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In this paper, an integrated model combining interval deep belief network (IDBN) and neural network with nonlinear weights, called IDBN-NN, is proposed for interval-valued data modeling. Firstly, the IDBN with variable learning rate is designed to initialize parameters of each sub-model. Based on a modified contrastive divergence algorithm the least square method is adopted to identify the coefficients of nonlinear weights in the output layer. Then, to improve the modeling accuracy, the Fuzzy C-Means (FCM) method and the Particle Swarm Optimization (PSO) algorithm are applied to tune the weights of sub-models. Though each sub-model can capture the nonlinear feature of the original system, by intersecting cut sets the synthesizing modeling scheme can further improve the performance of the proposed model. Some numerical examples show that the IDBN-NN with nonlinear output structure can achieve higher accuracy than some interval-valued data modeling methods. [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]

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]

In the framework of Axiomatic Fuzzy Set (AFS) theory, we propose a new approach to data clustering. The objective of this clustering is to adhere to some principles of grouping exercised by humans when determining a structure in data. Compared with other clustering approaches, the proposed approach offers more detailed insight into the cluster's structure and the underlying decision making process. This contributes to the enhanced interpretability of the results via the representation capabilities of AFS theory. The effectiveness of the proposed approach is demonstrated by using real-world data, and the obtained results show that the performance of the clustering is comparable with other fuzzy rule-based clustering methods, and benchmark fuzzy clustering methods FCM and K -means. Experimental studies have shown that the proposed fuzzy clustering method can discover the clusters in the data and help specify them in terms of some comprehensive fuzzy rules. [ABSTRACT FROM AUTHOR]

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

Abstract: Fuzzy clustering being focused on the discovery of structure in multivariable data is of relational nature in the sense of not distinguishing between the natures of the individual variables (features) encountered in the problem. In this study, we revisit the generic approach to clustering by studying situations in which there are families of features of descriptive and functional nature whose semantics needs to be incorporated into the clustering algorithm. While the structure is determined on the basis of all features taken en-block, it is anticipated that the topology revealed in this manner would aid the effectiveness of determining values of functional features given the vector of the corresponding descriptive features. We propose an augmented distance in which the families of descriptive and predictive features are distinguished through some weighted version of the distance between patterns. The optimization of this distance is guided by a reconstruction criterion, which helps minimize the reconstruction error between the original vector of functional features and their reconstruction realized by means of descriptive features. Experimental results are offered to demonstrate the performance of the clustering and quantify the effect of reaching balance between semantically distinct families of features. [ABSTRACT FROM AUTHOR]

Granular Computing (GrC) is a human-centric way to discover the fundamental structure of data sets. The resulting information granules can be efficiently exploited to organize knowledge and reveal data descriptions, which can play a pivotal role in the classification problems. Furthermore, information granules are abstract collections of data entities and exhibit flexibility and tolerance when it comes to the representation of incomplete data. However, most of the existing methods focused on the data imputation and classification separately. They also require better interpretability. The crux of this study is to develop a novel information granule-based classification method for incomplete data and a way of representing missing entities and regarding them as information granules in a unified framework. The first aspect focuses on revealing the structural backbone of multiple labeled subspaces of data by fuzzy clustering of missing values. It emerges a classifier with interpretable "IF-THEN" rules by the refinement of fuzzy prototypes in a supervised mode to capture the critical relationship of the multi-class incomplete data. The second aspect concerns the construction of some information granules to impute and represent missing values according to the refined prototypes and classification findings. The experimental studies involved synthetic and publicly available datasets in quantifying the advantages of the classification and representation abilities of the proposed methods on incomplete data. • We proposed a novel information granule-based classifier to reveal the structural of the subspaces of data, which is easier to be interpreted. • We considered incomplete data among classification modeling, so that this classifier can deal with the incomplete data straightforwardly. • We present a refinement mechanism for the information granule-based classifier to optimize the prototypes of the classification rules. • As a byproduct of the classification, the imputed information granules are distinguished from present data and have more tolerance to the imputation error. [ABSTRACT FROM AUTHOR]