Reconstructing complex physical fields from sparse observations remains a central challenge in science and engineering. Conventional data-driven approaches often depend on large, high-quality datasets and typically overlook underlying physical principles, whereas physics-informed neural networks (PINNs) are constrained by the requirement of predefined governing equations, which are frequently difficult to specify or may fail to capture the true system dynamics. This work presents a physics-guided spectral neural network framework that autonomously infers governing partial differential equations (PDEs) directly from data, enabling mathematical formulations to emerge without prior specification. The framework integrates a multiresolution spectral neural architecture to effectively represent intricate patterns and sharp gradients, together with an adaptive loss training strategy that enforces consistency with the inferred dynamics. The proposed approach demonstrates high reconstruction accuracy across diverse benchmark cases, including combustion temperature fields, unsteady wake flows, 3-D turbulence, and geophysical datasets. Overall, the framework provides a systematic methodology for discovering and applying PDEs, offering a generalizable paradigm for reconstructing physical fields from sparse data.

ABSTRACT Time series clustering is an important research problem in machine learning and data mining. With the rapid increase in the amount of time series data, many traditional clustering algorithms cannot directly deal with large‐scale time series due to some limitations in the memory capacity and the execution time. In this study, we suggest a distributed information granulation method for large‐scale time clustering problem. First, a distributed time series partitioning method is designed to randomly divide the original time series dataset into some data blocks. Then, the distributed time series granulation method is developed in the map‐reduce framework by the principle of reasonable granularity, where each time series can be described by some representative data points to show the trend state information. Finally, we introduce the large‐scale time series clustering method in terms of the fuzzy C‐means clustering algorithm. The experimental studies demonstrate the feasibility and the effectiveness on several UCR publicly benchmark time series datasets. Compared with the classical clustering methods, the proposed method can achieve a 4.86–9.65% improvement in average clustering accuracy. Meanwhile, the proposed method exhibits more advantages in both unequal length time series clustering and execution time.

Despite the impressive few-shot performance of in-context learning (ICL) in knowledge-based visual question answering (VQA), existing research often prioritizes addressing the image information gap in VQA, while placing less emphasis on organizing appropriate demonstrations (e.g., in-context examples) to support this task. Recent studies, however, have shown that ICL performance is sensitive to the organization of demonstrations. To address this, we introduce axiomatic fuzzy set (AFS) theory into knowledge-based VQA, leveraging its unsupervised and interpretable nature to effectively organize demonstrations by describing each candidate with semantic concepts, thereby enhancing both the understanding and trustworthiness of the decision-making process. In this article, we propose AFSPrompt, a train-free example selection and ranking framework based on AFS theory for knowledge-based VQA tasks. After filtering irrelevant examples using multimodal embeddings, we apply AFS logic to integrate comparison information from candidates with multidimensional features. Furthermore, to reduce reliance on large-scale language model APIs such as OpenAI and facilitate model deployment, we employ a smaller 7B LLM as the knowledge engine to answer questions based on the optimized prompt. Through extensive evaluations of two datasets, we demonstrate the effectiveness of AFSPrompt within a lightweight pipeline for knowledge-based VQA tasks. Our code is publicly available at https://github.com/afs001/AFSPrompt.

RSI-based (remote sensing image) target detection holds a significant position in the field within the domain of visual computing. However, recognizing small target objects within RSI poses a significant hurdle due to their often-elusive nature. To confront this challenge, the paper introduces a novel neural network YOLO-PDNet. This network is an extension of YOLOv8, integrating a rapid neural architecture, proficient convolution operator, and a loss function based on normalized Wasserstein distance. This novel strategy seeks to improve the precision and effectiveness of target detection in remote sensing image. First, in order to boost the inference speed and minimize parameter load, we introduce a partial and effective convolution module (CPF) and enhanced SPPFCSPC module. Second, the Convolutional Depthwise Efficient Feature (CDEF) is designed to enhance feature extraction ability. Simultaneously, it ensures shorter inference time by utilizing separate convolutions. Finally, to alleviate the strong sensitivity of joint intersection (IoU)-based metrics to slight positional deviations of tiny objects, we apply the NWDL (normalized Wasserstein distance loss) loss function. Thorough comparisons and analysis showcase the heightened detection performance of our YOLO-PDNet algorithm, surpassing contemporary models in terms of detection precision. YOLO-PDNet has notably achieved an impressive 69.6% mAP on the DOTA v1.5 dataset and an outstanding 93.6% mAP on the NWPUVHR-10 dataset, surpassing YOLOv8 by an average of more than 1%. This significant enhancement in performance underscores the efficacy of the innovative YOLO-PDNet architecture in identifying targets within images captured using advanced remote sensing technology with high spatial detail.

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.

Open domain question answering (QA) refers to the model that can retrieve multiple supporting documents related to the answer from comprehensive knowledge bases. The difficulties lie in discovering the semantic logic relationship between supporting documents and providing explainable reasoning processes. Current retrieval models rely heavily on the scale of parameters in text embedding and iterative optimization, resulting in high training costs and a lack of interpretability. Given the challenges outlined above, this article proposes an unsupervised tree-shaped axiomatic fuzzy set (AFS) clustering model with semantic framework, tailored for open-domain answer prediction. Unlike traditional rule-based methods that require repeated iterations leading to complex computational overhead, the hierarchical single-feature clustering model proposed in this article achieves high retrieval efficiency and semantic interpretability. Moreover, a retrieval strategy based on tree-structured clustering semantic descriptions for unsupervised question alignment reasoning paths is introduced, effectively enhancing the capacity for intricate reasoning of the model. The application of the AFS clustering theory to information retrieval with the single feature tree clustering method is original. Experimental results on three open domain QA datasets show the superiority of the proposed model.

Knowledge graphs are regarded as structured knowledge bases that embody various facts coming from the real world. Their completeness is still far from satisfactory. Relational learning models in link prediction can automatically find the missing relationships between entities to increase the integrality of the knowledge bases, which form two categories purely embedding-based and hybrid embedding-based. Several models including HolE and RotatE belong to purely embedding-based with inefficient performers and few element interactions. Based on the above, this paper advances a novel Knowledge Graph Embedding relational model that leverages a circular correlation operation in the complex domain and dubs as CircularE, which increases interactions between entities and relations to a great extent by this compressed operator without expanding the dimension of space. It gives expression of the interactions between element semantics to achieve good performance in relational learning. Besides, a self-adaption adversarial negative sampling scheme is proposed on account of the KGs structure and the probability semantic of the triples. This negative sampler efficiently optimizes the knowledge representation capability of CircularE and far more than enhances the outputs of several relational original models in embedding-based. Experiments indicate that the competitive properties of CircularE on the four large-scale benchmarks of knowledge base completion tasks are superior to the state-of-the-art methods.

Open-domain question answering (QA) tasks require a model to retrieve inference chains associated with the answer from massive documents. The core of a QA model is the information filtering ability and reasoning ability. This article proposes a semantic knowledge reasoning graph model based on the multidimensional axiomatic fuzzy set (AFS), which can generate the knowledge graph (KG) and build reasoning paths for reading comprehension tasks through unsupervised learning. Moreover, taking advantage of the interpretable AFS framework enables the proposed model to have the ability to learn and analyze the semantic relationships between candidate documents. Meanwhile, the utilization of the multidimensional AFS acquires semantic descriptions of candidate documents more concise and flexible. The similarity degree between paragraphs is calculated according to the AFS description to generate the graph. Interpretable chains of reasoning provided by the AFS knowledge graph (AFS Graph) will serve as the basis for the answer prediction. Compared with the previous methods, the AFS Graph model presented in this article improves interpretability and reasoning ability. Experimental results show that the proposed model can achieve the state-of-the-art performance on datasets of HotpotQA, SQuAD, and Natural Questions Open.

Data classification problem in the field of machine learning and pattern recognition is an important research content, but the classic data classification algorithms mainly use numeric as the basis of classification modeling or put the high accuracy as the only index of classification model, and these classification algorithms usually are unable to construct classification model for easy understanding. As a basic abstract structure supporting human-centered granular computing methods, information granules can construct easy-to-understand geometric structures for different types of data. In order to solve the above problems, this paper proposes a classification algorithm based on AFS(axiomatic fuzzy set) and information granules to enhance the interpretability of models so that people can better understand the relationship between different types of data. The model can be divided into four stages. In the first stage, each class of data is divided into different data subsets. In the second stage, the AFS membership degrees of the corresponding prototypes are calculated based on axiomatic fuzzy set theory and a series of data blocks are generated on each class based on these degrees. The third stage mainly generates supersphere information granules according to the reasonable granularity principle theory and confidence levels. In the final stage, the activation level is calculated by the distance between samples and the hypersphere information granules and confidence levels to determine the "If-Then" rules.

Granular computing has been an intense research area over the past two decades, focusing on acquiring, processing, and interpreting information granules. In this study, we focus on the granulation of time series and discover the overall structure of the original time series by clustering the granular time series. During the granulation process, when time series exhibit some trend (up trend, equal trend, or down trend) or consist of a variety of tendencies, the trend is essential to be involved to construct the granular time series. Following the principle of justifiable granularity, we propose to form a series of trend-based information granules to describe the original time series and effectively reduce its dimensionality. Then, the similarity measure between trend-based information granules is provided, and considering the dynamic feature of time-series data, dynamic time warping (DTW) distance is generalized to measure the distance for granular time series. In sum, we show here a novel way of forming trend-based granular time series and the corresponding similarity measure, then based on this, the hierarchical clustering of granular time series is realized. The proposed approach can capture the main essence of time series and help to reduce the computing overhead. Experimental results show that the designed approach can reveal meaningful trend-based information granules, and provide promising clustering results on UCR and real-world datasets.

In this article, we present a semantic semisupervised learning (Semantic SSL) approach targeted at unifying two machine-learning paradigms in a mutually beneficial way, where the classical support vector machine (SVM) learns to reveal primitive logic facts from data, while axiomatic fuzzy set (AFS) theory is utilized to exploit semantic knowledge and correct the wrongly perceived facts for improving the machine-learning model. This novel semisupervised method can easily produce interpretable semantic descriptions to outline different categories by forming a fuzzy set with semantic explanations realized on the basis of the AFS theory. Besides, it is known that disagreement-based semisupervised learning (SSL) can be viewed as an excellent schema so that a co-training approach with SVM and the AFS theory can be utilized to improve the resulting learning performance. Furthermore, an evaluation index is used to prune descriptions to deliver promising performance. Compared with other semisupervised approaches, the proposed approach can build a structure to reflect data-distributed information with unlabeled data and labeled data, so that the hidden information embedded in both labeled and unlabeled data can be sufficiently utilized and can potentially be applied to achieve good descriptions of each category. Experimental results demonstrate that this approach can offer a concise, comprehensible, and precise SSL frame, which strikes a balance between the interpretability and the accuracy.

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.

Wang–Mendel (WM) fuzzy system is an effective and interpretable model for solving tabular data classification problem. However, original WM fuzzy system is weak in handling dataset with high dimensionality or large volume. Meanwhile, its capability of characterizing data is narrow, which results from lacking hierarchical transformation of features like deep learning-based models. In this article, we propose a deep fuzzy rule-based classification system (DFRBCS) based on improved WM method, in which fuzzy technique and deep learning strategy are combined to make a desirable tradeoff between model’s interpretability and prediction accuracy. We first redefine the consequent part of fuzzy rule in WM fuzzy system with class probability vector, which endows the improved WM fuzzy system with capacity for serving as building block of deep model. The model structure of DFRBCS is designed in layer-by-layer manner, where raw features can be transformed hierarchically. For every layer in DFRBCS, it contains many improved WM fuzzy systems whose input spaces are generated by shuffling and sliding window operation on concatenated outputs of fuzzy systems in previous layer. Comparative experiments are conducted on 45 real-world datasets with various sizes and dimensionality between our method, five baseline models, and the other deep fuzzy classifiers (D-TSK-FC, HID-TSK-FC, FCCI-TSK, DSA-FC, and MEEFIS). The experimental results show that DFRBCS is competitive in classification performance and promising in model’s interpretability.

Information granulation is an effective vehicle to explore data structure and has become a timely research topic. In this article, a top-down granulation model (T-GrM) adhering to the two fundamental requirements (coverage and specificity) in the principle of justifiable granularity is designed. Here, this principle is employed as a fundamental method for constructing information granules in each layer, and as an indicator to determine whether the obtained partitions need to be further divided in the next layer. For capturing the correlation among features and reflecting the data structure, an improved version of a T-GrM is offered by incorporating the principal component analysis. The proposed granulation models are evaluated on synthetic datasets, where the obtained experimental results offer some insights into the feasibility of the designed models and describe the effect of parameter values on the constructed information granules. The improved version of the T-GrM has the advantage of being transparent by generating information granules on several principal components, and thereby, the description of information granules is simpler and descriptive. As application examples, two real-world datasets are analyzed to exhibit the advantage of the constructed information granules.

Numerous learning methods for fuzzy cognitive maps (FCMs), such as the Hebbian-based and the population-based learning methods, have been developed for modeling and simulating dynamic systems. However, these methods are faced with several obvious limitations. Most of these models are extremely time consuming when learning the large-scale FCMs with hundreds of nodes. Furthermore, the FCMs learned by those algorithms lack robustness when the experimental data contain noise. In addition, reasonable distribution of the weights is rarely considered in these algorithms, which could result in the reduction of the performance of the resulting FCM. In this article, a straightforward, rapid, and robust learning method is proposed to learn FCMs from noisy data, especially, to learn large-scale FCMs. The crux of the proposed algorithm is to equivalently transform the learning problem of FCMs to a classic-constrained convex optimization problem in which the least-squares term ensures the robustness of the well-learned FCM and the maximum entropy term regularizes the distribution of the weights of the well-learned FCM. A series of experiments covering two frequently used activation functions (the sigmoid and hyperbolic tangent functions) are performed on both synthetic datasets with noise and real-world datasets. The experimental results show that the proposed method is rapid and robust against data containing noise and that the well-learned weights have better distribution. In addition, the FCMs learned by the proposed method also exhibit superior performance in comparison with the existing methods.

In this article, we propose a two-stage time-series clustering approach to cluster time series with different shapes. The first step is to represent the time series by a suite of information granules following the principle of justifiable granularity to perform dimensionality reduction, while the second step is to realize the fuzzy clustering of the time series in the transformed representation space (viz., the space of information granules). In the dimensionality reduction process, the numerical data are granulated using a collection of information granules forming a new sequence that can well describe the original time series. Then, when clustering the time series, dynamic time warping (DTW) is employed to measure the similarity between time series and DTW barycenter averaging (DBA) is generalized to weighted DBA to be involved in the fuzzy C -means (FCMs) algorithm. Finally, the experiments are conducted on the datasets coming from UCR time-series database and Chinese stocks to demonstrate the effectiveness and advantages of the proposed fuzzy clustering approach.

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

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

In this article, a novel interpretable classifier is proposed on the basis of axiomatic fuzzy set (AFS) theory and semantic entropy. AFS theory provides a unified and coherent way to deal with uncertainty of randomness and imprecision of fuzziness in data mining and knowledge discovery, which support many investigations in classification area. However, the existing AFS-based classifiers are weak in obtaining the optimal semantic description. To address this drawback, a new measure, named semantic entropy extended in Shannon's entropy, is developed to evaluate the discriminatory capabilities of semantic descriptions for each category. Moreover, the semantic entropy is utilized to design a classifier in the framework of AFS theory, called axiomatic fuzzy set and semantic entropy (AFSSE), which is capable of achieving sound classification performance and interpretability. Meanwhile, it provides a new framework of classifier design that can adapt more human-oriented recognition mechanisms. Furthermore, an evaluation index is used to prune descriptions to deliver a promising performance. Compared to the previous AFS-based classifiers, the proposed approach offers a semantic entropy to measure the information that is derived from semantic descriptions of data, so that the optimal semantic descriptions of each class can be obtained. For the purpose of illustrating the effectiveness of the classifier, several datasets are utilized to facilitate a comparative analysis of the proposed approach and other state-of-the-art classifiers. The experimental studies demonstrate that the proposed approach can achieve the semantic descriptions of each class and the performance of AFSSE is comparable with the performance of other approaches.

The learning of fuzzy cognitive maps (FCMs) is a timely issue pursued by numerous researchers. Many learning methods, such as population-based algorithms and some hybrid algorithms, have been developed and applied to many fields resulting in better performance. However, those learning algorithms also exhibit obvious limitations: some of them either are difficult to handle the learning problem of large-scale FCM or extremely time-consuming with high computational overload. Furthermore, the learning problem of FCM with noisy data is rarely considered in existing algorithms. In this article, a fast and efficient method for learning FCM is proposed. It first transforms the learning problem of the FCM into a convex optimization problem with constraints, and then, the classic interior-point methods are invoked to solve the optimization problem to obtain the optimized weight matrix of the FCM. A series of experiments involving synthetic data, noisy synthetic data, real data, and publicly available data with noise are considered to demonstrate that the proposed method can rapidly and efficiently learn small-scale and large-scale FCMs and deal with the problem of learning FCM from noisy historical data.

Summary Fuzzy decision trees are one of the most important extensions of decision trees for symbolic knowledge acquisition by fuzzy representation. Many fuzzy decision trees employ fuzzy information gain as a measure to construct the tree node splitting criteria. These criteria play a critical role in the construction of decision trees. However, many of the criteria can only work well on small‐scale or medium‐scale data sets, and cannot directly deal with large‐scale data sets on the account of some limiting factors such as memory capacity, execution time, and data complexity. Parallel computing is one way to overcome these problems; in particular, MapReduce is one mainstream solution of parallel computing. In this paper, we design a parallel tree node splitting criterion (MR‐NSC) based on fuzzy information gain via MapReduce, which is completed equivalent to the traditional unparallel splitting rule. The experimental studies verify the equivalency between the proposed MR‐NSC algorithm and the traditional unparallel way through 22 UCI benchmark data sets. Furthermore, the feasibility and parallelism are also studied on two large‐scale data sets.

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

Presents the introductory welcome message from the conference proceedings. May include the conference officers' congratulations to all involved with the conference event and publication of the proceedings record.

Summary To address the time‐consuming problem for the confirmation of splitting attributes and splitting points in classic rank mutual information based decision trees, this paper establishes a fast rank mutual information based decision tree (FRMIDT) for classification problems. First, the proposed FRMIDT algorithm improves the velocity by a max‐relevance and min‐redundancy criterion to remove the redundant attributes in each tree node building. Then, the fuzzy c‐means algorithm is employed to confirm the splitting points for further acceleration. Meanwhile, a parallel implementation is developed in the framework of Map‐Reduce (MR‐FRMIDT) for medium or large‐scale data classification. Several comparative studies are conducted on UCI benchmark data sets. In contrast to the classic rank mutual information based decision tree on 12 data sets, the proposed FRMIDT model effectively reduces the computational time on the premise of keeping testing accuracy. Furthermore, the proposed FRMIDT algorithm is comparable through comparing FRMIDT with other traditional decision tree classifiers including BFT, C4.5, LAD, NBT, and SC. Meanwhile, the comparison with 7 different popular splitting measures based monotonic decision trees on several data sets illustrates the effectiveness of FRMIDT in monotonic classification. At last, the experimental analysis on other 6 data sets shows that the proposed MR‐FRMIDT is feasible and has a good parallel performance on reducing execution time and avoiding memory restrictions.

In this study, we present a comprehensible classifier AFSNN that embeds a new type of coherence membership function, which builds upon the theoretical findings of the axiomatic fuzzy set (AFS) theory into the hidden layer of neural network with random weights (NNRWs). Borrowing from the idea of NNRWs that employs the random initialization technique, the relation among attributes, simple concepts, and complex concepts are randomly determined. Complex concepts are generated through the combination of randomly selected simple concepts by AFS logic operation. The output weights of NNRWs are utilized to evaluate the confidence of each complex concept for every target class, which means that the feasibility of complex concepts for every class is determined analytically rather than through the tuning parameters of constraint conditions such as in conventional AFS-based classifiers. For the proposed method, compared to other neural-network-based classification methods, the fuzzy descriptions generated from complex concepts in hidden layer make classification result human understandable. We have experimented with several benchmark datasets and compared the results with other neural network-based classifiers. We show that our method outperforms Ensemble, EvRBFN, NNEP, LVQ, and iRProp+ in the seven out of ten datasets. The results show that the performance of AFSNN is competitive in terms of classification accuracy and the network shows a distinctive capability of providing explicit knowledge in the form of linguistic description.

In time series forecasting, a challenging and important task is to realize long-term forecasting that is both accurate and transparent. In this study, we propose a long-term prediction approach by transforming the original numerical data into some meaningful and interpretable entities following the principle of justifiable granularity. The obtained sequences exhibiting sound semantics may have different lengths, which bring some difficulties when carrying out predictions. To equalize these temporal sequences, we propose to adjust their lengths by involving the dynamic time warping (DTW) distance. Two theorems are included to ensure the correctness of the proposed equalization approach. Finally, we exploit hidden Markov models (HMM) to derive the relations existing in the granular time series. A series of experiments using publicly available data are conducted to assess the performance of the proposed prediction method. The comparative analysis demonstrates the performance of the prediction delivered by the proposed model.

Summary In the supervised classification, large training data are very common, and decision trees are widely used. However, as some bottlenecks such as memory restrictions, time complexity, or data complexity, many supervised classifiers including classical C4.5 tree cannot directly handle big data. One solution for this problem is to design a highly parallelized learning algorithm. Motivated by this, we propose a parallelized C4.5 decision tree algorithm based on MapReduce (MR‐C4.5‐Tree) with 2 parallelized methods to build the tree nodes. First, an information entropy‐based parallelized attribute selection method (MR‐A‐S) on several subsets for MR‐C4.5‐Tree is proposed to confirm the best splitting attribute and the cut points. Then, a data splitting method (MR‐D‐S) in parallel is presented to partition the training data into subsets. At last, we introduce the MR‐C4.5‐Tree learning algorithm that grows in a top‐down recursive way. Besides, the depth of the constructed decision tree, the number of samples and the maximal class probability in each tree node are used as the termination conditions to avoid the over‐partitioning problem. Experimental studies show the feasibility and the good performance of the proposed parallelized MR‐C4.5‐Tree algorithm.

Linguistic variables can better approximate the fuzziness of man’s thinking, which are important tools for multiple attribute decision-making problems. This paper establishes the possibility-based ELECTRE II model under the environment of uncertain linguistic fuzzy variables and uncertain weight information. By introducing the degree of possibility to ELECTRE II model, the concordance set, the discordance set and the indifferent set are obtained, respectively. Furthermore, the concordance index is redefined by considering deviation index under the same attribute, by which the strong and weak relationships are constructed, and then the rank of alternatives is obtained. A numerical example about the evaluation of socio-economic systems is employed to illustrate the convenience and applicability of the proposed method.

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

This paper deals with the problem of observer design for continuous-time Takagi-Sugeno fuzzy models with unmeasurable premise variables. First, in order to improve the existing results of observer design, a new method is proposed to bound the time derivatives of the membership function. Then, by applying the nonquadratic Lyapunov function and the matrix decoupling technique, the controller gains and observer gains are designed to guarantee that the error system is asymptotically stale. Furthermore, better H ∞ performance can be obtained by solving an optimization problem. All of the results are presented as linear matrices inequalities and three examples are provided to demonstrate the merits of the proposed approach.

In this paper, a fuzzy c-means clustering algorithm based on interval-valued weights is proposed for improving clustering performance. In the proposed algorithm, the interval-valued weights are first constructed by synergy of the ReliefF algorithm and the analytic hierarchy process (AHP) method, and then they are transformed into a constraint condition associating with each weight variable in the weighted clustering objective function. In the sequence, the weighted clustering objective function is solved by combining the Lagrange multiplier method with the gradient-based iteration computation. In the whole process of algorithm iteration, a compulsion strategy with human–computer cooperation is adopted to ensure each weight variable satisfies interval constraint itself. Three well-known data set are used to perform profound experiments. Experimental results clearly show that the proposed algorithm has better clustering performance than other the weighted fuzzy c-means clustering algorithm.

In this study, a two-stage method which extracts fuzzy rules directly from samples is proposed for classification. First, we introduce a neighborhood based attribute significance algorithm to select r of the most important attributes from the original attribute set. Second, the proposed algorithm generates fuzzy rule from each sample described by the selected attribute subset and finally simplifies the returned fuzzy rule-base. A confidence degree is assigned for each of the extracted fuzzy rules by counting the number of training samples covered by the rule to solve the conflicts among the rules and then the rule-base is pruned. The performance of the proposed classification method have been compared with other five classification approaches including C4.5, DTable, OneR, NNge, and PART on seven UCI data sets. The experimental results show that the proposed method is better than other methods in two aspects: the higher classification accuracy and the smaller rule-base.

In spite of the truly remarkable diversity of models of time series, there is still an evident need to develop constructs whose accuracy and interpretability are carefully identified and reconciled subsequently leading to highly interpretable (human-centric) constructs. While a great deal of research has been devoted to the design of nonlinear numeric models of time series (with an evident objective to achieve high accuracy of prediction), an issue of interpretability (transparency) of models of time series becomes an evident and ongoing challenge. The user-friendliness of models of time series comes with an ability of humans to perceive and process abstract constructs rather than dealing with plain numeric entities. In perception of time series, information granules (which are regarded as realizations of interpretable entities) play a pivotal role. This gives rise to a concept of granular models of time series or granular time series, in brief. This study revisits generic concepts of information granules and elaborates on a fundamental way of forming information granules (both sets—intervals as well as fuzzy sets) through applying a principle of justifiable granularity encountered in granular computing. Information granules are discussed with regard to the granulation of time series in a certain predefined representation space (viz. a feature space) and granulation carried out in time. The granular representation and description of time series is then presented. We elaborate on the fundamental hierarchically organized layers of processing supporting the development and interpretation of granular time series, namely (a) formation of granular descriptors used in their visualization, (b) construction of linguistic descriptors used afterwards in the generation of (c) linguistic description of time series. The layer of the linguistic prediction models of time series exploiting the linguistic descriptors is outlined as well. A number of examples are offered throughout the entire paper with intent to illustrate the main functionalities of the essential layers of the granular models of time series.

Most researchers of time series forecasting devote to design and develop quantitative models for pursuing high accuracy of forecasting on the numerical level. However, in real world, the numerical accuracy is sometimes not necessary for human cognition and decision-making and the numerical results of forecasting based on quantitative model are deficient in interpretability, thus the development of qualitative forecasting model of time series becomes an evident challenge. In this paper, the improved fuzzy cognitive map (IFCM) are proposed first, and then it is applied to develop qualitative model for linguistic forecasting of time series together with fuzzy c-means clustering technology and real-coded genetic algorithm (RCGA). Two real life time series are used to test the developed forecasting model and compare with another method based on FCM, whose results show the developed FCM forecasting model is more simpler and high quality on the linguistic level.

In this paper, a novel method is developed for the stability problem of a class of neural networks with time-varying delay. New delay-dependent stability criteria in terms of linear matrix inequalities for recurrent neural networks with time-varying delay are derived by the newly proposed augmented simple Lyapunov-Krasovski functional. Different from previous results by using the first-order convex combination property, our derivation applies the idea of second-order convex combination and the property of quadratic convex function which is given in the form of a lemma without resorting to Jensen's inequality. A numerical example is provided to verify the effectiveness and superiority of the presented results.

With the rapid increase of Web Service applications, the reliability of web service and service composition has drawn particular attention from researchers and industries. Many methods for testing and verifying the reliability have been discussed, however, the existing methods are weak in test automation and therefore difficult in tackling the dynamic features of modern SOA based application. The traditional method of model checking and the technique of test suite generation have a large potential for the reliability verification of web services and service composition. In this paper, an approach to integrating the test suite generation technique with model checking is presented. The approach takes advantage of model checking to verify BPEL script at the logical level, and to generate test suite automatically based on the model description, and finally to select test cases with respect to the counterexamples of model checking output. The approach contributes a set of algorithms and its implementation to support a translation from BPEL to LOTOS, and LTS(Labeled Transition Systems, LTS in short) to TTCN(Test and Testing Control Notation, TTCN in short) behavior Tree. Finally, a case study is presented to demonstrate and verify the proposed approach.

Near sets result from a generalization of rough sets, which introduced by Peters in 2006, and later formally defined in 2007. Near set theory provides a new framework for representation of objects characterized by the features that describe them. AFS (Axiomatic Fuzzy Set) theory was proposed by Liu (1998), which is a semantic methodology relating to the fuzzy theory. In this paper, a new version of near sets based on AFS theory is established, in which every object has an AFS fuzzy description with definitely semantics. The proposed approach to assessing the nearness (closeness) of objects is not defined directly using a distance metric, but depend on similarity of their fuzzy descriptions. It is also a natural linguistic description that is similar to humans perception. Moreover, an approach to set approximation based on the union of families of objects with similar fuzzy descriptions is given. The near sets based on AFS theory can be viewed as a new development of near sets within the fuzzy context.

The notion of a rough set was originally proposed by Pawlak underwent a number of extensions and generalizations. Dubois and Prade (1990) introduced fuzzy rough sets which involve the use of rough sets and fuzzy sets within a single framework. Radzikowska and Kerre (2002) proposed a broad family of fuzzy rough sets, referred to as ( t)-fuzzy rough sets which are determined by some implication operator (implicator), and a certain t-norm. In order to describe the linguistically represented concepts coming from data available in some information system, the concept of fuzzy rough sets are redefined and further studied in the setting of the Axiomatic Fuzzy Set (AFS) theory. Compared with the ( t)-fuzzy rough sets, the advantages of AFS fuzzy rough sets are twofold. They can be directly applied to data analysis present in any information system without resorting to the details concerning the choice of the implication, t-norm and a similarity relation S. Furthermore such rough approximations of fuzzy concepts come with a well-defined semantics and therefore offer a sound interpretation. Some examples are included to illustrate the effectiveness of the proposed construct. It is shown that the AFS fuzzy rough sets provide a far higher flexibility and effectiveness in comparison with rough sets and some of their generalizations.

Nowadays, component-based development (CBD) has been widely used in software engineering. As a core activity in the CBD process, component retrieval has attracted extensive research attention. Despite the research efforts so far, one major problem still exists in component retrieval, that is, the mismatch between user's query and identified components. This paper proposes a semantics-base approach to tackle the mismatch issue in component retrieval process effectively. A review is first presented on the current state of relevant technologies involved in semantics-based component retrieval, and on the current development in this field. Component retrieval rating factors and their calculating method are proposed. The rating factors are used to illustrate and improve the precision of component retrieval in quantitatively.

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In the framework of axiomatic fuzzy sets theory, we first study how to impersonally and automatically determine the membership functions for fuzzy sets according to original data and facts, and a new algorithmic framework of determining membership functions and their logic operations for fuzzy sets has been proposed. Then, we apply the proposed algorithmic framework to give a new clustering algorithm and show that the algorithm is feasible. A number of illustrative examples show that this approach offers a far more flexible and effective means for the intelligent systems in real-world applications. Compared with popular fuzzy clustering algorithms, such as c-means fuzzy algorithm and k-nearest-neighbor fuzzy algorithm, the new fuzzy clustering algorithm is more simple and understandable, the data types of the attributes can be various data types or subpreference relations, even descriptions of human intuition, and the distance function and the class number need not be given beforehand.

In this paper, we put forward the AFS (Axiomatics Fuzzy Sets) fuzzy logic which is a new logic system based on AFS theory. Instead of using T-norm, S-norm and negative operator, AFS fuzzy logic is set up by uniform algorithm according to the original data of the real world problems. It is not only similar to human thinking logic, but also convenient for the computers to process and can convert information in database to the fuzzy sets which can been easily apprehended and utilized by human.

In this paper, the problems of quadratic stability conditions and H/sub /spl infin// control designs for Takagi-Sugeno (T-S) fuzzy systems have been studied. First, a new quadratic stability condition, which is more simple than that in a previous paper, has been proposed. Second, two new sufficient conditions in the terms of linear matrix inequalities (LMIs) which guarantee the existence of the state feedback H/sub /spl infin// control for the T-S fuzzy systems have been proposed. The conditions are not only simple but also consider the interactions among the fuzzy subsystems. Finally, based on the LMIs, the H/sub /spl infin// controller designing methods for the T-S fuzzy systems have been given.

This paper advocates that extracting formal specification semantically consistent to the original legacy system will facilitate further redesign and forward engineering greatly. The three parts of reengineering could be integrated on the basis of a wide spectrum language. The key approach to comprehension and the production of formal specification is a notion of abstraction. Transformation can help to change the original source code into alternative forms, but with the same semantics. Abstraction is often interpreted as the act of hiding irrelevant details. What constitute as relevant details is often left open to different interpretations. A unified approach for reverse engineering is described within which the notion of abstraction is classified and precisely defined. Abstraction rules are given and applied to various case studies. Copyright © 2000 John Wiley & Sons, Ltd.

We present a calculus which can transfer specifications to objects for the development of real-time systems. The object model is based on a practical OO development technique-HRT-HOOD. A real-time logic is specified by extending a sound formal method for real-time systems-TAM, to formalise the object model. With integration of HRT-HOOD and TAM, the advantages of object-oriented structured methods with the stepwise refinement techniques are combined. The result is illustrated on a case study.

It is widely accepted that reverse engineering has three components: restructuring, comprehension and production of formal specification. In this paper, we advocate that the three components could be achieved in a systematic approach by successfully applying a series of sound rules. The key approach to comprehension and the production of formal specification is a notion of abstraction. Abstraction is often interpreted as the act of hiding irrelevant details. What constitute as relevant details is often left open to different interpretations. A unified approach for reverse engineering is described within which the notion of abstraction is classified and precisely defined. Abstraction rules are given and applied to various small examples.

Discusses the current situation of formal methods and their use in the re-engineering of computing systems, especially real-time systems. Based on the analysis results, a solution which uses a consistent 4-sector Wide Spectrum Language (WSL) is proposed, which presently includes the general architecture and work flow, the structure of the Object-Action Model, the syntax and semantics of the Object-oriented Temporal Agent Model (ObTAM) and the Timed Guarded Command Language (TGCL). A small case study shows an optimistic future for our WSL technique. Further research will aim to build the complete semantic kernel of the WSL and its associated algebraic laws, including transformation rules and abstraction rules.

This paper presents a short review of formal methods and their use in the re-engineering of computing systems. Five classes of formal notations and theories, namely state/model-based, logic-based, algebraic-based, process algebra and net-based formalisms together with combined formalisms are considered, analyzed and compared. Conclusions obtained focus on proposing an approach to building a practical re-engineering system.