Fuzzy Partition Research Articles

Fuzzy Coalition Graphs: A Framework for Understanding Cooperative Dominance in Uncertain Networks

In a fuzzy graph G, a fuzzy coalition is formed by two disjoint vertex sets V1 and V2, neither of which is a strongly dominating set, but the union V1∪V2 forms a strongly dominating set. A fuzzy coalition partition of G is defined as Π={V1,V2,⋯,Vk}, where each set Vi either forms a singleton strongly dominating set or is not a strongly dominating set but forms a fuzzy coalition with another non-strongly dominating set in Π. A fuzzy graph with such a fuzzy coalition partition Π is called a fuzzy coalition graph, denoted as FG(G,Π). The vertex set of the fuzzy coalition graph consists of {V1,V2,⋯,Vk}, corresponding one-to-one with the sets of Π, and the two vertices are adjacent in FG(G,Π) if and only if Vi and Vj are fuzzy coalition partners in Π. This study demonstrates how fuzzy coalition graphs can model and optimize cybersecurity collaborations across critical infrastructures in smart cities, ensuring comprehensive protection against cyber threats. This study concludes that fuzzy coalition graphs offer a robust framework for optimizing collaboration and decision-making in interconnected systems like smart cities.

Efficient algorithms for computing bisimulations for nondeterministic fuzzy transition systems

Nondeterministic fuzzy transition systems (NFTSs) offer a robust framework for modeling and analyzing systems with inherent uncertainties and imprecision, which are prevalent in real-world scenarios. Wu et al. (2018) provided an algorithm for computing the crisp bisimilarity (the greatest crisp bisimulation) of a finite NFTS S=〈S,A,δ〉, with a time complexity of order O(|S|4⋅|δ|2) under the assumption that |δ|≥|S|. Qiao et al. (2023) provided an algorithm for computing the fuzzy bisimilarity (the greatest fuzzy bisimulation) of a finite NFTS S under the Gödel semantics, with a time complexity of order O(|S|4⋅|δ|2⋅l) under the assumption that |δ|≥|S|, where l is the number of fuzzy values used in S plus 1. In this work, we provide efficient algorithms for computing the partition corresponding to the crisp bisimilarity of a finite NFTS S, as well as the compact fuzzy partition corresponding to the fuzzy bisimilarity of S under the Gödel semantics. Their time complexities are of the order O((size(δ)log⁡l+|S|)log⁡(|S|+|δ|)), where l is the number of fuzzy values used in S plus 2. When |δ|≥|S|, this order is within O(|S|⋅|δ|⋅log2⁡|δ|). The reduction of time complexity from O(|S|4⋅|δ|2) and O(|S|4⋅|δ|2⋅l) to O(|S|⋅|δ|⋅log2⁡|δ|) is a significant contribution of this work.

Fuzzy clustering with Barber modularity regularization

In this paper, we propose a new algorithm for the joint clustering of two sets of statistical units N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}$$\\end{document} and M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document} which are also equipped with an adjacency structure which is represented by a bipartite network. Our model is based on the fuzzy Partition Around Medoids, and it combines it with techniques for community detection in bipartite complex networks based on Barber modularity maximization. The goal is to produce a partition of N∪M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}\\cup \\mathcal {M}$$\\end{document} into clusters, each of which is also identified by two medoids, one in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}$$\\end{document} and one in M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document}, which represent the typical units in the cluster for each set. Such clusters are optimized so that units in the same cluster both have similar values on their attributes and are likely to be adjacent. We test the algorithm on both simulated and real data, to show how it is able to capture a wide range of different interactions between the distribution of the attributes and the network structure.

MvWECM: Multi-view Weighted Evidential [formula omitted]-Means clustering

Traditional multi-view clustering algorithms, designed to produce hard or fuzzy partitions, often neglect the inherent ambiguity and uncertainty in the cluster assignment of objects. This oversight may lead to performance degradation. To address these issues, this paper introduces a novel multi-view clustering method, termed MvWECM, capable of generating credal partitions within the framework of belief functions. The objective function of MvWECM is introduced considering the uncertainty in the cluster structure included in the multi-view dataset. We take into account inter-view conflict to effectively leverage coherent information across different views. Moreover, the effectiveness is heightened through the incorporation of adaptive view weights, which are customized to modulate their smoothness in accordance with their entropy. The optimization method to get the optimal credal membership and class prototypes is derived. The view wights can be also provided as a by-product. Experimental results on several real-word datasets demonstrate the effectiveness and superiority of MvWECM by comparing with some state-of-the-art methods.

ECM+: An improved evidential c-means with adaptive distance

Evidential c-means (ECM) is a prototype-based clustering algorithm that generates a credal partition. Such a partition encompasses the notions that can be encountered with a hard, fuzzy or possibilistic partition, allowing the representation of various situations concerning the class membership of an object. The ECM method provides a prototype for each subset of the possible classes, calculated by averaging the prototypes of the classes included in the subsets. Although this definition perfectly suits ECM when employing a Euclidean distance, it becomes inappropriate when using a Mahalanobis distance. In this context, a new definition of prototypes for the subsets is proposed. The ECM objective function is then optimized using the new definition of prototypes. The subsequent algorithm, named ECM+, is finally tested on various synthetic and real data sets to demonstrate its interest compared to ECM.

A Novel Method Based on the Fuzzy Entropy Measure to Optimize the Fuzziness in Trapezoidal Strong Fuzzy Partitions

Analyzing the uncertainty of outcomes based on estimates of the data’s membership degrees to fuzzy sets is essential for making decisions. These fuzzy sets are often designated by experts as strong fuzzy partitions of the data domain with trapezoidal fuzzy numbers. Some indices of the fuzzy set’s fuzziness provide an assessment of the degree of uncertainty of the results. It is feasible to bring the fuzzy sets’ fuzziness below a tolerable level by suitably redefining the strong fuzzy partition. Significant differences in the original fuzzy partition, however, result in disparities concerning the decision maker’s approximative reasoning and the interpretability of the results. In light of this, we provide in this study a technique applied to trapezoidal strong fuzzy partitions that, while not appreciably altering the original fuzzy partition, reduces the fuzziness of its fuzzy sets. The fuzziness of the fuzzy sets is assessed using the De Luca and Termini fuzzy entropy. An iterative process is then executed, with the aim of modifying the cores of the trapezoidal fuzzy partitions to decrease their fuzziness. This technique is tested on datasets containing average daily temperatures measured in various cities. The findings demonstrate that this approach strikes a great balance between the goal of lessening the fuzziness of the fuzzy sets and the goal of not appreciably altering the original fuzzy partition.

One-Step Multiview Fuzzy Clustering With Collaborative Learning Between Common and Specific Hidden Space Information.

Multiview data are widespread in real-world applications, and multiview clustering is a commonly used technique to effectively mine the data. Most of the existing algorithms perform multiview clustering by mining the commonly hidden space between views. Although this strategy is effective, there are two challenges that still need to be addressed to further improve the performance. First, how to design an efficient hidden space learning method so that the learned hidden spaces contain both shared and specific information of multiview data. Second, how to design an efficient mechanism to make the learned hidden space more suitable for the clustering task. In this study, a novel one-step multiview fuzzy clustering (OMFC-CS) method is proposed to address the two challenges by collaborative learning between the common and specific space information. To tackle the first challenge, we propose a mechanism to extract the common and specific information simultaneously based on matrix factorization. For the second challenge, we design a one-step learning framework to integrate the learning of common and specific spaces and the learning of fuzzy partitions. The integration is achieved in the framework by performing the two learning processes alternately and thereby yielding mutual benefit. Furthermore, the Shannon entropy strategy is introduced to obtain the optimal views weight assignment during clustering. The experimental results based on benchmark multiview datasets demonstrate that the proposed OMFC-CS outperforms many existing methods.

Belief rule learning and reasoning for classification based on fuzzy belief decision tree

The belief rules which extend the classical fuzzy IF-THEN rules with belief consequent parts have been widely used for classifier design due to their capabilities of building linguistic models interpretable to users and addressing various types of uncertainty. However, in the rule learning process, a high number of features generally results in a belief rule base with large size, which degrades both the classification accuracy and the model interpretability. Motivated by this challenge, the decision tree building technique which implements feature selection and model construction jointly is introduced in this paper to learn a compact and accurate belief rule base. To this end, a new fuzzy belief decision tree (FBDT) with fuzzy feature partitions and belief leaf nodes is designed: a fuzzy information gain ratio is first defined as the feature selection criterion for node fuzzy splitting and then the belief distributions are introduced to the leaf nodes to characterize the class uncertainty. Based on the initial rules extracted from the constructed FBDT, a joint optimization objective considering both classification accuracy and model interpretability is then designed to further reduce the rule redundancy. Experimental results based on real datasets show that the proposed FBDT-based classification method has much smaller rule base and better interpretability than other rule-based methods on the premise of competitive accuracy.

Applications of Fuzzy Logic and Probabilistic Neural Networks in E-Service for Malware Detection

The key point in the process of agent-based management in e-service for malware detection (according to accuracy criteria) is a decision-making process. To determine the optimal e-service for malware detection, two concepts were investigated: Fuzzy Logic (FL) and Probabilistic Neural Networks (PNN). In this study, three evolutionary variants of fuzzy partitioning, including regular, hierarchical fuzzy partitioning, and k-means, were used to automatically process the design of the fuzzy partition. Also, this study demonstrates the application of a feature selection method to reduce the dimensionality of the data by removing irrelevant features to create fuzzy logic in a dataset. The behaviors of malware are analyzed by fuzzifying relevant features for pattern recognition. The Apriori algorithm was applied to the fuzzified features to find the fuzzy-based rules, and these rules were used for predicting the output of malware detection e-services. Probabilistic neural networks were also used to find the ideal agent-based model for numerous classification problems. The numerical results show that the agent-based management performances trained with the clustering method achieve an accuracy of 100% with the PNN-MCD model. This is followed by the FL model, which classifies on the basis of linguistic variables and achieves an average accuracy of 82%.

Enhancements of evidential [formula omitted]-means algorithms: A clustering framework via feature-weight learning

As a core paradigm of the evidential clustering algorithm, evidential c-means (ECM) offers a more flexible credal partition to characterize uncertainty and imprecision in cluster assignment, which generalizes hard and fuzzy partitions. Nonetheless, ECM and its derivatives equally consider the importance of each feature in clustering, which can lead to suboptimal clustering performance, especially in high-dimensional datasets where some features are more informative than others. In this paper, we propose a series of new evidential clustering algorithms by integrating feature-weight learning to simultaneously characterize uncertainty and imprecision in cluster assignment and automatically learn the different importance of distinct features in clustering. We consider ECM with vector feature-weight learning and ECM with matrix feature-weight learning, and introduce two different constraints on feature weights, respectively. We develop four objective functions and derive the corresponding optimization process. Extensive experimental results using synthetic and real-world datasets demonstrate the feasibility and effectiveness of the proposed algorithms.

Three-way decision-based Takagi–Sugeno–Kang fuzzy classifier for partially labeled data

The salient features of Takagi–Sugeno–Kang (TSK) fuzzy classifiers are their superior nonlinear fitting capability and better interpretability, rendering them widely applicable in the domains of data mining and machine learning. However, using TSK fuzzy classifiers for partially labeled data has received little attention. Based on the three-way decision (TWD) theory, this study proposed a TSK fuzzy model to learn from partially labeled data. First, an adaptive sample weight strategy is introduced for the fuzzy partition of the antecedent part and the cross-entropy loss of the consequent part, respectively. Subsequently, a prototype-based firing strength loss mechanism is proposed to constrain the optimization of the model in the fuzzy representation space. It employs a TWD strategy based on the entropy criterion to classify samples as useful, useless, and uncertain, where the proposed model enhances the performance by iteratively utilizing a certain number of useful pseudo-labeled samples. Finally, the validity of the model is theoretically analyzed based on the principle of noise learning. The experimental results on UCI datasets demonstrate that the proposed model exhibits superior performance compared with classical semi-supervised methods, even attaining a performance comparable to that of the model trained on all data with ground-truth labels.

Effectiveness of Using Candlestick Charts to Forecast Ethereum Price Direction: A Machine Learning Approach

Cryptocurrency is a form of decentralized digital currency. Ethereum is the second-largest cryptocurrency by market capitalization and the largest altcoin. Cryptocurrencies including Ethereum are highly volatile. Hence, shortterm directional forecasts in the cryptocurrency market have become a widely discussing topic. Candlestick charts are useful visualizations of the open, high, low and close prices which can identify patterns and gauge the near-term direction of prices. This research explores the effectiveness of forecasting hourly Ethereum closing price direction based on candlestick charts within a short time horizon. The proposed forecasting algorithm incorporates clustering methods such as fuzzy K-means, K-means and partition around medoids clustering to cluster candlestick chart properties namely upper shadow length, body length and lower shadow length. Classification methods such as random forest, support vector machine and K-nearest neighbour were used to forecast closing price direction using 16 different predictor variable sets including open, high, low and close prices, candlestick chart price direction, USL, BL and LSL. The accuracy for all considered cases was around 50%. Clustering improved the accuracy slightly and including the CPD with the predictor variable sets under consideration can increase the accuracy slightly. However, this approach is performing better in predicting the Down cases to the total number of actual Down cases because there is a higher sensitivity of 81.20% based on the SVM with Open, High, Low and Close at t in the clustering ignored method.

Model reference adaptive controller with interpretable fuzzy rules for linear MIMO systems

This paper presents a method for fuzzy model reference adaptive control system design for a partially known multi-input multi-output (MIMO) linear dynamic plant. Differential or convolutional equations can describe the plant to be controlled, and the fuzzy controller is assumed to be from the MIMO sector-bounded nonstationary nonlinearities class. The task is to obtain robust adaptive stabilization. Some absolute stability theorems and integral evaluations of the state and control vectors are applied in the MIMO fuzzy adaptive controller design procedure for the first time. A method for automatically obtaining a near-optimal nonlinear dynamic reference model is described. A practical step-by-step procedure of the MIMO adaptive fuzzy controller design is proposed, resulting in highly interpretable MIMO fuzzy control rules based on triangular fuzzy sets and strong triangular fuzzy partition. An example of the fuzzy controller design according to the proposed procedure is given.

Development of the fuzzy grid partition methods in generating fuzzy rules for the classification of data set

The main weakness of complex and sizeable fuzzy rule systems is the complexity of data interpretation in terms of classification. Classification interpretation can be affected by reducing rules and removing important rules for several reasons. Based on the results of experiments using the fuzzy grid partition (FGP) approach for high-dimensional data, the difficulty in generating many fuzzy rules still increases exponentially as the number of characteristics increases. The solution to this problem is a hybrid method that combines the advantages of the rough set method and the FGP method, which is called the fuzzy grid partition rough set (FGPRS) method. In the Irish data, the rough set approach reduces the number of characteristics and objects so that data with excessive values can be minimized, and the fuzzy rules produced using the FGP method are more concise. The number of fuzzy rules produced using the FGPRS method at K=2 is 50%; at K=K+1, it is reduced by 66.7% and at K=2 K, it is reduced by 75%. Based on the findings of the data collection classification test, the FGPRS method has a classification accuracy rate of 83.33%, and all data can be classified.

IFKMHC: Implicit Fuzzy K-Means Model for High-Dimensional Data Clustering.

The graph-information-based fuzzy clustering has shown promising results in various datasets. However, its performance is hindered when dealing with high-dimensional data due to challenges related to redundant information and sensitivity to the similarity matrix design. To address these limitations, this article proposes an implicit fuzzy k-means (FKMs) model that enhances graph-based fuzzy clustering for high-dimensional data. Instead of explicitly designing a similarity matrix, our approach leverages the fuzzy partition result obtained from the implicit FKMs model to generate an effective similarity matrix. We employ a projection-based technique to handle redundant information, eliminating the need for specific feature extraction methods. By formulating the fuzzy clustering model solely based on the similarity matrix derived from the membership matrix, we mitigate issues, such as dependence on initial values and random fluctuations in clustering results. This innovative approach significantly improves the competitiveness of graph-enhanced fuzzy clustering for high-dimensional data. We present an efficient iterative optimization algorithm for our model and demonstrate its effectiveness through theoretical analysis and experimental comparisons with other state-of-the-art methods, showcasing its superior performance.

Unifying credal partitions and fuzzy orthopartitions

This work focuses on fuzzy orthopartitions and credal partitions, which are distinct mathematical models representing partitions where the membership of elements to classes is only partially known. Firstly, we show that fuzzy orthopartitions and credal partitions are special cases of generalized fuzzy orthopartitions, which we introduce in this article as a new structure for modelling partitions with uncertainty. Next, we examine the connections between credal partitions and fuzzy orthopartitions, considering that both can be seen as types of fuzzy partitions (in particular, we deal with fuzzy probabilistic and Ruspini partitions). Moreover, we find that each generalized fuzzy orthopartition corresponds to a collection of zero, one, or infinitely many credal partitions; conversely, a credal partition maps to at most one generalized fuzzy orthopartition. Finally, we identify the class of all credal partitions that coincide with fuzzy orthopartitions.

An Evaluation of Fuzzy in Image Enhancement: Design and Comparison for Penicillium and Aspergillus Species

The main focus in this study is to enhance and classify the image of a type of filamentous fungi named Penicillium and Aspergillus. For image enhancement, fuzzy-partition gamma adaptive histogram equalization (FpGAHE) is proposed to improve the quality of an image, in particular the low quality of a microscopic image. Two stages have been considered in this technique. In the first stage, a fuzzy partition is developed to handle the inconsistency of the grey level values of the images by introducing a fuzzy set. In the second stage, surrounding neighbourhood is employed to avoid the imbalance data and reduce the drastically changes of brightness values of the image. The performances are evaluated into two parts i.e., image processing and image classification by using the collected microscopic images of fungi species. To evaluate the effectiveness of the proposed technique, the existing techniques, HE, AHE, CLAHE, GC and AGC is compared to the FpGAHE. In image processing, the result attained shows that the proposed technique has a better performance by obtaining the highest value for the PSNR, SSIM and FSIM evaluation for the species of A. terreus in clean condition. Meanwhile, in image classification, five different nearest neighbour classifiers have been tested. The results show the proposed FpGAHE with Improved Fuzzy-Based k Nearest Centroid Neighbour (IFkNCN) classifier perform the best result compare to other nearest neighbour classifier by obtaining the value of 92.59 and 93.95 for the salt and pepper and Gaussian noise corrupted images respectively.

Parsimonious consensus hierarchies, partitions and fuzzy partitioning of a set of hierarchies

Methodology is described for fitting a fuzzy partition and a parsimonious consensus hierarchy (ultrametric matrix) to a set of hierarchies of the same set of objects. A model defining a fuzzy partition of a set of hierarchical classifications, with every class of the partition synthesized by a parsimonious consensus hierarchy is described. Each consensus includes an optimal consensus hard partition of objects and all the hierarchical agglomerative aggregations among the clusters of the consensus partition. The performances of the methodology are illustrated by an extended simulation study and applications to real data. A discussion is provided on the new methodology and some interesting future developments are described.

Adaptive weighted multi-view evidential clustering with feature preference

Multi-view clustering has attracted substantial attention thanks to its ability to integrate information from diverse views. However, the existing methods can only generate hard or fuzzy partitions, which cannot effectively represent the uncertainty and imprecision when facing objects in overlapping clusters, thus increasing the risk of error. To solve the above problems, in this paper, we propose an adaptive weighted multi-view evidential clustering (WMVEC) method based on the theory of belief functions to characterize the uncertainty and imprecision in cluster assignment. Technically, we integrate view weight assignments and credal partition between objects and cluster prototypes into a joint learning framework. The credal partition offers a more comprehensive insight into the data by enabling objects to be associated with not only singleton clusters but also subsets of these clusters (termed meta-clusters) and the empty set, which represents a noise cluster. To avoid the interference of irrelevant and redundant features, we further present a weighted multi-view evidential clustering with feature preference (WMVEC-FP) to learn the importance of each feature under different views. We suggest the objective functions of WMVEC and WMVEC-FP and design alternating optimization schemes to obtain the optimal solutions, respectively. Through an extensive array of experiments, it has been demonstrated that our proposed clustering methods outperform other related and state-of-the-art methods in terms of their advantages and overall effectiveness.

Real estate price estimation through a fuzzy partition-driven genetic algorithm

Evaluating the actual price of a residential property is a critical issue in the real estate market. Real estate market practitioners gauge a property's price by considering features such as property type and residential area. Subsequently, they evaluate the property's intrinsic features, such as condition, sun exposure, scenic views, and ancillary amenities. Finally, extrinsic features such as the proximity of services and infrastructure are assessed. This paper proposes a new genetic approach for selecting residential properties that meet the purchase offer and the intrinsic and extrinsic characteristics desired by the client. Since the real estate market's changes can influence extrinsic features, the method introduces price fluctuations of properties. Extrinsic features are modelled as fuzzy partitions: each fuzzy set describes a qualitative aspect of the corresponding feature that, expressed in a linguistic term, has a human-like interpretation. Then, a deviation value (fluctuation) from the average price of the property is considered for each fuzzy set in the partition. All the property features, extrinsic and intrinsic, are encoded in the chromosome genes of the genetic algorithm. The fitness function calculates the distance between the unit price of the property and the purchase offer. Some case studies were conducted in various Italian municipalities, using the average price per square meter of residential properties the Osservatorio del Mercato Immobiliare (OMI) assigned. Depending on customer requirements and preferences, different OMI zones were selected using additional characteristics such as type, location, conservation, and proximity to various urban services. The results demonstrated the effectiveness of the proposed approach for all the case studies, showing how the optimal solution represents a good compromise between customer preferences and market offerings.