Correct Number Of Clusters Research Articles

Cluster Detection and Centers of Mass Prediction in Images

This paper provides an exhaustive evaluation of different algorithms for the accurate detection and localization of dense clusters on complex image data with a resolution to maintain real centroids in obscure, overlapping, or densely packed situations. Our approach differs by its use of the local maximum algorithm to quickly identify potential cluster centers (by detecting local maxima in pixel intensity) as a first step. The algorithm starts by initializing cluster centers, which are then refined using the k-Means clustering method that iteratively relocates the positions of these clusters until an optimal configuration is reached. We tighten our approach by adding custom loss functions to Convolutional Neural Networks (CNNs) that encode the predicted number of clusters and their center positions. An early stopping mechanism is introduced to mitigate the issues of noise and overfitting, with performance measured against a test set. Furthermore, we examine the average pooling and center-of-gravity approaches for noise reduction to predict cluster centers without unnecessary outliers. This combination, along with the Hungarian algorithm, which is used to optimally match predicted and real centers, has shown better results in detecting the correct number of clusters across different datasets. These validation results show that our combined method effectively addresses challenges related to overlapping clusters, increasing robustness and accuracy in identification. The approach seems to have potential for applications in high-precision image segmentation and analysis.

Manhattan Metric Technique in K-Means Clustering for Data Grouping

Clustering can be defined as a method commonly applied in data mining to group objects into clusters. Clusters consist of data objects that are similar to each other in a group but different from objects in other clusters. In this study, the data used is the data of KIP scholarship recipients for the 2016-2023 period. Various clustering metric measurement techniques have been frequently used by researchers, especially those focusing on distance and similarity metrics, such as Euclidean Distance, Manhattan, and Minkowski. In general, K-Means is an unsupervised learning method used in the clustering process to group data based on similarity. The elbow method is used to determine the optimal number of clusters, so that the clustering results obtained can be maximized to achieve better results. This study aims to analyze the use of Manhattan technique in K-Means clustering for data grouping. The research problem is how to analyze the Manhattan metric technique in K-Means clustering for effective data grouping. Applying the K-Means method shows that the existing data is successfully divided into four specified clusters. After determining the correct number of clusters, the K-Means method is used to sort the data in the dataset. From 3172 data, the final results obtained cluster 0 as many as 774 data, cluster 1 as many as 417 data, cluster 2 as many as 1244 data, and cluster 3 as many as 737 data. The results of the clustering process obtained a davies-bouldin index value of 1.4568.

An active multiphase probabilistic power flow based on a clustering approach

Probabilistic methods for analyzing power systems have become more widespread, mainly due to the increase in load and generation uncertainties and the current need for planning problems that are sufficiently accurate to take appropriate actions. However, some characteristics need to be better investigated in the Distribution Systems (DS) analysis, such as the input random variables’ modeling using real databases and, mainly, the probabilistic treatment of unbalance and scalability studies. This paper proposes an active Multiphase Probabilistic Power Flow (MPPF) evaluation, considering real measurement databases to represent customer demand uncertainties. It is called Multiphase Probabilistic Power Flow Based on K-Means (MPPFKM), and it encompasses a mixture of a clustering technique and an analytical method. The K-Means algorithm divides the input database into a set of clusters that models the scenarios to be evaluated in the multiphase power flow problem, which considers the distribution network with its inherent unbalance. Subsequently, the output variables’ higher order statistical moments are calculated using the obtained responses for each cluster. Finally, their PDFs and CDFs are recovered using the Gram-Charlier Expansion analytical method. While the clustering algorithm allows working with real data, giving the proposed work greater proximity to real-world results, the analytical tool provides an accurate recovery of the probability functions, because it detects their distortions caused by the nature of the multiphase problem. Estimating the correct number of clusters is a key factor in obtaining the precision required by power systems engineers and achieving significant computational time reduction, especially for large-scale unbalanced systems. Simulations are carried out using several DS feeders to test the proposed method, which specifically addresses DS issues not yet well explored in the literature, such as probabilistic simulations under different unbalance situations, applications with single-phase DGs, and scalability studies are conducted using the IEEE 8500 test feeder. Comparisons are made with usual MPPF algorithms, including Monte Carlo Simulation (MCS) and Point Estimation Methods (PEM). Simulations using large-scale unbalanced systems proved that the output variables' probability functions might present distortions, becoming more challenging for other methods. Under this perspective, MPPFKM presents better exactness in the calculation of higher order statistical moments and, consequently, can estimate the PDFs of the MPPF output variables with great precision, which is crucial for statistical models that use actual input data.

A Theoretical Analysis of Density Peaks Clustering and the Component-Wise Peak-Finding Algorithm.

Density peaks clustering detects modes as points with high density and large distance to points of higher density. Each non-mode point is assigned to the same cluster as its nearest neighbor of higher density. Density peaks clustering has proved capable in applications, yet little work has been done to understand its theoretical properties or the characteristics of the clusterings it produces. Here, we prove that it consistently estimates the modes of the underlying density and correctly clusters the data with high probability. However, noise in the density estimates can lead to erroneous modes and incoherent cluster assignments. A novel clustering algorithm, Component-wise Peak-Finding (CPF), is proposed to remedy these issues. The improvements are twofold: 1) the assignment methodology is improved by applying the density peaks methodology within level sets of the estimated density; 2) the algorithm is not affected by spurious maxima of the density and hence is competent at automatically deciding the correct number of clusters. We present novel theoretical results, proving the consistency of CPF, as well as extensive experimental results demonstrating its exceptional performance. Finally, a semi-supervised version of CPF is presented, integrating clustering constraints to achieve excellent performance for an important problem in computer vision.

Weather types for the seasonal transitions in Central America

Unsupervised learning techniques are employed to study the relationship between atmospheric circulation and precipitation over Central America and its surrounding areas. Specifically, the clustering algorithm k-means++ is applied to three coarse-grained datasets from ERA-interim reanalysis that are the candidates for representing the atmospheric state vector, each candidate contains its full temporal variability. Datasets are composed of: a) wind fields at 925, 800 and 200 hPa, b) same as “a)” plus convective available potential energy and c) same as “a)” plus total column water vapor. Clustering metrics, namely the variance ratio criterion, the silhouette criterion and the mean squared error, are computed to quantify clustering quality. Clusters are interpreted as weather types, recurrent configurations of the atmospheric state vector associated with observable weather states. The correct number of clusters for each dataset is determined with a Monte Carlo test of normality, to assure cluster existence. The main objective is to obtain a set of weather types containing elements that characterize the transition from and to the rainy season over the Pacific side of Central America as well as other elements of the seasonal cycle of regional precipitation, such as the Mid-Summer Drought. Besides the statistical metrics, in order to select between candidate datasets and plausible number of clusters, focus is given to the temporal characteristics of the clusters. Existing literature does not provide a set of weather types suitable to analyze seasonal transitions and the differences in the mechanisms associated with rainfall maxima.

Efficient estimation of the number of clusters for high-dimension data

The exponential growth of digital image data has given rise to the need of efficient content management and retrieval tools. Currently, there is a lack of tools for processing the collected unlabeled data in a schematic manner. K-means is one of the most widely used clustering methods and has been applied in a variety of fields, one of them being image sorting. Although a useful tool for image management, the K-means method is heavily influenced by initializations, the most important one being the need to know the number of clusters a priori. A number of different methods have been proposed for identifying the correct number of clusters for K-means, one of them being the variance ratio criterion (VRC). Despite its popularity, the VRC method comes with two very important shortcomings: it only yields good results when the data dimensionality is low and it does not scale well for a high number of clusters, making it very difficult to use in computer vision applications. We propose an extension to the VRC method that works for increased cluster number and high-dimensionality data sets and therefore is fit for image data sets.

Clustering approximation via a fusion of multiple random samples

In big data clustering exploration, the situation is paradoxical because there is no prior or insufficient domain knowledge. Moreover, clustering a big dataset is a challenging task in distributed computing framework. To address this, we propose a new distributed clustering approximation framework for big data with quality guarantees. This innovative framework uses multiple disjoint random samples instead of a single random sample to compute an ensemble result as the estimation of the true result of the entire big dataset. To begin, we modeled a large dataset as a collection of random sample data blocks stored in a distributed file system. Henceafter, a subset of data blocks is randomly selected, and to generate the component clustering results, the serial clustering algorithm is executed in parallel on the distributed computing framework. In each selected random sample, the number of clusters and initial centroids is identified using a density peak-based I-niceDP clustering algorithm, and then the k-means sweep refines them. Since the random samples are disjoint and traditional consensus functions cannot be used, we propose two new methods, a graph similarity and a naturally inspired firefly-based algorithm, to integrate the component clustering results into the final ensemble result. The entire clustering process is displayed through systematic support, extensive measures of clusterability, and quality evaluation. The methods are verified in a series of experiments using synthetic and real-world datasets. Our comprehensive experimental results demonstrate that the proposed methods vividly (1) recognize the correct number of clusters by analyzing a subset of samples and (2) exhibit better scalability, efficiency, and clustering stability.

A Fuzzy Clustering Validity Index Induced by Triple Center Relation.

The existing clustering validity indexes (CVIs) show some difficulties to produce the correct cluster number when some cluster centers are close to each other, and the separation processing mechanism appears simple. The results are imperfect in case of noisy data sets. For this reason, in this study, we come up with a novel CVI for fuzzy clustering, referred to as the triple center relation (TCR) index. The originality of this index is twofold. On the one hand, a new fuzzy cardinality is built on the strength of the maximum membership degree, and a novel compactness formula is constructed by combining it with the within-class weighted squared error sum. On the other hand, starting from the minimum distance between different cluster centers, the mean distance as well as the sample variance of cluster centers in the statistical sense are further integrated. These three factors are combined by means of product to form a triple characterization of the relationship between cluster centers, and hence a 3-D expression pattern of separability is formed. Subsequently, the TCR index is put forward by combining the compactness formula with the separability expression pattern. By virtue of the degenerate structure of hard clustering, we show an important property of the TCR index. Finally, based on the fuzzy C -means (FCMs) clustering algorithm, experimental studies were conducted on 36 data sets (incorporating artificial and UCI data sets, images, the Olivetti face database). For comparative purposes, 10 CVIs were also considered. It has been found that the proposed TCR index performs best in finding the correct cluster number, and has excellent stability.

ACQC: Apollonius Circle‐based Quantum Clustering

The Quantum Clustering (QC) method begins by generating a wave function for estimating the data points’ density distribution, which is the sum of Gaussian kernels to the data points’ centers. It then forms the corresponding potential function and finally locates clusters, emphasizing the potential function’s minimums. This process is highly sensitive to the kernel bandwidth in the Schrödinger equation that controls the shape of the Gaussian kernel and affects the clustering result. This paper proposes an Apollonius Circle-based Quantum Clustering (ACQC) method, which adaptively and automatically sets the kernel bandwidth without any prior knowledge regarding the data points and clusters. ACQC is the first attempt to achieve such an adaptive kernel bandwidth through the Apollonius region’s neighborhood construction. The wave function is estimated based on data points in the neighborhood group constructed by Apollonius circles to optimize ACQC calculations. The experimental results of ACQC compared to the original QC method indicate an improvement in calculation efficiency by approximately a 38.3% reduction in terms of running time and a 41.17% improvement in detecting the correct number of clusters. Extensive experiments on four synthetic and 20 real-world datasets show that ACQC outperforms state-of-the-art methods significantly. All the implementation source codes of ACQC are made publicly available at https://github.com/NAbdolmaleki/ACQC-Apollonius-Circle-based-Quantum-Clustering.

Entia Non Sunt Multiplicanda … Shall I look for clusters in my cognitive data?

Unsupervised clustering methods are increasingly being applied in psychology. Researchers may use such methods on multivariate data to reveal previously undetected sub-populations of individuals within a larger population. Realistic research scenarios in the cognitive science may not be ideally suited for a successful use of these methods, however, as they are characterized by modest effect sizes, limited sample sizes, and non-orthogonal indicators. This combination of characteristics even presents a high risk of detecting non-existing clusters. A systematic review showed that, among 191 studies published in 2016-2020 that used different clustering methods to classify human participants, the median sample size was only 322, and a median of 3 latent classes/clusters were detected. None of them concluded in favor of a one-cluster solution, potentially giving rise to an extreme publication bias. Dimensionality reduction techniques are almost never used before clustering. In a subsequent simulation study, we examined the performance of popular clustering techniques, including Gaussian mixture model, a partitioning, and a hierarchical agglomerative algorithm. We focused on their ability to detect the correct number of clusters, and on their classification accuracy. Under a reasoned set of scenarios that we considered plausible for the cognitive research, none of the methods adequately discriminates between one vs two true clusters. In addition, non-orthogonal indicators lead to a high risk of incorrectly detecting multiple clusters where none existed, even in the presence of only modest correlation (a frequent case in psychology). In conclusion, it is hard for researchers to be in a condition to achieve a valid unsupervised clustering for inferential purposes with a view to classifying individuals.

Factor models for large and incomplete data sets with unknown group structure

Most economic applications rely on a large number of time series, which typically have a remarkable clustering structure and they are available over different spans. To handle these databases, we combined the expectation–maximization (EM) algorithm outlined by Stock and Watson (JBES, 2002) and the estimation algorithm for large factor models with an unknown number of group structures and unknown membership described by Ando and Bai (JAE, 2016; JASA, 2017) . Several Monte Carlo experiments demonstrated the good performance of the proposed method at determining the correct number of clusters, providing the appropriate number of group-specific factors, identifying error-free group membership, and obtaining accurate estimates of unobserved missing data. In addition, we found that our proposed method performed substantially better than the standard EM algorithm when the data had a grouped factor structure. Using the Federal Reserve Economic Data FRED-QD, our method detected two distinct groups of macroeconomic indicators comprising the real activity indicators and nominal indicators. Thus, we demonstrated the usefulness of our group-specific factor model for studies of business cycle chronology and for forecasting purposes.

Selection of the number of clusters in functional data analysis

Identifying the number K of clusters in a dataset is one of the most difficult problems in clustering analysis. A choice of K that correctly characterizes the features of the data is essential for building meaningful clusters. In this paper we tackle the problem of estimating the number of clusters in functional data analysis by introducing a new measure that can be used with different procedures in selecting the optimal K. The main idea is to use a combination of two test statistics, which measure the lack of parallelism and the mean distance between curves, to compute criteria such as the within and between cluster sum of squares. Simulations in challenging scenarios suggest that procedures using this measure can detect the correct number of clusters more frequently than existing methods in the literature. The application of the proposed method is illustrated on several real datasets.

Adaptive Mixed‐Attribute Data Clustering Method Based on Density Peaks

The clustering of mixed‐attribute data is a vital and challenging issue. The density peaks clustering algorithm brings us a simple and efficient solution, but it mainly focuses on numerical attribute data clustering and cannot be adaptive. In this paper, we studied the adaptive improvement method of such an algorithm and proposed an adaptive mixed‐attribute data clustering method based on density peaks called AMDPC. In this algorithm, we used the unified distance metric of mixed‐attribute data to construct the distance matrix, calculated the local density based on K‐nearest neighbors, and proposed the automatic determination method of cluster centers based on three inflection points. Experimental results on real University of California‐Irvine (UCI) datasets showed that the proposed AMDPC algorithm could realize adaptive clustering of mixed‐attribute data, can automatically obtain the correct number of clusters, and improved the clustering accuracy of all datasets by more than 22.58%, by 24.25%, by 28.03%, by 22.5%, and by 10.12% for the Heart, Cleveland, Credit, Acute, and Adult datasets compared to that of the traditional K‐prototype algorithm, respectively. It also outperformed a modified density peaks clustering algorithm for mixed‐attribute data (DPC_M) algorithms.

Cooperation and compressed data exchange between multiple gliders used to map oil spills in the ocean

This work describes a cooperation strategy and method to compress data to be transmitted between multiple underwater gliders used to delineate oil spills in the ocean. In the cooperation strategy, the survey area is discretized into a series of blocks. One glider is used as a scout to map the oil in each block. The scout glider then sends the locations of potential information-rich oil patches to follower gliders to conduct detailed surveys of the oil patch areas. To compress the data to be transmitted between gliders, the Support Vector Machine (SVM) method was used to classify the boundaries of the oil patches. Measurements from the scout glider were mapped onto a grid map to decrease the quantity of data being processed and a clustering method was proposed to be applied prior to the SVM to simplify the characteristics of the data sets. Two clustering methods were compared through simulations, of which a Density-Based Spatial Clustering of Applications with Noise method was found to be more accurate and reliable than a mean-shift method in classifying the correct number of clusters. Based on the classified boundaries of the oil patches, the scout glider could then send compressed data to its followers so that they could further investigate the patches. The proposed method compresses the information communicated between gliders which in turn was found in simulations to improve the efficiency of real-time underwater communication and cooperation between networked gliders.

Clustering functional data using forward search based on functional spatial ranks with medical applications.

Cluster analysis of functional data is finding increasing application in the field of medical research and statistics. Here we introduce a functional version of the forward search methodology for the purpose of functional data clustering. The proposed forward search algorithm is based on the functional spatial ranks and is a data-driven non-parametric method. It does not require any preprocessing functional data steps, nor does it require any dimension reduction before clustering. The Forward Search Based on Functional Spatial Rank (FSFSR) algorithm identifies the number of clusters in the curves and provides the basis for the accurate assignment of each curve to its cluster. We apply it to three simulated datasets and two real medical datasets, and compare it with six other standard methods. Based on both simulated and real data, the FSFSR algorithm identifies the correct number of clusters. Furthermore, when compared with six standard methods used for clustering and classification, it records the lowest misclassification rate. We conclude that the FSFSR algorithm has the potential to cluster and classify functional data.

A New Fuzzy Cluster Validity Index for Hyperellipsoid or Hyperspherical Shape Close Clusters With Distant Centroids

Determining the correct number of clusters is essential for efficient clustering and cluster validity indices are widely used for the same. Generally, the effectiveness of a cluster validity index relies on two factors: first, separation, defined by the distance between a pair of cluster centroids or a pair of data points belonging to different clusters and second, compactness, which is determined in terms of the distance between a data point and a centroid or between a pair of data points belonging to the same cluster. However, the existing cluster validity indices for centroid-based clustering are unreliable when the clusters are too close, but corresponding centroids are distant. To mitigate this, a new cluster validity index, Saraswat-and-Mittal index, has been proposed in this article for hyperellipsoid or hyperspherical shape close clusters with distant centroids, generated by fuzzy c-means. The proposed index computes compactness in terms of the distance between data points and corresponding centroids, whereas the distance between data points of disjoint clusters defines separation. These parameters benefit the proposed index in the analysis of close clusters with distinct centroids efficiently. The performance of the proposed index is validated against ten state-of-the-art cluster validity indices on artificial, UCI, and image datasets, clustered by the fuzzy c-means.

A novel cluster detection of COVID-19 patients and medical disease conditions using improved evolutionary clustering algorithm star

With the increasing number of samples, the manual clustering of COVID-19 and medical disease data samples becomes time-consuming and requires highly skilled labour. Recently, several algorithms have been used for clustering medical datasets deterministically; however, these definitions have not been effective in grouping and analysing medical diseases. The use of evolutionary clustering algorithms may help to effectively cluster these diseases. On this presumption, we improved the current evolutionary clustering algorithm star (ECA*), called iECA*, in three manners: (i) utilising the elbow method to find the correct number of clusters; (ii) cleaning and processing data as part of iECA* to apply it to multivariate and domain-theory datasets; (iii) using iECA* for real-world applications in clustering COVID-19 and medical disease datasets. Experiments were conducted to examine the performance of iECA* against state-of-the-art algorithms using performance and validation measures (validation measures, statistical benchmarking, and performance ranking framework). The results demonstrate three primary findings. First, iECA* was more effective than other algorithms in grouping the chosen medical disease datasets according to the cluster validation criteria. Second, iECA* exhibited the lower execution time and memory consumption for clustering all the datasets, compared to the current clustering methods analysed. Third, an operational framework was proposed to rate the effectiveness of iECA* against other algorithms in the datasets analysed, and the results indicated that iECA* exhibited the best performance in clustering all medical datasets. Further research is required on real-world multi-dimensional data containing complex knowledge fields for experimental verification of iECA* compared to evolutionary algorithms.

Object-based cluster validation with densities

Clustering validity indices are typically used as tools to find the correct number of clusters in a data set and/or to evaluate the quality of the clusters formed by clustering algorithms. Clustering validity indices measure separation and compactness of clusters. Typically, when applying a clustering algorithm, the input includes the number of clusters. After applying the algorithm with several different numbers of clusters, we determine the number of clusters to be the one with the best validity index. There are two types of clustering validity indices: external indices that are supervised, and internal indices that are unsupervised. The focus of this paper is on internal validity indices. Some existing internal validity indices capture the properties of the clusters by using representative statistics such as mean, variance, diameter, etc., however, these do not perform well when clusters have arbitrary shapes. One approach to overcome this issue is to use the density of the data objects in each cluster. That provides the advantage of capturing the full characteristics of the cluster which is most beneficial when there are clusters with arbitrary shapes. In the literature, a few density-based clustering validity indices have been proposed. However, some of them show poor performance when the clusters are not perfectly separated. Some others perform poorly because they use only representative objects from each cluster instead of all objects. The contribution of this paper is an internal validity index named the object-based clustering validity index with densities (OCVD). OCVD is a single number that averages the density-based contribution of individual data objects to both separation and compactness of clusters. The methodology behind calculating the density-based contributions of the objects is kernel density estimation. We show through several experiments that OCVD performs well in detecting the correct number of clusters in data sets with different cluster shapes including arbitrary shapes.

Improved Self-Adaptive ACS Algorithm to Determine the Optimal Number of Clusters

A fundamental problem in data clustering is how to determine the correct number of clusters. The k-adaptive medoid set ant colony optimization (ACO) clustering (METACOC-K) algorithm is superior in solving clustering problems. However, METACOC-K does not guarantee in finding the best number of clusters. It assumed the number of clusters based on an adaptive parameter strategy that lacks feedback learning. This has restrained the algorithm in producing compact clusters and the optimal number of clusters. In this paper, a self-adaptive ACO clustering (S-ACOC) algorithm is proposed to produce the optimal number of clusters by incorporating a self-adaptive parameter strategy. The S-ACOC algorithm is a centroid-based algorithm that automatically adjusts the number of clusters during the algorithm run. The selection of the number of clusters is based on a construction graph that reflects the influence of a pheromone in algorithm learning. Experiments were conducted on real-world datasets to evaluate the performance of the proposed algorithm. The external evaluation metrics (purity, F-measure, and entropy) were used to compare the results of the proposed algorithm with other swarm clustering algorithms, including a genetic algorithm (GA), particle swarm optimization (PSO), and METACOC-K. Results showed that S-ACOC provides higher purity (50%) and lower entropy (40%) than GA, PSO, and METACOC-K. Experiments were also performed on several predefined clusters, and results demonstrate that the S-ACOC algorithm is superior to GA, PSO, and METACOC-K. Based on the superior performance, S-ACOC can be used to solve clustering problems in various application domains.

An improved density peaks clustering algorithm based on CURE

As a new density-based clustering algorithm, clustering by fast search and find of Density Peaks (DP) algorithm regards each density peak as a potential clustering center when dealing with a single cluster with multiple density peaks, therefore it is difficult to determine the correct number of clusters in the data set. To solve this problem, a mixed density peak clustering algorithm namely C-DP was proposed. Firstly, the density peak points were considered as the initial clustering centers and the dataset was divided into sub-clusters. Then, learned from the Clustering Using Representatives algorithm (CURE), the scattered representative points were selected from the sub-clusters, the clusters of the representative point pairs with the smallest distance were merged, and a parameter contraction factor was introduced to control the shape of the clusters. The experimental results show that the C-DP algorithm has better clustering effect than the DP algorithm. The comparison of the F-measure Index shows that the C-DP algorithm improves the accuracy of clustering when datasets contain multiple density peaks in a single cluster.