Fuzzy clustering optimal k selection method based on multi-objective optimization

In order to overcome the dimension problem of the traditional fuzzy clustering, we use kernel-based fuzzy c-means clustering (KFCM) to construct first-order TSK fuzzy models. The proposed algorithm is composed of two phases. In the first phase, the antecedent fuzzy sets are obtained by KFCM. We present the expression of the cluster prototypes of KFCM with different kernel functions in original input space. The use of cluster validity indices is a standard approach to determine an appropriate number of clusters in a data set. However, cluster validity index demands running the clustering algorithm for different number of clusters repeatedly. Therefore, a novel method specifying the number of clusters automatically is given for the purpose of reducing the computational complexity and eliminating the outliers. In the second phase, the consequent parameters can be identified by the least squares method. Experiment results show that the proposed method improves the generalization ability and robustness of fuzzy models compared with the traditional techniques.

PurposeThe theory of double jeopardy (DJ) is shown to hold across broad ranging geographies and physical product categories. However, there is very little research appertaining to the subject within an online environment. In particular, studies that investigate the presence of DJ and the contrasting view point to DJ, namely, that of negative double jeopardy (NDJ), are lacking. This study aims to contribute to this identified research gap and examines the presence of DJ and NDJ within a product category, utilising data from Twitter.Design/methodology/approachA total of 354,676 tweets are scraped from Twitter and their sentiment analysed and allocated into positive, negative and no-opinion clusters using fuzzy c-means clustering. The sentiment is then compared to the market share of brands within the beer product category to establish whether a DJ or NDJ effect is present.FindingsData reveal an NDJ effect with regards to original tweets (i.e. tweets which have not been retweeted). That is, when analysing tweets relating to brands within a defined beer category, the authors find that larger brands suffer by having an increased negativity amongst the larger proportion of tweets associated with them.Research limitations/implicationsThe clustering approach to analyse sentiment in Twitter data brings a new direction to analysis of such sentiment. Future consideration of different numbers of clusters may further the insights this form of analysis can bring to the DJ/NDJ phenomenon. Managerial implications discuss the uncovered practitioner’s paradox of NDJ and strategies for dealing with DJ and NDJ effects.Originality/valueThis study is the first to explore the presence of DJ and NDJ through the utilisation of sentiment analysis-derived data and fuzzy clustering. DJ and NDJ are under-explored constructs in the online environment. Typically, past research examines DJ and NDJ in separate and detached fashions. Thus, the study is of theoretical value because it outlines boundaries to the DJ and NDJ conditions. Second, this research is the first study to analyse the sentiment of consumer-authored tweets to explore DJ and NDJ effects. Finally, the current study offers valuable insight into the DJ and NDJ effects for practicing marketing managers.

Typically clustering algorithms provide clustering solutions with prespecified number of clusters. The lack of a priori knowledge on the true number of underlying clusters in the dataset makes it important to have a metric to compare the clustering solutions with different number of clusters. This article quantifies a notion of persistence of clustering solutions that enables comparing solutions with different number of clusters. The persistence relates to the range of dataresolution scales over which a clustering solution persists; it is quantified in terms of the maximum over two-norms of all the associated cluster-covariance matrices. Thus we associate a persistence value for each element in a set of clustering solutions with different number of clusters. We show that the datasets where natural clusters are a priori known, the clustering solutions that identify the natural clusters are most persistent - in this way, this notion can be used to identify solutions with true number of clusters. Detailed experiments on a variety of standard and synthetic datasets demonstrate that the proposed persistence-based indicator outperforms the existing approaches, such as, gap-statistic method, X-means, Gmeans, PG-means, dip-means algorithms and informationtheoretic method, in accurately identifying the clustering solutions with true number of clusters. Interestingly, our method can be explained in terms of the phase-transition phenomenon in the deterministic annealing algorithm, where the number of distinct cluster centers changes (bifurcates) with respect to an annealing parameter.

Cluster-randomized trials allow for the evaluation of a community-level or group-/cluster-level intervention. For studies that require a cluster-randomized trial design to evaluate cluster-level interventions aimed at controlling vector-borne diseases, it may be difficult to assess a large number of clusters while performing the additional work needed to monitor participants, vectors, and environmental factors associated with the disease. One such example of a cluster-randomized trial with few clusters was the "efficacy and risk of harms of repeated ivermectin mass drug administrations for control of malaria" trial. Although previous work has provided recommendations for analyzing trials like repeated ivermectin mass drug administrations for control of malaria, additional evaluation of the multiple approaches for analysis is needed for study designs with count outcomes. Using a simulation study, we applied three analysis frameworks to three cluster-randomized trial designs (single-year, 2-year parallel, and 2-year crossover) in the context of a 2-year parallel follow-up of repeated ivermectin mass drug administrations for control of malaria. Mixed-effects models, generalized estimating equations, and cluster-level analyses were evaluated. Additional 2-year parallel designs with different numbers of clusters and different cluster correlations were also explored. Mixed-effects models with a small sample correction and unweighted cluster-level summaries yielded both high power and control of the Type I error rate. Generalized estimating equation approaches that utilized small sample corrections controlled the Type I error rate but did not confer greater power when compared to a mixed model approach with small sample correction. The crossover design generally yielded higher power relative to the parallel equivalent. Differences in power between analysis methods became less pronounced as the number of clusters increased. The strength of within-cluster correlation impacted the relative differences in power. Regardless of study design, cluster-level analyses as well as individual-level analyses like mixed-effects models or generalized estimating equations with small sample size corrections can both provide reliable results in small cluster settings. For 2-year parallel follow-up of repeated ivermectin mass drug administrations for control of malaria, we recommend a mixed-effects model with a pseudo-likelihood approximation method and Kenward-Roger correction. Similarly designed studies with small sample sizes and count outcomes should consider adjustments for small sample sizes when using a mixed-effects model or generalized estimating equation for analysis. Although the 2-year parallel follow-up of repeated ivermectin mass drug administrations for control of malaria is already underway as a parallel trial, applying the simulation parameters to a crossover design yielded improved power, suggesting that crossover designs may be valuable in settings where the number of available clusters is limited. Finally, the sensitivity of the analysis approach to the strength of within-cluster correlation should be carefully considered when selecting the primary analysis for a cluster-randomized trial.

The MR brain image analysis is used to extract clinical information that improve diagnosis and treatment of disease. Brain tumors are one of the most common brain diseases. Clustering is a process for segmenting and classifying objects. There are many clustering strategies such as the hard clustering scheme and the fuzzy clustering scheme, each of them has its own special characteristics. In this work Fuzzy C-Mean clustering was implemented to segment three abnormal brain MR images, and the performance of it was analyised. This algorithms was applied to cluster the images into different clusters number: 5-9 with different values of membership grade: 0.50-0.90 with steps of 0,05 for each cluster number. The percentage of the unclassified pixels that were produced from implementing FCM algorithm with different configuration was calculated. The minimum values of the objective function of the FCM algorithm for different number of clusters and for different membership grade values were also calculated. The results showed that an optimal number of clusters that corresponds to optimal segmentation error is depending on the slice condition. In this experiment, the optimal cluster number was found to be 6. The fluctuation around this number is affected also by the anatomical structure of the slice. In addition, it can be concluded that the objective function may not be the superior criterion for the judgments of goodness, where it may be a few number of pixels with high uncertainty is the source of high error.

Clustering is a group of unsupervised statistical techniques commonly used in many disciplines. Considering their applications to fish abundance data, many technical details need to be considered to ensure reasonable interpretation. However, the reliability and stability of the clustering methods have rarely been studied in the contexts of fisheries. This study presents an intensive evaluation of three common clustering methods, including hierarchical clustering (HC), K-means (KM), and expectation-maximization (EM) methods, based on fish community surveys in the coastal waters of Shandong, China. We evaluated the performances of these three methods considering different numbers of clusters, data size, and data transformation approaches, focusing on the consistency validation using the index of average proportion of non-overlap (APN). The results indicate that the three methods tend to be inconsistent in the optimal number of clusters. EM showed relatively better performances to avoid unbalanced classification, whereas HC and KM provided more stable clustering results. Data transformation including scaling, square-root, and log-transformation had substantial influences on the clustering results, especially for KM. Moreover, transformation also influenced clustering stability, wherein scaling tended to provide a stable solution at the same number of clusters. The APN values indicated improved stability with increasing data size, and the effect leveled off over 70 samples in general and most quickly in EM. We conclude that the best clustering method can be chosen depending on the aim of the study and the number of clusters. In general, KM is relatively robust in our tests. We also provide recommendations for future application of clustering analyses. This study is helpful to ensure the credibility of the application and interpretation of clustering methods.

Recommendations for validating hierarchical clustering in consumer sensory projects

Learning simultaneous adaptive clustering and classification via MOEA

Determining the most proper number of cluster in fuzzy clustering by using artificial neural networks

In this paper, we propose a Multi-objective Five-element Cycle Optimization algorithm based on Multi-strategy fusion (MOFECO-MS) to address the Bi-objective Traveling Thief Problem (BITTP), an extension of the Traveling Thief Problem that incorporates two conflicting objectives. The novelty of our approach lies in a unique individual selection strategy coupled with an innovative element update mechanism rooted in the Five-element Cycle Model. To balance global exploration and local exploitation, the algorithm categorizes the population into distinct groups and applies crossover operations both within and between these groups, while also employing a mutation operator for local searches on the best individuals. This coordinated approach optimizes parameter settings and enhances the search capabilities of the algorithm. Extensive experiments were conducted on nine BITTP instances, comparing MOFECO-MS against eight state-of-the-art multi-objective optimization algorithms. The results show that MOFECO-MS excels in both Hypervolume (HV) and Spread (SP) indicators, while also maintaining a high level of Pure Diversity (PD). Overall, MOFECO-MS outperformed the other algorithms in most instances, demonstrating its superiority and robustness in solving complex multi-objective optimization problems.

In this paper, the problems of fuel cost and emission reduction based on multi-objective optimization are studied and solved. Various multi-objective optimization algorithms, such as the multi-objective ant lion optimization (MOALO), the multi-objective multiverse optimization (MOMVO), the multi-objective particle swarm optimization (MOPSO), the multi-objective thermal exchange optimization (MOTEO), and the multi-objective grasshopper optimization algorithm (MOGOA), are used to reduce the cost and emissions of generation fuel. With equality and inequality restrictions, the suggested work is carried out in an IEEE 30 bus system. To determine the lowest system-wide minimization of generating cost and emission, the results of several optimization strategies are compared. In terms of lowering fuel costs and emissions, the MOGOA performs better than any other multi-objective algorithms.

Clustering is an important technique for discovering the inherent structure in a given data set without any ‘priori’ knowledge. Fuzzy clustering analysis is to assign objects to a given number of clusters with respect to some criteria such that each object may belong to more than one cluster with different degrees of membership. In this article, a new fuzzy clustering method based on tabu search called Improved Tabu Search Fuzzy Clustering (ITSFC) is proposed to find the proper clustering of data sets. In the ITSFC approach, a fuzzy c-means operation is developed to fine-tune the clustering solution obtained in the process of iterations and a divide-and-merge operation is designed to establish the neighborhood. Experimental results on two artificial and four real life data sets are given to illustrate the superiority of the proposed algorithm over a tabu search clustering algorithm and an artificial bee colony clustering algorithm.   Key words: Fuzzy clustering, tabu search, artificial bee colony.

Battle Royale Optimizer (BRO) is a recently proposed optimization algorithm that has added a new category named game-based optimization algorithms to the existing categorization of optimization algorithms. Both continuous and binary versions of this algorithm have already been proposed. Generally, optimization problems can be divided into single-objective and multi-objective problems. Although BRO has successfully solved single-objective optimization problems, no multi-objective version has been proposed for it yet. This gap motivated us to design and implement the multi-objective version of BRO (MOBRO). Although there are some multi-objective optimization algorithms in the literature, according to the no-free-lunch theorem, no optimization algorithm can efficiently solve all optimization problems. We applied the proposed algorithm to four benchmark datasets: CEC 2009, CEC 2018, ZDT, and DTLZ. We measured the performance of MOBRO based on three aspects: convergence, spread, and distribution, using three performance criteria: inverted generational distance, maximum spread, and spacing. We also compared its obtained results with those of three state-of-the-art optimization algorithms: the multi-objective Gray Wolf optimization algorithm (MOGWO), the multi-objective particle swarm optimization algorithm (MOPSO), the multi-objective artificial vulture's optimization algorithm (MOAVAO), the optimization algorithm for multi-objective problems (MAOA), and the multi-objective non-dominated sorting genetic algorithm III (NSGA-III). The obtained results approve that MOBRO outperforms the existing optimization algorithms in most of the benchmark suites and operates competitively with them in the others.

Clustering is an unsupervised learning algorithm. k-Means algorithm is one of the well-known and promising clustering algorithms that can converge to a local optimum in few iterative. In our work, we will be hybridizing k-Means algorithm with Genetic Algorithms to look for the solution in the global search space in order to converge to a global optima. The problem for clustering is that when the number of clusters increases up to the same number of total records in the dataset, it leads to a scenario in which a cluster only contains a single record, and thus the cluster purity is maximized to the maximum value, 1. However, it will be useless since the common regularities among records will not be seen. Therefore, choosing the best number of clusters is trivial. Instead of choosing an inappropriate number of clusters and risking the main purpose of the clustering process, a Genetic Algorithm based k-Means ensemble is proposed in order to find the consensus result of several runs of clustering task using different number of clusters, k.

In this study, clustering based method for activation detection in functional magnetic resonance imaging (fMRI) is employed. Moreover, some features are obtained by fitting two models namely FIR filter and Gamma function, to hemodynamic response function (HRF). After applying clustering methods (that require number of clusters as an input) to feature space, our simulations show that number of clusters can affect activation detection significantly. Therefore a newly proposed clustering algorithm namely evolving neural gas (ENG) that gives optimal number of clusters is exploited. In addition to ENG, the result of four clustering algorithms namely k-means, fuzzy C-means, neural gas, and clara in different number of clusters are evaluated. The results show that the best activation detection is taken place using obtained optimal number of clusters.