Axiomatic Fuzzy Set Research Articles

Granular computing-based time series anomaly pattern detection with semantic interpretation

Time series analysis may suffer from the “curse of dimensionality” due to its high-dimensionality characteristics. In terms of this issue, information granulation offers an effective vehicle to process time series at a higher level of abstraction. To take this advantage, this study conducts an interpretable time series anomaly pattern detection under the framework of granular computing, where each time series is granulated into a series of semantics according to its patterns and the anomaly ones are identified based on the obtained semantics. First, the time series is partitioned into a predefined number of segments and trend-based information granules are formed for the data points in each time interval. Guided by the principle of justifiable granularity, the granular results maintaining the main features of the original time series realize informative feature extraction and meaningful dimensionality reduction. Then, to realize semantic anomaly pattern detection, the Axiomatic Fuzzy Set (AFS) theory is generalized to construct and compute with semantic time series, and an AFS-based anomaly score is proposed to discover anomaly patterns. In the experiments, the proposed method is conducted on both UCR data sets and real-world time series, where the detected anomaly patterns are equipped with well-defined semantics.

The fuzzy inference system based on axiomatic fuzzy sets using overlap functions as aggregation operators and its approximation properties

The fuzzy inference system based on axiomatic fuzzy sets using overlap functions as aggregation operators and its approximation properties

A Novel Fuzzy Bi-Clustering Algorithm with Axiomatic Fuzzy Set for Identification of Co-Regulated Genes

The identification of co-regulated genes and their Transcription-Factor Binding Sites (TFBSs) are the key steps toward understanding transcription regulation. In addition to effective laboratory assays, various bi-clustering algorithms for the detection of the co-expressed genes have been developed. Bi-clustering methods are used to discover subgroups of genes with similar expression patterns under to-be-identified subsets of experimental conditions when applied to gene expression data. By building two fuzzy partition matrices of the gene expression data with the Axiomatic Fuzzy Set (AFS) theory, this paper proposes a novel fuzzy bi-clustering algorithm for the identification of co-regulated genes. Specifically, the gene expression data are transformed into two fuzzy partition matrices via the sub-preference relations theory of AFS at first. One of the matrices considers the genes as the universe and the conditions as the concept, and the other one considers the genes as the concept and the conditions as the universe. The identification of the co-regulated genes (bi-clusters) is carried out on the two partition matrices at the same time. Then, a novel fuzzy-based similarity criterion is defined based on the partition matrices, and a cyclic optimization algorithm is designed to discover the significant bi-clusters at the expression level. The above procedures guarantee that the generated bi-clusters have more significant expression values than those extracted by the traditional bi-clustering methods. Finally, the performance of the proposed method is evaluated with the performance of the three well-known bi-clustering algorithms on publicly available real microarray datasets. The experimental results are in agreement with the theoretical analysis and show that the proposed algorithm can effectively detect the co-regulated genes without any prior knowledge of the gene expression data.

Hierarchical knowledge graph relationship prediction leverage of axiomatic fuzzy set graph structure

Knowledge graph embedding has found widespread application across various fields due to its inherent structured data. However, some non-graph-based models necessitate improvement in accurately describing complex hierarchical structures and connecting intricate relational paths. Graph neural networks embedding methods provide a promising direction for exploring complex hierarchical modeling, facilitating the flow of information from nodes along relational paths. Nevertheless, the deficient interpretable characterization by the saliency estimation in entities, which contributes to the efficient performance of the model, is difficult to comprehend. This paper proposes a novel model to mitigate the described phenomenon by intensifying the original structure of knowledge graphs. Specifically, it integrates the Axiomatic Fuzzy Set entity semantic extraction framework with a heterogeneous relationship self-attention mechanism to bolster node representation. Next, this framework incorporates a convolutional neural network model to increase the interaction capabilities of entities and relations. This interpretable entity-level attention mechanism complements the structural information of the knowledge graphs, enabling more comprehensive neighbor information capture and improving the ability to identify the correct entity location. Experimental validation on two relationally complex datasets demonstrates superior performance compared to current methods, obtaining a new Hit@1 value of 45.6% on WN18RR.

Interpretable classifier design by axiomatic fuzzy sets theory and derivative-free optimization

Axiomatic fuzzy sets (AFS) theory has seen extensive application in classification tasks in recent years. Yet, efficiently generating precise fuzzy descriptions for each target class from a multitude of complex concepts is challenging. This study aims to bridge this gap by transforming the challenge into a combinatorial optimization problem. We adopt an advanced model-based evolutionary optimization method, Randomized Coordinate Shrinking Classification (RACOS), to create an interpretable classifier within the AFS framework. By refining the definition of the complex concept set in AFS theory, we define a feasible search space for the optimization method. Innovative fitness functions have been developed focusing on semantic discrimination and prediction accuracy. Concurrently, we establish an encoding–decoding mechanism to link the solution vector with the complex concept for each fitness function. Ultimately, the complex concepts, guided by various fitness functions, are integrated into the class’s fuzzy description using AFS logical operations. Our method demonstrates competitive classification performance and superior interpretability compared to other evolutionary fuzzy rule-based classifiers.

AFS Graph: Multidimensional Axiomatic Fuzzy Set Knowledge Graph for Open-Domain Question Answering.

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.

Rough set model based on axiomatic fuzzy set

Both fuzzy set and rough set are important mathematical tools to describe incomplete and uncertain information, and they are highly complementary to each other. What is more, most fuzzy rough sets are obtained by combining Zadeh fuzzy sets and Pawlak rough sets. There are few reports about the combination of axiomatic fuzzy sets and Pawlak rough sets. For this reason, we propose the axiomatic fuzzy rough sets (namely rough set model with respect to the axiomatic fuzzy set) establishing on fuzzy membership space. In this paper, we first present a similarity description method based on vague partitions. Then the concept of similarity operator is proposed to describe uncertainty in the fuzzy approximation space. Finally, some characterizations concerning upper and lower approximation operators are shown, including basic properties. Furthermore, we give a algorithm to verify the effectiveness and efficiency of the model.

Fuzzy vs. Traditional Reliability Model for Inverse Weibull Distribution

In this paper, fuzzy stress strengths RF=P(Y≺X) and traditional stress strengths R=P(Y<X) are considered and compared when X and Y are independently inverse Weibull random variables. When axiomatic fuzzy set theory is taken into account in the stress–strength inference, it enables the generation of more precise studies on the underlying systems. We discuss estimating both conventional and fuzzy models of stress strength utilizing a maximum product of spacing, maximum likelihood, and Bayesian approaches. Simulations based on the Markov Chain Monte Carlo method are used to produce various estimators of conventional and fuzzy dependability of stress strength for the inverse Weibull model. To generate both conventional and fuzzy models of dependability, we use the Metropolis–Hastings method while performing Bayesian estimation. In conclusion, we will examine a scenario taken from actual life and apply a real-world data application to validate the accuracy of the provided estimators.

Rule-based models via the axiomatic fuzzy set clustering and their granular aggregation

Rule-based models have become a popular way to represent and analyze the main knowledge residing in data because of the increasing complexity and uncertainty. For extracting semantically sound rules in a multi-view perspective, a novel rule-based model combined with the axiomatic fuzzy set (AFS) algorithm is developed in distributed systems. Using the idea of the AFS algorithm, several clusters and accompanying fuzzy descriptions are formed in terms of data distribution; thereby, the input of rules exhibits well-defined semantics by the logic compound of the predefined linguistic terms. The output of the rule is approximated by the Takagi–Sugeno (T–S) model, in which an extended weighted least squares method is designed to optimize the parameter vectors simultaneously. In virtue of the diversity of these individual results, a granular aggregation procedure incorporates the weighted principle of justifiable granularity to summarize all the local results into compact and meaningful descriptors (information granules). Five experiments considering synthetic and publicly available datasets are carried out to demonstrate the performance of the proposed approach. In addition, the proposed approach is also shown effective in an application involving the credit dataset.

Multi-level retrieval with semantic Axiomatic Fuzzy Set clustering for question answering

This paper considers open-domain and multi-hop reading comprehension tasks that require complex multi-step reasoning processes. The study is particularly challenging because it requires a model to learn to explore “bridge” information to connect text snippets relevant to the answer. Unlike the usual neural-network-based retrieval models, which are difficult to interpret, this paper proposes a coarse-to-fine unsupervised evidence sentences retrieval model based on the Axiomatic Fuzzy Sets clustering with both reasoning ability and interpretability. According to the entities that appeared in a question, a chained inference retrieval was carried out to get the coarser candidate documents from knowledge bases. Then, sentence-level multi-feature scoring rules based on the part of speech and grammar are proposed. The Axiomatic Fuzzy Sets clustering algorithm based on the feature scores selects finer and sentence-level evidence by semantic descriptions. The retrieval process of the candidate sentences is unsupervised and straightforward, which does not require word embedding. Our model achieves state-of-the-art results in three open-domain QA datasets: HotpotQA, SQuAD Open and Natural Questions Open.

Hierarchical Axiomatic Fuzzy Set Granulation for Financial Time Series Clustering

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.

基于公理化模糊集合的投影相似度多属性决策方法

基于公理化模糊集合的投影相似度多属性决策方法

Semisupervised Learning via Axiomatic Fuzzy Set Theory and SVM.

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.

Facial expression description and recognition based on fuzzy semantic concepts

This paper proposes a novel approach to identifying various expressions using semantic concepts. Based on the framework of the axiomatic fuzzy set, facial features are transformed into semantic concepts, which are then considered as a ruleset to differentiate expression categories. This method has two main advantages. First, it bridges the descriptors between image features and semantic concepts, according to which facial geometric features can be mirrored directly. Second, it alters the description patterns of fuzzy rulesets, which can reduce the dimensionality of expression features. We establish optimization criteria for selecting salient semantic concepts to represent expression characteristics. Experiments are conducted using the proposed method on the FEI and CK+ databases. Semantic concepts are considered as a ruleset to describe the differences between various expressions. The performances of state-of-the art classifiers and the proposed method are compared and analyzed. The results demonstrate that the proposed method provides excellent interpretability and classification performance for facial expressions.

An advisable facial semantic characterization based on Axiomatic Fuzzy Set theory and information granules

With the ever-increasing popularity of social networks, a colossal amount of images are being uploaded to the digital world including human faces, and semantic facial descriptors would be very useful to describe some facial action units, which can be further used for affection analysis. In this paper, a novel facial semantic characterization method via Axiomatic Fuzzy Set (AFS) clustering scheme and information granules is proposed. Firstly, semantic descriptions for each facial component are extracted in the framework of the AFS theory. Then in order to further characterize the semantics obtained by the AFS clustering, the information granules are created around the constructed prototypes, that can specify the merits of the semantic facial descriptors. Multiple experiments on the Multi-PIE and BU-4DFE facial databases demonstrate that the proposed facial semantic characterization method not only can obtain more accurate and interpretable descriptions, but also the results are much closer to human perception.

The selection of wart treatment method based on Synthetic Minority Over-sampling Technique and Axiomatic Fuzzy Set theory

Wart disease is a kind of skin illness that is caused by Human Papillomavirus (HPV). Many medical studies are being carried out with the aid of machine learning and data mining techniques to find the most appropriate and effective treatment for a specific wart patient. However, the imbalanced distribution of medical data may lead to misclassification in this field. The purpose of this paper is to propose a algorithm to predict the response of the patients towards a specific treatment and choose an appropriate treatment method. In this paper, Synthetic Minority Over-sampling (SMOTE) method is adopted to deal with the unbalanced data and combined with Axiomatic Fuzzy Set (AFS) theory to predict whether patients can respond to treatment or not. Compared with other existing approaches, the proposed approach can provide descriptive information of the patients which can help to predict the response towards the treatment with an average prediction accuracy of 97.63% and 92.33% for cryotherapy and immunotherapy data, respectively. Furthermore, the experimental results demonstrate that it can assist doctors in treatment, save medical resources and improve the quality of treatment.

The Semantic Classification Approach Base on Neural Networks

This paper addresses the classification problem, and a semantic classification approach using neural networks is proposed. The approach embeds the theoretical findings of the axiomatic fuzzy set theory in neural networks. Complex concepts are extracted by neural networks, which means that the class description is formed analytically rather than by tuning parameters of constraint conditions. The experiments are carried out on five benchmark datasets and compared results with five other neural network-based classifiers. The experimental results show that the proposed approach produces high classification accuracy and has a better explanation.

Axiomatic fuzzy set theory-based fuzzy oblique decision tree with dynamic mining fuzzy rules

This paper proposes a novel classification technology—fuzzy rule-based oblique decision tree (FRODT). The neighborhood rough sets-based FAST feature selection (NRS_FS_FAST) is first introduced to reduce attributes. In the axiomatic fuzzy set theory framework, the fuzzy rule extraction algorithm is then proposed to dynamically extract fuzzy rules. And these rules are regarded as the decision function during the tree construction. The FRODT is developed by expanding the unique non-leaf node in each layer of the tree, which results in a new tree structure with linguistic interpretation. Moreover, the genetic algorithm is implemented on $$\sigma $$ to obtain the balanced results between classification accuracy and tree size. A series of comparative experiments are carried out with five classical classification algorithms (C4.5, BFT, LAD, SC and NBT), and recently proposed decision tree HHCART on 20 UCI data sets. Experiment results show that the FRODT exhibits better classification performance on accuracy and tree size than those of the rival algorithms.

AFSSE: An Interpretable Classifier With Axiomatic Fuzzy Set and Semantic Entropy

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.

Notice of Retraction: Interpretable Facial Semantic Extractor Based on Axiomatic Fuzzy Set

The main purpose of this paper is to label one face to which it refers using a commonly accepted concept or linguistic term that corresponds to a given face class. Therefore, a facial semantic description method based on the axiomatic fuzzy set (AFS) theory is proposed. Firstly, the facial geometric features are defined based on the landmarks, such as distance features, angle features and scale features, etc. And then the salient features are selected via AFS feature selection method, and the fuzzy linguistic terms are defined the selected features. Thirdly, the facial descriptions which can characterize the salient characteristics are extracted via a clustering scheme in the framework of AFS theory. Multiple experiments are done on Chinese ethnic face database including Korean, Uyghur and Zhuang. The experimental results demonstrate that AFS feature selection can obtain higher accuracies via AFS clustering algorithm for each ethnic group than that via mRMR. Furthermore, compared with k-means, FCM and Minmax k-means, AFS clustering algorithm obtains the highest accuracies for each ethnic group, which verifies the efficacy of the proposed semantic extractor. Consequently, one comparative experimental results verify the consistency between the extracted semantic description method and the facial physical characteristics in physical anthropology. Also the facial semantics would be very useful to further describe some facial action units or facial expressions, which can be further used for affection analysis of images as well as semi-automatic action unit/facial expression annotation.