Optimal cluster numbers of unsupervised classification in Minkowski spaces

Identification of coherent generators (CGs) is necessary for the area-based monitoring and protection system of a wide area power system. Synchrophasor has enabled smarter monitoring and control measures to be devised; hence, measurement-based methodologies can be implemented in online applications to identify the CGs. This paper presents a new framework for coherency identification that is based on the dynamic coupling of generators. A distance matrix that contains the dissimilarity indices between any pair of generators is constructed from the pairwise dynamic coupling of generators after the post-disturbance data are obtained by phasor measurement units (PMUs). The dataset is embedded in Euclidean space to produce a new dataset with a metric distance between the points, and then the support vector clustering (SVC) technique is applied to the embedded dataset to identify the final clusters of generators. Unlike other clustering methods that need a priori knowledge about the number of clusters or the parameters of clustering, this information is set in an automatic search procedure that results in the optimal number of clusters. The algorithm is verified by time-domain simulations of defined scenarios in 39 bus and 118 bus test systems. Finally, the clustering result of 39 bus systems is validated by cluster validity measures, and a comparative study investigates the efficacy of the proposed algorithm to cluster the generators with an optimal number of clusters and also its computational efficiency compared with other clustering methods.

With the development of shale gas reservoirs, various fracture propagation models have been developed to predict hydraulic fracture geometry by considering the stress shadow effects. Also, many researchers have been working on development of various production-simulation approaches to simulate production from the complex fracture geometries. The objective of this study is to combine the fracture propagation modeling and reservoir simulation to optimize production in shale gas reservoirs. First, we have developed a semi-analytical model to simulate shale gas production from the complex non-planar fracture geometry with varying fracture width and fracture permeability along fracture length. The important gas transport mechanisms including gas slippage, gas diffusion, and gas desorption are considered. Second, we integrate the semi-analytical model with a three-dimensional fracture propagation model, which fully couples elastic deformation of the rock and fluid flow to simulate complex hydraulic fracture propagation, to optimize fracture treatment design in Marcellus shale gas reservoirs. Specifically, the fracture propagation model is utilized to predict the more-realistic non-planar fracture geometry with varying fracture width and fracture permeability along fracture length. In this study, the effect of varying number of perforation clusters within a given stage ranging from 2 to 5 on the fracture propagation was investigated. Three values of stage spacing were considered including 100 ft, 200 ft, and 300 ft. After predicting the non-planar fracture geometry, the semi-analytical model is employed to simulate production from such fracture geometries. According to the well productivity, the optimal number of clusters per stage for varying stage spacing is discussed and determined. In addition, we compared the well performance with the optimal number of perforation clusters for the stage spacing of 100 ft, 200 ft, and 300 ft under a given horizontal well length of 2,500 ft. Furthermore, we used the integrated approach to analyze a well performance from Marcellus shale gas reservoirs. Through the field case study, the more-realistic non-planar fracture geometry was quantified. Also, the difference between the non-planar fracture geometry and ideal planar fracture geometry with equal fracture width and fracture half-length was compared. Finally, the optimal fracture treatment design for the field well development was recommended. This work can provide new insights into optimization of number of perforation clusters per stage within a given fracture stage and a better understanding of the non-planar fracture geometry in shale gas reservoirs.

BackgroundCluster analysis is the most common unsupervised method for finding hidden groups in data. Clustering presents two main challenges: (1) finding the optimal number of clusters, and (2) removing “outliers” among the objects being clustered. Few clustering algorithms currently deal directly with the outlier problem. Furthermore, existing methods for identifying the number of clusters still have some drawbacks. Thus, there is a need for a better algorithm to tackle both challenges.ResultsWe present a new approach, implemented in an R package called Thresher, to cluster objects in general datasets. Thresher combines ideas from principal component analysis, outlier filtering, and von Mises-Fisher mixture models in order to select the optimal number of clusters. We performed a large Monte Carlo simulation study to compare Thresher with other methods for detecting outliers and determining the number of clusters. We found that Thresher had good sensitivity and specificity for detecting and removing outliers. We also found that Thresher is the best method for estimating the optimal number of clusters when the number of objects being clustered is smaller than the number of variables used for clustering. Finally, we applied Thresher and eleven other methods to 25 sets of breast cancer data downloaded from the Gene Expression Omnibus; only Thresher consistently estimated the number of clusters to lie in the range of 4–7 that is consistent with the literature.ConclusionsThresher is effective at automatically detecting and removing outliers. By thus cleaning the data, it produces better estimates of the optimal number of clusters when there are more variables than objects. When we applied Thresher to a variety of breast cancer datasets, it produced estimates that were both self-consistent and consistent with the literature. We expect Thresher to be useful for studying a wide variety of biological datasets.

To find an optimal cluster size and number of clusters for a reasonable estimate of the prevalence of cataract blindness in people aged > or = 50 years in 19 rural districts of a state in India. Cluster sampling methodology was used in 19 rural districts of Karnataka State, India. In each district, 15 clusters were randomly selected and 90 people aged > or = 50 years were examined in each cluster. As a result the visual acuity and lens status of a total of 22,218 people were assessed. For each district, the design effect for cluster size ranging from 20 to 90 was calculated and the optimal cluster size and the required number of clusters to achieve an accuracy of 1% errors and 80% confidence was assessed. The age and gender adjusted prevalence of cataract blindness varied from 1.58% to 7.24%, which justifies district level surveys. The design effect is nearly 1.5 for clusters of sizes 30 and 40. With an average prevalence of 4.93% with 1% error and 80% confidence level, the optimal number of clusters is 37 and 28 for a cluster size of 30 and 40 respectively and the average sample size for a district around 1100. Rapid assessments for cataract blindness in those aged > or = 50 years can be conducted at district level in India with existing resources and at affordable costs. These provide reliable data, essential for effective monitoring and planning. Other parameters, for instance, surgical coverage can also be assessed. The availability of standardized software for data entry and analysis and strict adherence to survey procedures is essential.

Cluster validity measures are used in order to determine an appropriate number of clusters and evaluate cluster partitions obtained by clustering algorithms. When we handle a set of data, data contains inherent uncertainty e.g., errors, ranges or some missing value of attributes. The concept of tolerance has been proposed from the viewpoint of handling such uncertain data. In this paper, we introduce clustering algorithms for data with tolerance. Moreover, we propose new five measures for data with tolerance, that is, the determinants and the traces of fuzzy covariance matrices, the Xie-Beni's index, the Fukuyama-Sugeno's index, and the Davies-Bouldin's index. We compare the performance of conventional ones with their tolerance versions. We found that our proposed measures takes smaller value than conventional ones. These results indicate tolerance based clustering method is suitable for handling uncertain data.

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.

To the Editor: We read with great interest the article by Hadanny et al,1 published in Neurosurgery. In this study, the authors used an approach which augmented unsupervised and supervised machine learning methods to predict response to spinal cord stimulators. This paper is an excellent example of how both approaches can be used together to prognosticate on patient outcomes, and the methodological implications span beyond the impact on patients with spinal cord stimulators. Unsupervised machine learning (UML) operates on the premise that the ground truth is not known—for this reason, there are no training data involved in the model. The outcomes of UML algorithms are usually unique clusters identified in the data set. This varies from supervised machine learning (SML) algorithms, which construct a model based on known input and output data. In the literature, UML methods are commonly used to identify novel and emergent phenotypes, whereas SML has commonly been used to perform outcome prediction.2,3 When using UML methods, there are several important parameters to consider which may influence the outcomes seen. First parameter is the cluster number identification. In this paper, the authors used the K-nearest neighbor algorithm (KNN), preceded by an elbow-based cluster optimization step. The KNN algorithm itself is not a completely unsupervised algorithm and requires the inclusion of the number of preset clusters. It requires a cluster optimization step to identify this parameter. In their supplementary methods, they argue that, using the elbow method, a pseudoelbow was created at K = 3. As a result, the decision is made that two clusters exist in the data. We encourage the authors to consider the use of other approaches to validate this finding, including the silhouette method, which has been shown to be more robust and quantifiable in some cases and may be less prone to subjective examination of the analyst.4 The number of clusters is also not readily apparent in the reprojections to the principal component analysis domain space. We suggest the use of other dimensionality reduction approaches to visualize the clusters, namely t-distributed stochastic network embedding or uniform manifold approximation and projection. While the clustering should remain in the multidimensional space, dimensionality reduction approaches have been useful in visualizing multidimensional data otherwise difficult to appreciate. Second parameter is the algorithm itself. KNN is a very well-documented partitioning algorithm. However, it has several limitations. First, the algorithm requires the use of numerical data, which may distort the categorical data, such as patient biological sex, diagnosis, smoking status, and present inherent to the data set. In addition, the sample size required for use of the KNN method is also relatively large, with some studies suggesting the need for a sample size of 2^n, where n is the number of features (24 features in this study).5,6 Other clustering algorithms, such as hierarchical clustering, have been shown to be far more effective for smaller data sets, and we encourage the authors to explore the use of more robust algorithms that would better fit the constraints of the data. As more neurosurgical studies use machine learning, specifically UML, approaches for the analysis of data, it becomes very important for researchers to consider the aforementioned parameters to strengthen their observations and findings. We applaud Hadanny et al1 for an excellent presentation of how both UML and SML approaches can be augmented, and we look forward to additional novel contributions using both approaches in the future to better help understand complex disease processes and ultimately help patients.

Fuzzy clustering is a popular unsupervised learning method that is used in cluster analysis. Fuzzy clustering allows a data point to belong to two or more clusters. Fuzzy c-means is the most well-known method that is applied to cluster analysis, however, the shortcoming is that the number of clusters need to be predefined. This paper proposes a clustering approach based on Particle Swarm Optimization (PSO). This PSO approach determines the optimal number of clusters automatically with the help of a threshold vector. The algorithm first randomly partitions the data set within a preset number of clusters, and then uses a reconstruction criterion to evaluate the performance of the clustering results. The experiments conducted demonstrate that the proposed algorithm automatically finds the optimal number of clusters. Furthermore, to visualize the results principal component analysis projection, conventional Sammon mapping, and fuzzy Sammon mapping were used

This paper proposes a method of determining the optimal number of clusters dividing the multiple transformations for the purpose of the efficient processing of query against the results of applying the transformations to time series. In this paper, the moving average is used as a transformation for simplicity. The model of query time to the number of clusters is constructed for determining the optimal number of clusters. As the query time could be represented with the concave function of the number of clusters, it is shown that the optimal number of clusters for the best query time can be obtained. The verification experiment confirms the validity of the model constructed. It is revealed that the optimal number of clusters could be determined by the times obtained from a single query execution.

In this paper, a new multi-objective (MO) fuzzy-clustering technique is developed for unsupervised land-cover classification in remote-sensing images. This technique automatically partitions the pixels in the spectral domain into several clusters. The new MO clustering algorithm uses the newly developed simulated-annealing-based multi-objective optimization technique (AMOSA) as the underlying optimization strategy. Centre-based cluster encoding is used. Each cluster is divided into several small hyperspherical sub-clusters, and the centres of all these small sub-clusters are encoded in a string to represent the whole cluster. For membership-value computation of points to different clusters, these sub-clusters are considered individually. However, for the purpose of objective-function evaluation, these sub-clusters are merged appropriately to form some variable number of whole clusters. Three objective functions, one reflecting the total compactness of the partitioning based on the Euclidean distance, one reflecting the total point-symmetrical compactness of the obtained partitioning and one reflecting the total line-symmetrical compactness of the obtained partitioning are considered here. These are optimized simultaneously using AMOSA in order to detect the appropriate number of clusters and the appropriate partitioning from remote-sensing image datasets. The point symmetry present in a particular partitioning is measured using a newly developed point-symmetry-based distance, whereas the line symmetry present in a particular partitioning is measured using a newly developed line-symmetry-based distance. Since AMOSA, as well as any other MO optimization technique, provides a set of Pareto-optimal solutions, a new method is also developed to determine a single solution from this set. Different land-cover regions in remote-sensing imagery have been classified using the proposed technique to establish its efficiency. Results are compared with those obtained by fuzzy C-means (FCM) clustering technique and a recently developed symmetry-based automatic clustering technique, the variable string length genetic point symmetry (VGAPS) technique, both qualitatively and quantitatively. Qualitative comparisons are also carried out with the segmentation results obtained by the mean-shift-based segmentation technique.

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.

In solving the hard clustering problem, the number of clusters in general is unknown for most real-world applications. Therefore, clustering becomes a trial-and-error task and the clustering result is often not very promising especially when the number of clusters is difficult to guess. In this paper, we propose an unsupervised clustering approach which utilizes a hierarchical differential evolution algorithm. The proposed approach can effectively search the proper number of clusters and simultaneously determine the cluster centers. The performance of the proposed approach is evaluated, in conjunction with three cluster validity indices, namely Davies-Bouldin index, Dunnpsilas index, and Calinski-Harabasz index. Experimental results are provided to illustrate the feasibility of the proposed approach.

The Standardized Precipitation Index (SPI) is computed over CONUS using daily precipitation estimates from the NOAA Daily U.S. Climate Gridded Dataset (NClimGrid-Daily). From the NClimGrid-SPI (1951-present; 0.05°x0.05°), we derive historical hydro-climatological conditions and drought information from the Drought Severity and Coverage Index (DSCI) which combines drought levels into a single areal value (from 0 to 500). One of our objective is to better understand drought dynamic and particularly how drought episodes evolve from short term rainfall deficit (i.e., less than three months) to persistent drought condition (i.e., beyond nine months). To investigate how those cascading effect work, we use a Machine Learning (ML) approach to identify spatio-temporal patterns of drought episodes over CONUS. Several unsupervised ML clustering algorithms are tested using an ensemble of features including drought duration, rainfall accumulation, drought severity (maximum DSCI, time of maximum DSCI), seasonality (drought beginning and end dates), location (latitude, longitude). Results show that the most severe drought events (i.e., DSCI > 350) are those that have the longest durations and for which drought relief is associated with higher rainfall accumulation regardless of the location considered. Furthermore, there is an apparent consistency across accumulation scales and the number of parameters selected with an optimum number of clusters around four. The Euclidian distance ML models tested seems to be able to define spatiotemporal areas of similar drought patterns. Differences between models are observed in terms of spatial definitions and predominance of a cluster at a given location. The strongest prevalence of a given cluster has allowed to isolate areas of coherence such as the Pacific Northwest, the PNW, the Eastern Seaboard and the Southeast, and the Southwest area along the MX-US border. Domain delineations are weaker for areas such as the Rockies, the Midwest, and the Great Plains. While the SPI algorithm assumes a Gamma (McKee et al., 1993) or a Pearson III (Guttman, 1998) distribution for monthly rainfall accumulation periods, results show that this assumption might not be optimal depending on the domain considered and the accumulation period when applied to daily drought monitoring.

Abstract. The model's ability to reproduce the state of the simulated object or particular feature or phenomenon is always a subject of discussion. Multidimensional model quality assessment is usually customized to the specific focus of the study and often to a limited number of locations. In this paper, we propose a method that provides information on the accuracy of the model in general, while all dimensional information for posterior analysis of the specific tasks is retained. The main goal of the method is to perform clustering of the multivariate model errors. The clustering is done using the K-means algorithm of unsupervised machine learning. In addition, the potential application of the K-means clustering of model errors for learning and predicting is shown. The method is tested on the 40-year simulation results of the general circulation model of the Baltic Sea. The model results are evaluated with the measurement data of temperature and salinity from more than one million casts by forming a two-dimensional error space and performing a clustering procedure in it. The optimal number of clusters that consist of four clusters was determined using the Elbow cluster selection criteria and based on the analysis of the different number of error clusters. In this particular model, the error cluster of good quality of the model with a bias of 0.4 °C (std = 0.8 °C) for temperature and 0.6 g kg−1 (std = 0.7 g kg−1) for salinity made up 57 % of all comparison data pairs. The prediction of centroids from a limited number of randomly selected data showed that the obtained centroids gained a stability of at least 100 000 error pairs in the learning dataset.

Clustering is an unsupervised learning approach used to group similar features using specific mathematical criteria. This mathematical criterion is known as the objective function. Any clustering is done depending on some objective function.K-means is one of the widely used partitional clustering algorithms whose performance depends on the initial point and the value of K. In this paper, we have combined both these parameters. We have defined an entropy-based objective function for the initialization process, which is better than other existing initialization methods of K-means clustering. Here, we have also designed an algorithm to calculate the correct number of clusters of datasets using some cluster validity indexes. In this paper, the entropy-based initialization algorithm has been proposed and applied to different 2D and 3D data sets. The comparison with other existing initialization methods has been represented in this paper.