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The highly dimensional characteristic of time series brings many challenges on direct mining time series, such as high cost in time and space. Granular computing provides a potential strategy for representing and dealing with time series at a higher level of abstraction. In this study, we propose an information granulation-based weighted fuzzy ${C}$ -means (wFCM) method to realize time-series clustering, which could avoid high dimensionality processing and provide a concise and visible granular prototype for each cluster. In this method, each time series is first transformed into a series of information granules with trend following the principle of justifiable granularity. The formed granular time series can well capture the main features lying in the original time series and help realize dimensionality reduction. Then, the wFCM method is developed to complete time-series clustering in the granular space. Here, the dynamic time warping (DTW) is extended to capture the similarity for trend-based granular time series. Furthermore, the weighted DTW barycenter averaging is introduced to derive prototypes presented in a granular format, capturing the level, the fluctuation, and the changing trend, which are meaningful and understandable clustering results. The experiments conducted on real-world datasets coming from the UCR time-series database and Chinese stocks are presented to illustrate the effectiveness and practicality of the designed time-series clustering model.

Fuzzy clustering-based neural networks (FCNNs) based on information granulation techniques have been shown to be effective Takagi–Sugeno (TS)-type fuzzy models. However, the existing FCNNs could not cope well with sequential learning tasks. In this study, we introduce incremental FCNNs (IFCNNs), which could dynamically update themselves whenever new learning data (e.g., single datum or block data) are incorporated into the dataset. Specifically, we employ dynamic (incremental) fuzzy C-means (FCMs) clustering algorithms to reveal a structure in data and divide the entire input space into several subregions. In the aforementioned partition, the dynamic FCM adaptively adjusts the position of its prototypes by using sequential data. Due to the time-sharing arrival of training data, compared with batch learning models, incremental learning methods may lose classification (prediction) accuracy. In order to tackle this challenge, we utilize quasi-fuzzy local models (QFLMs) based on modified Schmidt neural networks to replace the popular linear functions in TS-type fuzzy models to refine and enhance the ability to represent the behavior of fuzzy subspaces. Meanwhile, the recursive least square error (LSE) estimation is utilized to update the weights of QFLMs from one-by-one or block-by-block (fixed or varying block size) learning data. In addition, the $L_{2}$ regularization is considered to ameliorate the deterioration of generalization abilities caused by potential overfitting when carrying out weight estimation. The proposed method leads to the construction of FCNNs in a new way, which can effectively deal with incremental data as well as deliver sound generalization capability. Extensive machine-learning datasets and a real-world application are employed to show the validity and performance of the presented methods. From the experimental results, we show that the proposal can maintain sound classification accuracy when effectively processing sequential data.

Information granularity and information granules are fundamental concepts that permeate the entire area of granular computing. With this regard, the principle of justifiable granularity was proposed by Pedrycz, and subsequently a general two-phase framework of designing information granules based on Fuzzy C-means clustering was successfully developed. This design process leads to information granules that are likely to intersect each other in substantially overlapping clusters, which inevitably leads to some ambiguity and misperception as well as loss of semantic clarity of information granules. This limitation is largely due to imprecise description of boundary-overlapping data in the existing algorithms. To address this issue, the rough $k$ -means clustering is introduced in an innovative way into Pedrycz’s two-phase information granulation framework, together with the proposed local boundary fuzzy metric. To further strengthen the characteristics of support and inhibition of boundary-overlapping data, an augmented parametric version of the principle is refined. On this basis, a local boundary fuzzified rough $k$ -means-based information granulation algorithm is developed. In this manner, the generated granules are unique and representative whilst ensuring clearer boundaries. The validity and performance of this algorithm are demonstrated through the results of comparative experiments.

Thanks to the efficient retrieval speed and low storage consumption, learning to hash has been widely used in visual retrieval tasks. However, the known hashing methods assume that the query and retrieval samples lie in homogeneous feature space within the same domain. As a result, they cannot be directly applied to heterogeneous cross-domain retrieval. In this article, we propose a generalized image transfer retrieval (GITR) problem, which encounters two crucial bottlenecks: 1) the query and retrieval samples may come from different domains, leading to an inevitable domain distribution gap and 2) the features of the two domains may be heterogeneous or misaligned, bringing up an additional feature gap. To address the GITR problem, we propose an asymmetric transfer hashing (ATH) framework with its unsupervised/semisupervised/supervised realizations. Specifically, ATH characterizes the domain distribution gap by the discrepancy between two asymmetric hash functions, and minimizes the feature gap with the help of a novel adaptive bipartite graph constructed on cross-domain data. By jointly optimizing asymmetric hash functions and the bipartite graph, not only can knowledge transfer be achieved but information loss caused by feature alignment can also be avoided. Meanwhile, to alleviate negative transfer, the intrinsic geometrical structure of single-domain data is preserved by involving a domain affinity graph. Extensive experiments on both single-domain and cross-domain benchmarks under different GITR subtasks indicate the superiority of our ATH method in comparison with the state-of-the-art hashing methods.

Since a noisy image has inferior characteristics, the direct use of Fuzzy ${C}$ -Means (FCM) to segment it often produces poor image segmentation results. Intuitively, using its ideal value (noise-free image) benefits FCM’s robustness enhancement. Therefore, the realization of accurate noise estimation in FCM is a new and important task. To date, only two noise-estimation-based FCM algorithms have been proposed for image segmentation, that is: 1) deviation-sparse FCM (DSFCM) and 2) our earlier proposed residual-driven FCM (RFCM). In this article, we make a thorough comparative study of DSFCM and RFCM. We demonstrate that an RFCM framework can realize more accurate noise estimation than DSFCM when different types of noise are involved. It is mainly thanks to its utilization of noise distribution characteristics instead of noise sparsity used in DSFCM. We show that DSFCM is a particular case of RFCM, thus signifying that they are the same when only impulse noise is involved. With a spatial information constraint, we demonstrate RFCM’s superior effectiveness and efficiency over DSFCM in terms of supporting experiments with different levels of single, mixed, and unknown noise.

In this study, we establish a new design methodology of granular models realized by augmenting the existing numeric models through analyzing and modeling their associated prediction error. Several novel approaches to the construction of granular architectures through augmenting existing numeric models by incorporating modeling errors are proposed in order to improve and quantify the numeric models’ prediction abilities. The resulting construct arises as a granular model that produces granular outcomes generated as a result of the aggregation of the outputs produced by the numeric model (or its granular counterpart) and the corresponding error terms. Three different architectural developments are formulated and analyzed. In comparison with the numeric models, which strive to achieve the highest accuracy, granular models are developed in a way such that they produce comprehensive prediction outcomes realized as information granules. In virtue of the granular nature of results, the coverage and specificity of the constructed information granules express the quality of the results of prediction in a more descriptive and comprehensive manner. The performance of the granular constructs is evaluated using the criteria of coverage and specificity, which are pertinent to granular outputs produced by the granular models.

Fuzzy utility (FU) pattern mining with an advantage in human reasoning has become one of the interesting topics in studies of knowledge discovery. The discovered information in FU pattern mining from real-life quantitative databases with item profits is suitable for interpreting data from a human perspective because it is not expressed using numerical values but linguistic terms which consist of natural languages. State-of-the-art approaches in this literature provide extended results by considering temporal factors, such as seasons, which can be influential in real-life situations. However, they still suffer from scalability issues because they are based on level-wise approaches which generate a number of candidates. In this article, we propose a scalable and efficient approach with a novel data structure for mining high temporal FU patterns without generating candidates. Efficient pruning techniques and algorithms are presented to improve the performance of the proposed approach. Performance experiments on both real and synthetic datasets show that the suggested algorithm has better performance than the state-of-the-art algorithms in terms of runtime, memory usage, and scalability.

This article aims to design a trend-oriented-granulation-based fuzzy $C$ -means (FCM) algorithm that can cluster a group of time series at an abstract (granular) level. To achieve a better trend-oriented granulation of a time series, ${l}_{1}$ trend filtering is firstly carried out to result in segments which are then optimized by the proposed segment merging algorithm. By constructing a linear fuzzy information granule (LFIG) on each segment, a granular time series which well reflects the linear trend characteristic of the original time series is produced. With the novel designed distance that can well measure the trend similarity of two LFIGs, the distance between two granular time series is calculated by the modified dynamic time warping (DTW) algorithm. Based on this distance, the LFIG-based FCM algorithm is developed for clustering time series. In this algorithm, cluster prototypes are iteratively updated by the specifically designed granule splitting and merging algorithm, which allows the lengths of prototypes to change in the process of iteration. This overcomes the serious drawback of the existing approaches, where the lengths of prototypes cannot be changed. Experimental studies demonstrate the superior performance of the proposed algorithm in clustering time series with different shapes or trends.

A visible trend in representing knowledge through information granules manifests in the developments of information granules of higher type and higher order, in particular, type-2 fuzzy sets and order-2 fuzzy sets. All these constructs are aimed at the formalization and processing data at a certain level of abstraction. Along the same line, in the recent years, we have seen intensive developments in fuzzy clustering, which are not surprising in light of a growing impact of clustering on fundamentals of fuzzy sets (as supporting ways to elicit membership functions) as well as algorithms (in which clustering and clusters form an integral functional component of various fuzzy models). In this study, we investigate order-2 information granules (fuzzy sets) by analyzing their formal description and properties to cope with structural and hierarchically organized concepts emerging from data. The design of order-2 information granules on a basis of available experimental evidence is discussed and a way of expressing similarity (resemblance) of two order-2 information granules by engaging semantically oriented distance is discussed. In the sequel, the study reported here delivers highly original contributions in the realm of order-2 clustering algorithms. Formally, the clustering problem under discussion is posed as follows: given is a finite collection of reference information granules. Determine a structure in data defined over the space of such granules. Conceptually, this makes a radical shift in comparison with data defined in the ${p}$ -dimensional space of real numbers Rp. In this situation, expressing distance between two data deserves prudent treatment so that such distance properly captures the semantics and consequently, the closeness between any two information granules to be determined in cluster formation. Following the proposal of the semantically guided distance (and its ensuing design process), we develop an order-2 variant of the fuzzy C-means (FCM), discuss its detailed algorithmic steps, and deliver interpretation of the obtained clustering results. Several relevant applied scenarios of order-2 FCM are identified for spatially and temporally distributed data, which deliver interesting motivating arguments and underline the practical relevance of this category of clustering. Experimental studies are provided to further elicit the performance of the clustering method and discuss essential ways of interpreting results.

This article proposes a new multiattribute group decision-making (MAGDM) method with probabilistic linguistic information that considers the following three aspects: an allocation of ignorance information, a realization of group consensus, and an aggregation of assessments. To allocate ignorance information, an optimization model based on minimizing the distances among experts is developed. To measure the consensus degree, a consensus index that considers the information granules of linguistic terms (LTs) is defined. On this basis, a suitable optimization model is established to realize the group consensus adaptively by optimizing the allocation of information granules of LTs with the particle swarm optimization (PSO) algorithm. With an objective to reduce the information loss during aggregation phases, the process of generating comprehensive assessments of alternatives with the evidential reasoning (ER) algorithm is presented. Therefore, a new method is developed based on the adaptive consensus reaching (ACR) model and the ER algorithm. Finally, the applicability of the proposed method is demonstrated by solving a selection problem of a financial technology company. Comparative analyses are conducted to show the advantages of the proposed method.

Network embedding which aims to learn a low dimensional representation of nodes is a powerful technique for network analysis. While network embedding for networks with complete attributes has been widely investigated, in many real-world applications the attributes of partial nodes are unobserved (i.e., missing) due to privacy concern or resource limit. Very recently, several network embedding methods have been proposed for attribute-missing networks. They first complete the missing attributes and then use the complemented network to learn network embedding. The parameters of these two processes cannot be adjusted by each other, resulting in compromised results. To address this problem, we propose a unified model in which the process of completing missing attributes and the process of learning embedding are not separated but closely intertwined. Being specific, completing missing attributes is under the guidance of learning network representation via mutual information maximization, and the complemented attributes directly enter network representation module which will generate further feedback for completing missing attributes. We further impose attribute-structure relationship constraint for completing missing attributes by designing a new generative adversarial networks (GANs) model. To the best of our knowledge, this is the first unified model for attribute-missing network embedding. Empirical results on real-world datasets show the superiority of our new method over other state-of-the-art methods on four network analysis tasks, including node classification, node clustering, link prediction, and network visualization.

A key issue in social network group decision making (SNGDM) is to determine the weights (i.e., social influences) of individuals. Notably, in some SNGDM scenarios, the social influences of individuals may evolve over time. Meanwhile, consensus reaching is another important issue in SNGDM. In this article, we are dedicated to disclosing the natural evolution process of social influence, and further to discussing the consensus reaching issue in SNGDM. First, we establish the social influence evolution model, where the individual’s social influence is obtained by combining his/her intrinsic influence and network influence. Afterward, we design the consensus reaching process based on social influence evolution (CRP-SIE) to assist the individuals to reach a consensus. Furthermore, we use a hypothetical application to show the applicability of the proposed CRP-SIE. Finally, simulation analysis is adopted to investigate the effects of social influence evolution on consensus reaching in SNGDM, and comparative analysis is conducted to demonstrate the advantages of our proposal.

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

Consensus reaching process (CRP) is a key topic in the area of group decision making (GDM). When the consensus level is not high enough, it becomes necessary to adjust the original opinions of decision makers (DMs). To offer the adjustment reference for DMs, we build the programming models to determine the minimum modification to be carried out from the individual and global perspectives. Meanwhile, all DMs are divided into two subgroups: DMs with acceptable and unacceptable consensus levels. If some DMs with unacceptable consensus level do not accept the relevant modifications, the Nash bargaining game-based programming model is built for the fairness and efficiency of modifications. When some DMs refuse to make any modifications or tend to modify the opinions in their way, with respect to different group consensus situations, we make the minimum hybrid penalty mechanism by the Nash bargaining game-based programming models. For each case, we determine the corresponding optimal modification mechanism in view of the fixed individual total modification and the maximum consensus level. Furthermore, we study the arrangements of weights of DMs according to their cardinal and ordinal consensus contributions. Based on these results, we present a new algorithm and illustrate its application by a numerical example. Moreover, we carry out the sensitivity and comparison analysis. We summarize the conclusions and future research directions in the end. The main originality of the new method includes: the fairness and efficiency of modifications, and the determination of the hybrid penalty mechanism.