Cluster Validity Index Research Articles

APPLICATION OF AGGLOMERATIVE HIERARCHICAL CLUSTERING IN GROUPING TOURISM INDUSTRY SMES IN EAST JAVA

This study addresses a significant research gap in the classification and evaluation of Small and Medium Enterprises (SMEs) within the tourism industry in East Java, specifically focusing on their marketing performance. The primary objective is to categorize these SMEs based on their marketing effectiveness, a critical factor for their competitiveness in the tourism sector. To achieve this, the research employs cluster analysis, utilizing Agglomerative Hierarchical Clustering techniques. The optimal clustering method was determined using the cophenetic correlation coefficient and cluster validation indices. Data for the analysis were collected through questionnaires, with a sample comprising 30 tourism SMEs across 11 regions in East Java, with responses from three employees per SME. The findings revealed two distinct clusters: SMEs with fairly good marketing performance and those with good marketing performance. The results suggest that while many SMEs in the tourism industry in East Java exhibit competitive marketing capabilities, there remains a subset that requires strategic improvements, particularly in spiritual marketing practices. This study's novelty lies in its approach to classifying tourism SMEs by marketing performance, providing a framework for targeted performance enhancement and industry strengthening in the region.

PyCVI: A Python package for internal Cluster Validity Indices, compatible with time-series data

PyCVI: A Python package for internal Cluster Validity Indices, compatible with time-series data

K-PCD: A new clustering algorithm for building energy consumption time series analysis and predicting model accuracy improvement

Clustering algorithms are often applied to building energy consumption data analysis to mine representative patterns of building energy usage. This paper proposes a new clustering algorithm: K-PCD, which is particularly suitable for building energy consumption time series. K-PCD utilizes a Pearson correlation coefficient-based distance measure (PCD), and a novel centroid calculation method that takes into account both the PCD and the traditional Euclidean distance (ED) between time series. This study also proposes a new clustering validity index (CVI) tailored to the predictability of building energy consumption: Energy Prediction Clustering Performance Index (EPCPI). The K-PCD and EPCPI are practically analyzed and validated using one year of data from 29 real buildings. The results indicate that comparing with traditional clustering algorithms, the K-PCD achieves better clustering results. This EPCPI index thoroughly explores building operation patterns, and after adding clustering labels, it can maximize the prediction accuracy of the Back Propagation Neural Network (BPNN) model for building energy consumption.

A survey of genetic algorithms for clustering: Taxonomy and empirical analysis

Clustering, an unsupervised learning technique, aims to group patterns into clusters where similar patterns are grouped together, while dissimilar ones are placed in different clusters. This task can present itself as a complex optimization problem due to the extensive search space generated by all potential data partitions. Genetic Algorithms (GAs) have emerged as efficient tools for addressing this task. Consequently, significant advancements and numerous proposals have been developed in this field.This work offers a comprehensive and critical review of state-of-the-art mono-objective Genetic Algorithms (GAs) for partitional clustering. From a more theoretical standpoint, it examines 22 well-known proposals in detail, covering their encoding strategies, objective functions, genetic operators, local search methods, and parent selection strategies. Based on this information, a specific taxonomy is proposed. In addition, from a more practical standpoint, a detailed experimental study is conducted to discern the advantages and disadvantages of approaches. Specifically, 22 different cluster validation indices are considered to compare the performance of clustering techniques. This evaluation is performed across 94 datasets encompassing diverse configurations, including the number of classes, separation between classes, and pattern dimensionality. Results reveal interesting findings, such as the key role of local search in optimizing results and reducing search space. Additionally, representations based on centroids and labels demonstrate greater efficiency and crossover and mutation operators do not prove to be as relevant. Ultimately, while the results are satisfactory, real-world clustering problems introduce additional complexity, especially for algorithms aiming to determine the number of clusters, resulting in diminished performance and the need for new approaches to be explored. Code, datasets and instructions to run algorithms in the LEAL library are available in an associated repository, in order to facilitate future experiments in this environment.

Quantitative evaluation of internal cluster validation indices using binary data sets

AbstractAimsDifferent clustering methods often classify the same data set differently. Selecting the “best” clustering solution from alternatives is possible with cluster validation indices. Because of the large variety of cluster validation indices (CVIs), choosing the most suitable index concerning the data set and clustering algorithms is challenging. We aim to assess different internal clustering validation indices.MethodsArtificial binary data sets with equal‐ and unequal‐sized well‐separated a priori clusters were simulated and three levels of noise were then added. Twenty replications of each of the six types of data sets (two group sizes × three levels of noise) were created and analyzed by three clustering algorithms with Jaccard dissimilarity. Twenty‐seven clustering validation indices are evaluated including both geometric and non‐geometric indices.ResultsAlthough, in theory, all CVIs could differentiate between good and wrong classifications, only a few perform as expected with noisy data. Tau and silhouette widths proved to be the best geometric CVIs both for equal and unequal cluster sizes. Among non‐geometric indices, crispness and OptimClass performed best.ConclusionWe recommend using these best‐performing CVIs. We suggest plotting the CVI value against the number of clusters because the lack of a sharp peak means that the position of the maximum is uncertain.

A Bayesian cluster validity index

Selecting the appropriate number of clusters is a critical step in applying clustering algorithms. To assist in this process, various cluster validity indices (CVIs) have been developed. These indices are designed to identify the optimal number of clusters within a dataset. However, users may not always seek the absolute optimal number of clusters but rather a secondary option that better aligns with their specific applications. This realization has led us to introduce a Bayesian cluster validity index (BCVI), which builds upon existing indices. The BCVI utilizes either Dirichlet or generalized Dirichlet priors, resulting in the same posterior distribution. The proposed BCVI is evaluated using the Calinski-Harabasz, CVNN, Davies–Bouldin, silhouette, Starczewski, and Wiroonsri indices for hard clustering and the KWON2, Wiroonsri–Preedasawakul, and Xie–Beni indices for soft clustering as underlying indices. The performance of the proposed BCVI with that of the original underlying indices has been compared. The BCVI offers clear advantages in situations where user expertise is valuable, allowing users to specify their desired range for the final number of clusters. To illustrate this, experiments classified into three different scenarios are conducted. Additionally, the practical applicability of the proposed approach through real-world datasets, such as MRI brain tumor images are presented. These tools are published as a recent R package ‘BayesCVI’.

Damage identification of plain-woven composites at T > Tg using AE: Damage clustering and initiation detection

In this work, the damage modes and their initiation detections of plain-woven composites under tensile and compressive loads at T > Tg are investigated by conducting a comprehensive analysis of AE signals, including the feature selection, algorithm comparison and clustering validity. According to a newly proposed Pearson correlation coefficient cumulative criterion, the PF and PA are the most representative features of AE signals. A systematic comparison of K-Means, GMM and Hierarchical model, reveals that the Hierarchical model is the most suitable clustering algorithm for plain-woven composites at T > Tg. The cluster validity indexes combined with microscopic observations identify matrix cracking, interface debonding and fiber breakage as the three dominant damage modes, with PF ranges of [0–200 kHz], [200–350 kHz] and [350–600 kHz], respectively, and independent of different loading conditions. Three damage initiation criteria, sentry function, b-value and cumulative AE counts, are compared and the sentry function is determined to be optimal. The damage initiations of matrix cracking, interface debonding and fiber breakage within plain-woven composites at T > Tg can be effectively detected by AE.

From A-to-Z review of clustering validation indices

Data clustering involves identifying latent similarities within a dataset and organizing them into clusters or groups. The outcomes of various clustering algorithms differ as they are susceptible to the intrinsic characteristics of the original dataset, including noise and dimensionality. The effectiveness of such clustering procedures directly impacts the homogeneity of clusters, underscoring the significance of evaluating algorithmic outcomes. Consequently, the assessment of clustering quality presents a significant and complex endeavor. A pivotal aspect affecting clustering validation is the cluster validity metric, which aids in determining the optimal number of clusters. The main goal of this study is to comprehensively review and explain the mathematical operation of internal and external cluster validity indices, but not all, to categorize these indices and to brainstorm suggestions for future advancement of clustering validation research. In addition, we review and evaluate the performance of internal and external clustering validation indices on the most common clustering algorithms, such as the evolutionary clustering algorithm star (ECA*). Finally, we suggest a classification framework for examining the functionality of both internal and external clustering validation measures regarding their ideal values, user-friendliness, responsiveness to input data, and appropriateness across various fields. This classification aids researchers in selecting the appropriate clustering validation measure to suit their specific requirements.

Spatially constrained statistical approach for determining the optimal number of regions in regionalization

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.

Partitioning of wind pressure coefficients on a tall building under interference effect: A cluster analysis approach

This study proposed a cluster analysis-based approach for partitioning wind pressure on tall buildings in densely-packed areas, considering interference effects. By utilizing a modified GK cluster algorithm and leveraging the coordinate positions of pressure taps along with their corresponding wind pressure coefficients (WPC), the study aimed to distinguish different parts of the building surface exhibiting varying wind load characteristics. The adoption of cluster validation indices (CVIs) allowed for the determination of the optimal number of clusters, enhancing the accuracy and reliability of the partitioning process. The resulting clusters effectively captured the distribution of wind pressure, providing valuable insights for wind-resistant design considerations. Overall, the proposed approach demonstrates promising potential in facilitating the design and analysis of tall buildings subjected to complex wind loads in urban environments.

DEA-based internal validity index for clustering

Internal validity indices are crucial in evaluating the quality of clustering results, serving as valuable tools for comparing various clustering algorithms and determining the optimal number of clusters for datasets. Most existing internal validity indices use the worst-case scenario to represent the overall validity. Moreover, some indices assign equal weights to distances among different clusters, even when these distances might have varying degrees of influence on the overall validity. Data envelopment analysis (DEA) is an effective technique for evaluating the performance of decision-making units through the computation of the ratio of the weighted sum of outputs to the weighted sum of inputs. The weight assigned to each indicator signifies its degree of influence on efficiency. Furthermore, DEA can be viewed as a multiple-criteria evaluation methodology, wherein inputs and outputs are two sets of performance criteria. We propose a DEA-based internal validity index (DEAI) to evaluate the validity of the clustering results. In this approach, intra-cluster compactness and inter-cluster separation are employed for determining the input(s) and output(s). The DEAI is then applied to the artificial datasets and empirical examples. Experimental results illustrate that DEAI outperforms six classic internal validity indices in accurately identifying the optimal cluster across all 10 datasets.

Ship trajectory segmentation and semisupervised clustering via geospatial background knowledge

Cluster analysis is a common application of maritime traffic pattern recognition. However, most current methods are unsupervised clustering algorithms that rely solely on the features of the trajectories themselves for clustering, making them susceptible to uneven density distributions. To address this issue, we propose a semisupervised clustering algorithm that utilizes geospatial data to segment trajectories and generate clustering constraints. First, shoreline and anchorage information are employed for trajectory segmentation to enhance the consistency of local trajectory features. Second, cannot-link constraints derived from shoreline data are used to modify the distance between two trajectories, thereby preventing the clustering of trajectories separated by nonnavigable waters into a single cluster. Last, experimental verification is conducted using automatic identification system data, and the results are compared with those of existing algorithms. The results demonstrate more concentrated trajectory segmentation points, easier selection of clustering parameters, and significantly improved evaluation outcomes based on external clustering validity indices.

Extended multivariate comparison of 68 cluster validity indices. A review

Clustering is an unsupervised machine learning methodology widely used in several sciences to find groups of similar patterns in complex data. The results generated by clustering algorithms generally depend on user-defined input parameters such as the number of expected clusters, which can have a great impact on the homogeneity of the identified clusters.Clustering validity indices (CVIs) are an effective method for determining the optimal number of clusters that best fit the natural partition of a dataset. They do not require any underlying assumption nor a priori knowledge about the true dataset structure. Since 1965, many cluster validity indices have been proposed in the literature and used in several different applications.In this paper, the performance of 68 cluster validity indices was evaluated on 21 real-life research and simulated datasets. CVIs were compared on the same partition for each dataset, which was searched for by the k-means clustering algorithm. Multivariate chemometric methods were applied to disclose mutual relationships among the indices and to select those that are more effective in terms of accuracy and reliability.

New method to identify optimal discontinuity set number of rock tunnel excavation face orientation based on Fisher mixed evaluation

Discontinuity is critical for strength, deformability, and permeability of rock mass. Set information is one of the essential discontinuity characteristics and is usually accessed by orientation grouping. Traditional methods of identifying optimal discontinuity set numbers are usually achieved by clustering validity indexes, which mainly relies on the aggregation and dispersion of clusters and leads to the inaccuracy and instability of evaluation. This paper proposes a new method of Fisher mixed evaluation (FME) to identify optimal group numbers of rock mass discontinuity orientation. In FME, orientation distribution is regarded as the superposition of Fisher mixed distributions. Optimal grouping results are identified by considering the fitting accuracy of Fisher mixed distributions, the probability monopoly and central location significance of independent Fisher centers. A Halley-Expectation-Maximization (EM) algorithm is derived to achieve an automatic fitting of Fisher mixed distribution. Three real rock discontinuity models combined with three orientation clustering algorithms are adopted for discontinuity grouping. Four clustering validity indexes are used to automatically identify optimal group numbers for comparison. The results show that FME is more accurate and robust than the other clustering validity indexes in optimal discontinuity group number identification for different rock models and orientation clustering algorithms.

A parametric design framework for the mass customization of bicycle helmet

Cluster analysis of 3D head shapes plays a crucial role in the mass customization design of products related to the head. Head shapes exhibit variations across different races, and designing helmets exclusively for Chinese individuals cannot solely rely on or reference foreign head models. Currently, research on cluster analysis of Chinese head shapes is limited, especially concerning shape variances. To address this, we developed an improved k-medoids algorithm and integrated Cluster Validity Index as an assessment metric. This enabled us to cluster 339 Chinese young males aged 18 to 30 into 7 groups based on their head shapes. By comparing our improved algorithm to the traditional k-medoids method, we affirmed its superiority in achieving higher sample participation rates and reducing inter-cluster sample disparities. To simplify the helmet design and editing process, and to improve the efficiency of mass customization, we have developed a parametric modeling program for bicycle helmets based on the head shape clustering results. Results from the Helmet Fit Index and stress simulation analysis demonstrate that our approach significantly enhances helmet fit and wearer comfort.

Ensemble CART surrogate-assisted automatic multi-objective rough fuzzy clustering algorithm for unsupervised image segmentation

Multi-objective clustering algorithms (MOCAs) are popular in unsupervised image segmentation due to their merit of meeting multiple segmentation requirements and the prospect of automatically estimating the number of clusters. However, most of them suffer from high time costs and are easily to be influenced by the uncertainty when handling real complex images. To address these issues, we propose an ensemble classification and regression tree (CART) surrogate-assisted automatic multi-objective rough fuzzy clustering (ECS-AMRFC) algorithm for unsupervised image segmentation. Firstly, a cluster medoid-based encoding scheme is employed to represent solutions with different number of clusters and meanwhile lessen the length of encoding. Then, we design an ensemble CART as the surrogate model to significantly reduce the computational burden. Moreover, a surrogate model management strategy is proposed to accelerate the optimization and enhance the quality of surrogate modeling. To handle the uncertainty in data, we extend the rough fuzzy clustering into MOCAs and construct three complementary objective functions to seek proper cluster medoids from multiple perspectives. In addition, the Gaussian kernel is introduced into the objective functions to handle image pixels that cannot separate linearly in the feature space. Finally, a kernelized rough fuzzy clustering validity index is defined to automatically select the optimal solution with no requirements of any prior knowledge. Experiments show that ECS-AMRFC not only identifies appropriate number of clusters on different kinds of images, but also obtains better segmentation results than state-of-the-art rough fuzzy clustering algorithms and automatic MOCAs.

Application of clustering strategy for automatic segmentation of tissue regions in mass spectrometry imaging.

Mass spectrometry imaging (MSI) has been widely used in biomedical research fields. Each pixel in MSI consists of a mass spectrum that reflects the molecule feature of the tissue spot. Because MSI contains high-dimensional datasets, it is highly desired to develop computational methods for data mining and constructing tissue segmentation maps. To visualize different tissue regions based on mass spectrum features and improve the efficiency in processing enormous data, we proposed a computational strategy that consists of four procedures including preprocessing, data reduction, clustering, and quantitative validation. In this study, we examined the combination of t-distributed stochastic neighbor embedding (t-SNE) and hierarchical clustering (HC) for MSI data analysis. Using publicly available MSI datasets, one dataset of mouse urinary bladder, and one dataset of human colorectal cancer, we demonstrated that the generated tissue segmentation maps from this combination were superior to other data reduction and clustering algorithms. Using the staining image as a reference, we assessed the performance of clustering algorithms with external and internal clustering validation measures, including purity, adjusted Rand index (ARI), Davies-Bouldin index (DBI), and spatial aggregation index (SAI). The result indicated that SAI delivered excellent performance for automatic segmentation of tissue regions in MSI. We used a clustering algorithm to construct tissue automatic segmentation in MSI datasets. The performance was evaluated by comparing it with the stained image and calculating clustering validation indexes. The results indicated that SAI is important for automatic tissue segmentation in MSI, different from traditional clustering validation measures. Compared to the reports that used internal clustering validation measures such as DBI, our method offers more effective evaluation of clustering results for MSI segmentation. We envision that the proposed automatic image segmentation strategy can facilitate deep learning in molecular feature extraction and biomarker discovery for the biomedical applications of MSI.

A deep clustering framework for load pattern segmentation

In recent years, the widespread use of smart meters in power networks has generated a wealth of data from electricity customers. However, much of this real-world smart meter dataset lacks labeled data, posing a significant challenge that can be addressed by clustering. Traditional clustering methods often struggle in high-dimensional spaces, leading to less accurate results and increased computational demands. In response to these challenges, this research introduces a framework that utilizes a deep-learning-based clustering approach to address the issue of accurate customer clustering based on usage patterns in unlabeled data. Using an autoencoder, our approach integrates dimensionality reduction and clustering into an end-to-end unsupervised learning framework. Our algorithm significantly improves load profiling by tackling challenges related to nonlinear decision boundaries at the autoencoder bottleneck. Unlike traditional approaches, we propose separating the optimization of reconstruction and cluster loss, bridging the gap between clustering quality and reconstruction efficiency. We rigorously analyze the performance of our approach by comparing classical and state-of-the-art algorithms using two real-world smart meter data. We provide a comprehensive comparative analysis of our method against five common dimension reduction techniques used in high-dimensional clustering. The experimental analysis concludes that the proposed algorithm significantly enhances load profiling more than others, as confirmed through detailed load curve analysis and clustering validity indexes. This comprehensive assessment highlights the effectiveness and versatility of our proposed methodology when compared to others. Moreover, this research advances load profiling in smart grid analytics, providing practical insights for utilities and stakeholders looking to optimize power network operations.

ICVI-ARTMAP: Using Incremental Cluster Validity Indices and Adaptive Resonance Theory Reset Mechanism to Accelerate Validation and Achieve Multiprototype Unsupervised Representations.

This article presents an adaptive resonance theory predictive mapping (ARTMAP) model, which uses incremental cluster validity indices (iCVIs) to perform unsupervised learning, namely, iCVI-ARTMAP. Incorporating iCVIs to the decision-making and many-to-one mapping capabilities of this adaptive resonance theory (ART)-based model can improve the choices of clusters to which samples are incrementally assigned. These improvements are accomplished by intelligently performing the operations of swapping sample assignments between clusters, splitting and merging clusters, and caching the values of variables when iCVI values need to be recomputed. Using recursive formulations enables iCVI-ARTMAP to considerably reduce the computational burden associated with cluster validity index (CVI)-based offline clustering. In this work, six iCVI-ARTMAP variants were realized via the integration of one information-theoretic and five sum-of-squares-based iCVIs into fuzzy ARTMAP. With proper choice of iCVI, iCVI-ARTMAP either outperformed or performed comparably to three ART-based and four non-ART-based clustering algorithms in experiments using benchmark datasets of different natures. Naturally, the performance of iCVI-ARTMAP is subject to the selected iCVI and its suitability to the data at hand; fortunately, it is a general model in which other iCVIs can be easily embedded.

Multi-fuzzy clustering validity index ensemble: A Dempster-Shafer theory-based parallel and series fusion

Clustering validity evaluation is a key part in clustering process. To adapt the complex data structure, the traditional fuzzy clustering validity index (FCVI) is designed more complex. The weighted combined validity evaluation method (WCVEM) is simple in structure but difficult in weight selection. Therefore, this paper proposed an ensemble method based on multi-fuzzy clustering algorithms and multi-FCVI. Firstly, multi-FCVI are calculated by using the multiple sets of cluster centers and membership degrees that obtained by multi-fuzzy clustering algorithms. This can improve the robustness of the multi-FCVI. Secondly, multi-FCVI are ensembled by Dempster-Shafer (DS) theory. The validity index basic probability assignment function can be obtained by calculating the credibility of each validity index with different clusters number. Finally, the decision module is used to output the optimal clusters number. This paper ensembles multi-fuzzy clustering algorithms, multi-FCVI, and the DS theory by using series and parallel structure to verify performance of the proposed model and the degree of information retention of the FCVI. The proposed method is simple in structure and does not need to be select weighted. 6 artificial datasets and 12 UCI datasets were selected to simulate and verify the method. When facing different data, the simulation results show that the parallel structure has the highest accuracy, and the series structure is even worse than the weighted method in some datasets. In addition, the paper changes the value of fuzzy weighted, and experimental results show that the ensemble method has better stability than other methods in the face of different fuzzy weighted strategy.