by means of satisfaction levels of the constraints). The role of information granules of order-2 is revealed with regard to results of clustering produced in the transformed space. A generalization of the proposed approach is also discussed in case the clustered data are not numeric but are provided in the form of information granules; in this case a special attention is paid to a way in which a representation (description) of information granules is realized through relational constraints. We elaborate on the formation of the space (induced by constraints) and original data as well as discuss the detailed algorithmic developments. [ABSTRACT FROM AUTHOR]

The study is focused on a development of a global structure in a family of distributed data realized on a basis of locally discovered structures. The local structures are revealed by running fuzzy clustering (Fuzzy C-Means), whereas building a global view is realized by forming global proximity matrices on a basis of the local proximity matrices implied by the partition matrices formed for the individual data sets. To capture the diversity of local structures, a global perspective at the structure of the data is captured in terms of a granular proximity matrix, which is built by invoking a principle of justifiable granularity with regard to the aggregation of individual proximity matrices. The three main scenarios are investigated: (a) designing a global structure among the data through building a granular proximity matrix, (b) refining a local structure (expressed in the form of a partition matrix) by engaging structural knowledge conveyed at the higher level of the hierarchy and provided in the form of the granular proximity matrix, (c) forming a consensus-building scheme and updating all local structures with the aid of the proximity dependences available at the upper layer of the hierarchy. While the first scenario delivers a passive approach to the development of the global structure, the two others are of an active nature by facilitating a structural feedback between the local and global level of the hierarchy of the developed structures. The study is illustrated through a series of experiments carried out for synthetic and publicly available data sets. [ABSTRACT FROM AUTHOR]

In spite of the impressive diversity of models of time series, there is still an acute need to develop constructs that are both accurate and transparent. Meanwhile, long-term time series prediction is challenging and of great interest to both practitioners and research community. The role of information granulation is to organize detailed numerical data into some meaningful, semantically sound entities. With this regard, the design of time series forecasting models used the information granulation is interpretable and easily comprehended by humans. In order to cluster information granules, a modified fuzzy c-means which does not require that data have the same dimensionality is proposed. Then, we develop forecasting model combining the modified fuzzy c-means and information granulation for solving the problem of time series long-term prediction. Synthetic time series, chaotic Mackey–Glass time series, power demand, daily temperatures, stock index, and wind speed are used in a series of experiments. The experimental results show that the proposed model produces better forecasting results than several existing models. [ABSTRACT FROM AUTHOR]

The study is concerned with a concept and a design of granular time series and granular classifiers. In contrast to the plethora of various models of time series, which are predominantly numeric, we propose to effectively exploit the idea of information granules in the description and classification of time series. The numeric (optimization-oriented) and interpretation abilities of granular time series and their classifiers are highlighted and quantified. A general topology of the granular classifier involving a formation of a granular feature space and the usage of the framework of relational structures (relational equations) in the realization of the classifiers is presented. A detailed design process is elaborated on along with a discussion of the pertinent optimization mechanisms. A series of experiments is covered leading to a quantitative assessment of the granular classifiers and their parametric analysis. [ABSTRACT FROM AUTHOR]

Clustering is a powerful vehicle to reveal and visualize structure of data. When dealing with time series, selecting a suitable measure to evaluate the similarities/dissimilarities within the data becomes necessary and subsequently it exhibits a significant impact on the results of clustering. This selection should be based upon the nature of time series and the application itself. When grouping time series based on their shape information is of interest (shape-based clustering), using a Dynamic Time Warping (DTW) distance is a desirable choice. Using stretching or compressing segments of temporal data, DTW determines an optimal match between any two time series. In this way, time series exhibiting similar patterns occurring at different time periods, are considered as being similar. Although DTW is a suitable choice for comparing data with respect to their shape information, calculating the average of a collection of time series (which is required in clustering methods) based on this distance becomes a challenging problem. As the result, employing clustering techniques like K-Means and Fuzzy C-Means (where the cluster centers – prototypes are calculated through averaging the data) along with the DTW distance is a challenging task and may produce unsatisfactory results. In this study, three alternatives for fuzzy clustering of time series using DTW distance are proposed. In the first method, a DTW-based averaging technique proposed in the literature, has been applied to the Fuzzy C-Means clustering. The second method considers a Fuzzy C-Medoids clustering, while the third alternative comes as a hybrid technique, which exploits the advantages of both the Fuzzy C-Means and Fuzzy C-Medoids when clustering time series. Experimental studies are reported over a set of time series coming from the UCR time series database. [ABSTRACT FROM AUTHOR]

Highlights: [•] Interval attribute weights are proposed and introduced for fuzzy clustering. [•] Genetic mechanism and gradient-based iteration constitute optimization strategy. [•] Data partition and weights can be obtained by minimizing the objective function. [•] Reasonable clustering results can be achieved more easily. [ABSTRACT FROM AUTHOR]

Abstract: In real-world problems we encounter situations where patterns are described by blocks (families) of features where each of these groups comes with a well-expressed semantics. For instance, in spatiotemporal data we are dealing with spatial coordinates of the objects (say, x–y coordinates) while the temporal part of the objects forms another collection of features. It is apparent that when clustering objects being described by families of features, it becomes intuitively justifiable to anticipate their different role and contribution to the clustering process of the data whereas the clustering is sought to be reflective of an overall structure in the data set. To address this issue, we introduce an agreement based fuzzy clustering—a fuzzy clustering with blocks of features. The detailed investigations are carried out for the well-known algorithm of fuzzy clustering that is fuzzy C-means (FCM). We propose an extended version of the FCM where a composite distance function is endowed with adjustable weights (parameters) quantifying an impact coming from the blocks of features. A global evaluation criterion is used to assess the quality of the obtained results. It is treated as a fitness function in the optimization of the weights through the use of particle swarm optimization (PSO). The behavior of the proposed method is investigated in application to synthetic and real-world data as well as a certain case study. [Copyright &y& Elsevier]

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]

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]

Several studies were devoted to the usage of the Fuzzy C-Means (FCM) algorithm to collaborative clustering, especially in the realm of data analysis, data mining, and pattern recognition. In this study, a novel interval-valued fuzzy set-based approach to realize collaborative clustering is presented. In collaborative clustering diagram, the local clustering results acquired locally (at a specific data site) impact clustering carried out at some other data sites. Those clustering methods endowed with interval-valued fuzzy sets help cope with uncertainties present in the data and the nature of the collaborative process itself. The validity indices such as fuzzy silhouette and SSE (Sum of Squared Error) are extended to quantify results produced by collaborative fuzzy clustering. Several experimental studies are presented using which we demonstrate the advantages of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper is concerned with a new architecture of a reinforced hybrid interval fuzzy neural networks (RHIFNN) classifier developed with aid of Fuzzy C-Means (FCM) clustering and Particle Swarm Optimization (PSO). The key objectives of this study concern the following: (a) selection of preprocessing techniques for the dimensionality reduction of input space. Linear discriminant analysis (LDA) or principal component analysis (PCA) algorithm forms a front end of the network to form the low-dimensional input variables. (b) The efficient process of dealing with uncertain information by interval type-2 fuzzy sets using Fuzzy C-Means (FCM) clustering. The premise (condition) part of the rules is realized by two FCM clustering algorithms, which are invoked by using different values of the fuzzification coefficient subsequently resulting in interval-valued type-2 membership functions. (c) The simultaneous structural and parametric optimization of network by evolutionary algorithm is completed. The parameters of the network including both the premise and consequent parts are optimized by means of the particle swarm optimization (PSO). The proposed classifier is applied to a variety of machine learning datasets and the results are compared with those provided by other classifiers reported in the literature. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel category of double fuzzy clustering-driven context neural networks (DFCCNNs). The study is focused on the development of advanced design methodologies for redesigning the structure of conventional fuzzy clustering-based neural networks. The conventional fuzzy clustering-based neural networks typically focus on dividing the input space into several local spaces (implied by clusters). In contrast, the proposed DFCCNNs take into account two distinct local spaces called context and cluster spaces, respectively. Cluster space refers to the local space positioned in the input space whereas context space concerns a local space formed in the output space. Through partitioning the output space into several local spaces, each context space is used as the desired (target) local output to construct local models. To complete this, the proposed network includes a new context layer for reasoning about context space in the output space. In this sense, Fuzzy C-Means (FCM) clustering is useful to form local spaces in both input and output spaces. The first one is used in order to form clusters and train weights positioned between the input and hidden layer, whereas the other one is applied to the output space to form context spaces. The key features of the proposed DFCCNNs can be enumerated as follows: (i) the parameters between the input layer and hidden layer are built through FCM clustering. The connections (weights) are specified as constant terms being in fact the centers of the clusters. The membership functions (represented through the partition matrix) produced by the FCM are used as activation functions located at the hidden layer of the “conventional” neural networks. (ii) Following the hidden layer, a context layer is formed to approximate the context space of the output variable and each node in context layer means individual local model. The outputs of the context layer are specified as a combination of both weights formed as linear function and the outputs of the hidden layer. The weights are updated using the least square estimation (LSE)-based method. (iii) At the output layer, the outputs of context layer are decoded to produce the corresponding numeric output. At this time, the weighted average is used and the weights are also adjusted with the use of the LSE scheme. From the viewpoint of performance improvement, the proposed design methodologies are discussed and experimented with the aid of benchmark machine learning datasets. Through the experiments, it is shown that the generalization abilities of the proposed DFCCNNs are better than those of the conventional FCNNs reported in the literature. [ABSTRACT FROM AUTHOR]

Data clustering has been widely applied to numerous real-world problems such as natural resource management, urban planning, and satellite image analysis. Especially, fuzzy clustering with its ability of handling uncertainty has been developed for image segmentation or image analysis e.g. in health image analysis, satellite image classification. Normally, image segmentation algorithms like fuzzy clustering use spatial information along with the color information to improve the cluster quality. This paper introduces an approach, which exploits local spatial information between the pixel and its neighbors to compute the membership degree by using an interval type-2 fuzzy clustering algorithm, called IIT2-FCM. Besides, a Semi-supervising Interval Type-2 Fuzzy C-Means algorithm using spatial information, called SIIT2-FCM, is proposed to move the prototype of clusters to the expected centroids which are pre-defined on a basis of available samples. The proposed algorithms are applied to the problems of satellite image analysis consisting of land cover classification and change detection. Experimental results are reported for various datasets of the LandSat7 imagery at multi-temporal points and compared with the results produced by some existing algorithms and obtained from some survey data. The clustering results assessed with regard to some validity indexes demonstrate that the proposed algorithms form clusters of better quality and higher accuracy in problems of land cover classification and change detection. [ABSTRACT FROM AUTHOR]

In this study, we introduce novel fuzzy neural networks designed with the aid of linear function prototype-driven fuzzy clustering (LFPFC) and layer reconstruction-based network design strategy to deal with the regression problem. The LFPFC constitutes a new clustering technique inspired by the fuzzy c-regression model (FCRM) clustering unlike fuzzy c-means (FCM) clustering LFPFC represents the prototypes of clusters as linear functions, and this can lead to more reliable data analysis of complex regression problems. We propose two types of LFPFC such as an estimated output-based LFPFC and a distance-based LFPFC. The estimated output-based LFPFC uses the output estimated on a basis of the simple model instead of the target output to calculate the centroid of LFPFC. A centroid of distance-based LFPFC is computed through the Euclidean distance between input data and the centroid of the cluster. By using two kinds of LFPFC approaches, we propose three different types of fuzzy neural networks: i) the fuzzy neural networks through layer reconstruction-based network design strategy consists of two models. The first model serves as an estimate of the desired output and the estimated output is used in the LFPFC of the second model. ii) In the fuzzy neural networks applied to the basic architecture of distance-based LFPFC, the hidden layer using the membership function changes to basic distance-based LFPFC, and the partition matrix obtained from LFPFC is used as the output of the hidden layer. iii) in the fuzzy neural network with the advanced architecture of distance-based LFPFC, an additional auxiliary layer is considered between the hidden and output layers to estimate the membership function of output space through LFPFC. In the experiments, we evaluate the performance index of the proposed models using publicly available machine learning datasets. The superiority of the proposed fuzzy neural networks designed by using LFPFC is demonstrated through the comparative analysis with the diverse regression models offered in the Weka data mining software. By conducting the Friedman test we show that the proposed model exhibits visible competitiveness from the viewpoint of performance. In addition, a real-world Portland cement dataset is dealt with to demonstrate the superiority of the models designed with the aid of LFPFC and reinforced layer reconstruction-based network design strategy. [ABSTRACT FROM AUTHOR]

Burn-through point (BTP) is an essential parameter in the iron ore sintering process. Operators usually judge whether the current production is stable by monitoring the BTP. It comes with significant application prospects to predict the BTP accurately. A prediction model of the BTP with fuzzy time series is designed in this paper. First, the fuzzy time series prediction method with the Fuzzy C-Means clustering is presented as the core modeling method. A prediction model of the response is constructed to obtain a timely response to the current BTP. The prediction model of the difference is established to estimate the present unmeasurable disturbance on the BTP. Then, a hybrid prediction model is built, which realizes the composition of these two models by an adjustment factor. Finally, a series of experiments is carried out using the raw time series data from an iron and steel plant. The experimental result shows that the designed model has better prediction performance for the BTP than existing models, which is an advantage resulting from the hybrid structure and the fuzzy time series prediction model with the Fuzzy C-Means clustering. This prediction model of the BTP implies the foundation for the stable control of the iron ore sintering process. • A fuzzy time series prediction model with the Fuzzy C-Means clustering is designed. • A hybrid prediction model of the burn-through point is developed. • The proposed model has a higher prediction accuracy than the existed models. [ABSTRACT FROM AUTHOR]