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Determining the optimal number of regions is a challenging issue in regionalization. Although cluster validity indices developed for non-spatial clustering have been used to determine the optimal number of regions, spatial contiguity constraints for regionalization are often neglected. Consequently, different regionalization results can share the same validity index value, which reduces the reliability of identifying the optimal number of regions in regionalization. To overcome this limitation, this study proposes a spatially constrained statistical approach for determining the optimal number of regions using two metrics: (i) a permutation-based variance for measuring the homogeneity within regions and (ii) a proportion index based on spatially constrained k-nearest neighbors to quantify the separation between regions. Furthermore, a distance-based method is employed to balance these two metrics to automatically determine the optimal number of regions. Experimental results on five synthetic datasets, the US presidential election and climate datasets show that the statistical approach developed in this study outperforms three widely used cluster validity indices in determining the optimal number of regions. The proposed statistical approach is straightforward to implement and can effectively reduce subjectivity in regionalization. [ABSTRACT FROM AUTHOR]

Recognition of spatio-temporal traffic patterns at the network-wide level plays an important role in data-driven intelligent transport systems (ITS) and is a basis for applications such as short-term prediction and scenario-based traffic management. Common practice in the transport literature is to rely on well-known general unsupervised machine-learning methods (e.g., k-means, hierarchical, spectral, DBSCAN) to select the most representative structure and number of day-types based solely on internal evaluation indices. These are easy to calculate but are limited since they only use information in the clustered dataset itself. In addition, the quality of clustering should ideally be demonstrated by external validation criteria, by expert assessment or the performance in its intended application. The main contribution of this paper is to test and compare the common practice of internal validation with external validation criteria represented by the application to short-term prediction, which also serves as a proxy for more general traffic management applications. When compared to external evaluation using short-term prediction, internal evaluation methods have a tendency to underestimate the number of representative day-types needed for the application. Additionally, the paper investigates the impact of using dimensionality reduction. By using just 0.1% of the original dataset dimensions, very similar clustering and prediction performance can be achieved, with up to 20 times lower computational costs, depending on the clustering method. K-means and agglomerative clustering may be the most scalable methods, using up to 60 times fewer computational resources for very similar prediction performance to the p-median clustering. [ABSTRACT FROM AUTHOR]

Fuzzy clustering algorithms are one of the most important techniques for analysing and extracting information from data when working with datasets containing overlapping clusters. Fuzzy clustering provides a more precise representation of complex data structures compared to conventional crisp (hard) clustering approaches. It accomplishes this by allowing data points to be assigned to multiple clusters with different degrees of membership. This paper provides an extensive review of various fuzzy clustering algorithms, such as Fuzzy C - Means (FCM) and its variations including Gustafson - Kessel (GK), Noise Clustering (NC), Possibilistic C - Means (PCM), Possibilistic Fuzzy C - Means (PFCM), Credibilistic Fuzzy C - Means (CFCM), and Kernel Fuzzy C - Means (KFCM). Their underlying mathematical theories, pseudocodes and applicability to different types of data are reviewed. Furthermore, the time complexity of these algorithms are analysed and a detailed comparison is presented to clarify their performance and scalability. By examining both theoretical aspects and empirical results, this study aims to provide a comprehensive knowledge of the trade-offs between computational efficiency and clustering accuracy. This analysis is intended to serve as a resource for researchers and practitioners in selecting appropriate fuzzy clustering techniques for their specific applications. [ABSTRACT FROM AUTHOR]

Fuzzy clustering methods allow the objects to belong to several clusters simultaneously, with different degrees of membership. However, a factor that influences the performance of fuzzy algorithms is the value of fuzzifier parameter. In this paper, we propose a fuzzy clustering procedure for data (time) series that does not depend on the definition of a fuzzifier parameter. It comes from two approaches, theoretically motivated for unsupervised and supervised classification cases, respectively. The first is the Probabilistic Distance clustering procedure. The second is the well known Boosting philosophy. Our idea is to adopt a boosting prospective for unsupervised learning problems, in particular we face with non hierarchical clustering problems. The global performance of the proposed method is investigated by various experiments. [ABSTRACT FROM AUTHOR]

In this work, we present a novel clustering approach which is inheriting the best characteristics of Unsupervised Functional Link Artificial Neural Network (UFLANN) and Genetic Algorithms (GAs) for uncovering clusters embedded in dataset represented through $(X)_{Nxd}$ , where X consists of $N$ data points with $d$ -dimensions. With an aim to realize natural clusters in a linear space UFLANN mapped the input vectors from lower dimension to higher dimension with a greater hope to achieve linearity in higher dimensional space. As a result, UFLANN introduces the problem of curse of dimensionality in the given datasets. However, it has been realized that the problems like sparse data and distance concentration associated with curse of dimensionality cast this problem to again a very complex problem. Hence to address some of the issues of curse of dimensionality, we have used GAs for selecting optimal number of features in the higher dimension for UFLANN to discover clusters embedded in the dataset. The proposed approach herein after named as GAUFLANN has been experimentally evaluated by using the metrics like (i) Davies-Bouldin Index, ii) Silhouette Score, and iii) Completeness score on different synthetic and real datasets. Our experimental study confirms that GAUFLANN is evidently scoring better in DB-index, Silhouette score, and Completeness score than the clustering methods like K-means, Hierarchical-Agglomerative (Average Linkage), and UFLANN across the datasets like Circles, Moons, Iris, and CORD-19.

To evaluate clustering results is a significant part of cluster analysis. Since there are no true class labels for clustering in typical unsupervised learning, many internal cluster validity indices (CVIs), which use predicted labels and data, have been created. Without true labels, to design an effective CVI is as difficult as to create a clustering method. And it is crucial to have more CVIs because there are no universal CVIs that can be used to measure all datasets and no specific methods of selecting a proper CVI for clusters without true labels. Therefore, to apply a variety of CVIs to evaluate clustering results is necessary. In this paper, we propose a novel internal CVI – the Distance-based Separability Index (DSI), based on a data separability measure. We compared the DSI with eight internal CVIs including studies from early Dunn (1974) to most recent CVDD (2019) and an external CVI as ground truth, by using clustering results of five clustering algorithms on 12 real and 97 synthetic datasets. Results show DSI is an effective, unique, and competitive CVI to other compared CVIs. We also summarized the general process to evaluate CVIs and created the rank-difference metric for comparison of CVIs' results. [ABSTRACT FROM AUTHOR]

Analyzing and processing mine wind speed monitoring data is the key to realizing intelligent ventilation and real-time calculation of the ventilation network. According to the characteristics of the artificial regulation of a mine ventilation system, a local regression fuzzy C clustering algorithm is proposed in this paper, which combines local outlier processing with global air volume state analysis. Firstly, the algorithm uses the robust local weighted regression principle to analyze and preprocess the data locally, determines the risk degree of the abnormal data according to the identified times of outliers, determines the clustering number according to the clustering validity function, and analyzes the global air volume fluctuation according to the clustering results. The results show that most outliers are identified in data preprocessing. Still, the processing of dense outliers is weak, related to the window width setting and weighting multiple. The number of clusters can represent the fluctuation of the ventilation state and the pre-processed cluster centers are 4.4% lower than the original data because most of the outliers are higher than the average data. According to the law of air volume balance, the clustering results can pave the way for the global deduction of mine wind speed. There is an implicit relationship between data preprocessing and the clustering process, and when intensive outliers are not eliminated, they may be identified as separate clusters. The research of this paper points out the direction of mine wind speed data analysis, which can provide a theoretical basis for intelligent mine ventilation and real-time calculation of the ventilation network. [ABSTRACT FROM AUTHOR]

The optimal number of clusters (Copt) is one of the determinants of clustering efficiency. In this article, we present a new method of quantifying Copt for centroid-based clustering. First, we propose a new clustering validity index named fRisk(C) based on the fuzzy set theory. It takes the role of normalization and accumulation of local risks coming from each action either splitting data from a cluster or merging data into a cluster. fRisk(C) exploits the local distribution information of the database to catch the global information of the clustering process in the form of the risk degree. Based on the monotonous reduction property of fRisk(C), which is proved theoretically, we present a fRisk-based new algorithm named fRisk4-bA for determining Copt. In the algorithm, the well-known L-method is employed as a supplemented tool to catch Copt on the graph of the fRisk(C). Along with the stable convergence trend of the method to be proved theoretically, numerical surveys are also carried out. The surveys show that the high reliability and stability, as well as the sensitivity in separating/merging clusters in high-density areas, even if the presence of noise in the databases, are the strong points of the proposed method. [ABSTRACT FROM AUTHOR]

Analyzing and processing mine wind speed monitoring data is the key to realizing intelligent ventilation and real-time calculation of the ventilation network. According to the characteristics of the artificial regulation of a mine ventilation system, a local regression fuzzy C clustering algorithm is proposed in this paper, which combines local outlier processing with global air volume state analysis. Firstly, the algorithm uses the robust local weighted regression principle to analyze and preprocess the data locally, determines the risk degree of the abnormal data according to the identified times of outliers, determines the clustering number according to the clustering validity function, and analyzes the global air volume fluctuation according to the clustering results. The results show that most outliers are identified in data preprocessing. Still, the processing of dense outliers is weak, related to the window width setting and weighting multiple. The number of clusters can represent the fluctuation of the ventilation state and the pre-processed cluster centers are 4.4% lower than the original data because most of the outliers are higher than the average data. According to the law of air volume balance, the clustering results can pave the way for the global deduction of mine wind speed. There is an implicit relationship between data preprocessing and the clustering process, and when intensive outliers are not eliminated, they may be identified as separate clusters. The research of this paper points out the direction of mine wind speed data analysis, which can provide a theoretical basis for intelligent mine ventilation and real-time calculation of the ventilation network.

In comparing clustering partitions, the Rand index (RI) and the adjusted Rand index (ARI) are commonly used for measuring the agreement between partitions. Such external validation indexes can be used to quantify how close the clusters are to a reference partition (or to prior knowledge about the data) by counting classified pairs of elements. To evaluate the solution of a fuzzy clustering algorithm, several extensions of the Rand index and other similarity measures to fuzzy partitions have been proposed. An extension of the ARI for fuzzy partitions based on the normalized degree of concordance is proposed. The performance of the proposed index is evaluated through Monte Carlo simulation studies. [ABSTRACT FROM AUTHOR]

Traditional decision trees for fault diagnosis often use an ID3 construction algorithm. For promoting the accuracy and efficiency of decision trees, considering the cluster validity and fault rates, this paper proposes two improved trees, CV-DTs and FR-DTs. This paper mainly has two highlights. The first highlight is to propose a CV-DT which is constructed by an improved ID3 algorithm considering the cluster validity index. A new cluster validity index which can compare the cluster validities of different attributes is proposed to modify the information gain. This method selects the splitting attributes with higher classification credibility and increases the diagnostic accuracy. The second highlight is to propose an FR-DT which is constructed by an improved ID3 algorithm considering the fault rates. This algorithm not only considers the partitioning ability of each attribute, but also considers the isolation priority of faults with higher fault rates. This method decreases the average diagnostic steps and promotes the diagnostic efficiency. Through a simulation case and a real board case, these decision trees are proved to be effective diagnostic tools which have higher accuracies or efficiencies in analog circuit.

This paper presents a modified intuitionistic fuzzy clustering (IFCM) algorithm for medical image segmentation. IFCM is a variant of the conventional fuzzy C-means (FCM) based on intuitionistic fuzzy set (IFS) theory. Unlike FCM, IFCM considers both membership and nonmembership values. The existing IFCM method uses Sugeno’s and Yager’s IFS generators to compute nonmembership value. But for certain parameters, IFS constructed using above complement generators does not satisfy the elementary condition of intuitionism. To overcome this problem, this paper adopts a new IFS generator. Further, Hausdorff distance is used as distance metric to calculate the distance between cluster center and pixel. Extensive experimentations are carried out on standard datasets like brain, lungs, liver and breast images. This paper compares the proposed method with other IFS based methods. The proposed algorithm satisfies the elementary condition of intuitionism. Further, this algorithm outperforms other methods with the use of various cluster validity functions.

The application of clustering involves the interpretation of objects placed in multi-dimensional spaces. The task of clustering itself is inherently submitted to subjectivity, the optimal solution can be extremely costly to discover and sometimes even unreachable or nonexistent. This fact introduces a trade-off between accuracy and computational effort, moreover given that engineering applications usually work well with suboptimal solutions. In such applied scenarios, cluster validation is mandatory to refine algorithms and ensure that solutions are meaningful. Validity indices are commonly intended to benchmark diverse clustering setups, therefore they are coefficients with a relative nature, i.e., useful when compared to one another. In this paper, we propose a validation methodology that enables absolute evaluations of clustering results. Our method performs geometric measurements of the solution space and provides a coherent interpretation of the data structure by using indices based on inter- and intra-cluster distances, density, and multimodality within clusters. Conducted tests and comparisons with well-known indices show that our validation methodology improves the robustness of the clustering application for knowledge discovery. While clustering is often performed as a black box technique, our index is construable and therefore allows for the implementation of systems enriched with self-checking capabilities. [ABSTRACT FROM AUTHOR]

Abstract Nuclear magnetic resonance T 1 -T 2 maps are popular for characterizing fluids in shale. The characterization, however, is often done manually, which is difficult for shale due to its complicated nature. This work investigates the clustering approach for fluid characterization on T 1 -T 2 maps by comparing 6 different algorithms: k-means, Gaussian mixture model, spectral clustering, and 3 hierarchical methods. T 1 -T 2 maps are collected on shale samples at as-received and dried conditions. We propose two cluster validity indices to select the optimal cluster number and best algorithm. Our results validate the capability of the two indices. Gaussian mixture model is found to be the best algorithm in most of the cases, as its fluid partitioning shows the highest consistency with theoretical fluid boundaries. In addition, 5 fluid components are identified from Gaussian mixture model, and their values are qualitatively validated by comparing with those in literature. Results also indicate that clustering is sensitive to the fluid distribution. Drying the sample producing better clustering by revealing the footprint of organic matter. This work provides a practical guide for applying cluster analysis in fluid characterization in Nuclear magnetic resonance T 1 -T 2 maps. Highlights • Six clustering algorithms are compared. • Two cluster validity indices are proposed. • Gaussian mixture model is the best algorithm for fluid characterization using NMR T 1 -T 2 maps. [ABSTRACT FROM AUTHOR]

The article presents both methods of clustering and outlier detection in complex data, such as rule-based knowledge bases. What distinguishes this work from others is, first, the application of clustering algorithms to rules in domain knowledge bases, and secondly, the use of outlier detection algorithms to detect unusual rules in knowledge bases. The aim of the paper is the analysis of using four algorithms for outlier detection in rule-based knowledge bases: Local Outlier Factor (LOF), Connectivity-based Outlier Factor (COF), K-MEANS, and SMALLCLUSTERS. The subject of outlier mining is very important nowadays. Outliers in rules If-Then mean unusual rules, which are rare in comparing to others and should be explored by the domain expert as soon as possible. In the research, the authors use the outlier detection methods to find a given number of outliers in rules (1%, 5%, 10%), while in small groups, the number of outliers covers no more than 5% of the rule cluster. Subsequently, the authors analyze which of seven various quality indices, which they use for all rules and after removing selected outliers, improve the quality of rule clusters. In the experimental stage, the authors use six different knowledge bases. The best results (the most often the clusters quality was improved) are achieved for two outlier detection algorithms LOF and COF.

Model-free methods are widely used for the processing of brain fMRI data collected under natural stimulations, sleep, or rest. Among them is the popular fuzzy c-mean algorithm, commonly combined with cluster validity (CV) indices to identify the ‘true’ number of clusters (components), in an unsupervised way. CV indices may however reveal different optimal c-partitions for the same fMRI data, and their effectiveness can be hindered by the high data dimensionality, the limited signal-to-noise ratio, the small proportion of relevant voxels, and the presence of artefacts or outliers. Here, the author investigated the behaviour of seven robust CV indices. A new CV index that incorporates both compactness and separation measures is also introduced. Using both artificial and real fMRI data, the findings highlight the importance of looking at the behavior of different compactness and separation measures, defined here as building blocks of CV indices, to depict a full description of the data structure, in particular when no agreement is found between CV indices. Overall, for fMRI, it makes sense to relax the assumption that only one unique c-partition exists, and appreciate that different c-partitions (with different optimal numbers of clusters) can be useful explanations of the data, given the hierarchical organization of many brain networks.

The paper deals with the problem of discovering fuzzy clusters with optimal number of elements in heuristic possibilistic clustering. The relational clustering procedure using a parameter that controls cluster sizes is considered and a technique for detecting the optimal number of elements in fuzzy clusters is proposed. The effectiveness of the proposed technique is illustrated through numerical examples. Experimental results are discussed and some preliminary conclusions are formulated.