Internal Cluster Validity Indices Research Articles

A Distance-based Separability Measure for Internal Cluster Validation

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.

Toolbox for Distance Estimation and Cluster Validation on Data With Missing Values

Missing data are unavoidable in the real-world application of unsupervised machine learning, and their nonoptimal processing may decrease the quality of data-driven models. Imputation is a common remedy for missing values, but directly estimating expected distances have also emerged. Because treatment of missing values is rarely considered in clustering related tasks and distance metrics have a central role both in clustering and cluster validation, we developed a new toolbox that provides a wide range of algorithms for data preprocessing, distance estimation, clustering, and cluster validation in the presence of missing values. All these are core elements in any comprehensive cluster analysis methodology. We describe the methodological background of the implemented algorithms and present multiple illustrations of their use. The experiments include validating distance estimation methods against selected reference methods and demonstrating the performance of internal cluster validation indices. The experimental results demonstrate the general usability of the toolbox for the straightforward realization of alternate data processing pipelines. Source code, data sets, results, and example macros are available on GitHub. <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/markoniem/nanclustering_toolbox</uri>

Cluster analysis of crude oils with k-means based on their physicochemical properties

The values of the physicochemical properties of crude oils vary significantly, depending on their geographical origins. A standard categorization of crude oils is grossly based on the density and sulfur content, not considering other properties that can have meaningful impacts on blending and in some refining processes. Cluster analysis is an unsupervised machine learning technique that categorizes observations based on their similarity. In this work, k-means clustering algorithm was applied to a wide range of physicochemical properties to identify groups of crudes oils with high affinity that possibly have similar behavior later on, in downstream operations.A data set from Galp SA refineries (located in Portugal) containing 454 observations, corresponding to values of 9 properties, from 45 different crude oil sources was used in the present analysis. After suitable preprocessing, k-means was applied using different cluster numbers, and their performance was evaluated through the internal validation metrics silhouette index and Local Cores-based Cluster Validity (LCCV) index. The recommend number of clusters was 3, which presented the best performance with a LCCV index of 0.39. Crude oils from the same source should be incorporated in the same cluster, and this was corroborated by external validation, with 1.8% of the observations were placed in a different cluster than the majority of same source crude oils. The proposed method was also able to identify observations with unusually high iron contents concerning the same source of crude oils when more clusters were considered.This work provides a methodology to obtain a better categorization of crude oils by using cluster analysis, allowing the refineries to know how similar crude oils and their sources are. This categorization is very useful for improving the formulation of crude blends and the crude oils quality control, with the goal to optimize further the refining operations.

Feature-based Groundwater Hydrograph Clustering Using Unsupervised Self-Organizing Map-Ensembles

Hydrograph clustering helps to identify dynamic patterns within aquifers systems, an important foundation of characterizing groundwater systems and their influences, which is necessary to effectively manage groundwater resources. We develope an unsupervised modeling approach to characterize and cluster hydrographs on regional scale according to their dynamics. We apply feature-based clustering to improve the exploitation of heterogeneous datasets, explore the usefulness of existing features and propose new features specifically useful to describe groundwater hydrographs. The clustering itself is based on a powerful combination of Self-Organizing Maps with a modified DS2L-Algorithm, which automatically derives the cluster number but also allows to influence the level of detail of the clustering. We further develop a framework that combines these methods with ensemble modeling, internal cluster validation indices, resampling and consensus voting to finally obtain a robust clustering result and remove arbitrariness from the feature selection process. Further we propose a measure to sort hydrographs within clusters, useful for both interpretability and visualization. We test the framework with weekly data from the Upper Rhine Graben System, using more than 1800 hydrographs from a period of 30 years (1986-2016). The results show that our approach is adaptively capable of identifying homogeneous groups of hydrograph dynamics. The resulting clusters show both spatially known and unknown patterns, some of which correspond clearly to external controlling factors, such as intensive groundwater management in the northern part of the test area. This framework is easily transferable to other regions and, by adapting the describing features, also to other time series-clustering applications.

A practical approach to cluster validation in the energy sector

With increasing digitization, new opportunities emerge concerning the availability and use of data in the energy sector. A comprehensive literature review shows an abundance in available unsupervised clustering algorithms as well as internal, relative and external cluster validation indices (cvi) to evaluate the results. Yet, the comparison of different clustering results on the same dataset, executed with different algorithms and a specific practical goal in mind still proves scientifically challenging. A large variety of cvi are described and consolidated in commonly used composite indices (e.g. Davies-Bouldin-Index, silhouette-Index, Dunn-Index). Previous works show the challenges surrounding these composite indices since they serve a generalized cluster quality evaluation. However, this does not suit individual clustering goals in many cases. The presented paper introduces the current state of science, existing cluster validation indices and proposes a practical method to combine them to an individual composite index, using Multi Criteria Decision Analysis (mcda). The methodology is applied on two energy economic use cases for clustering load profiles of bidirectional electric vehicles and municipalities.

Evaluating Evolving Structure in Streaming Data With Modified Dunn's Indices

Dunn's internal cluster validity index is used to assess partition quality and identify a “best” crisp c-partition of n objects built from static data sets. This index is quite sensitive to inliers and outliers in the input data, so a subsequent study developed a family of 17 generalized Dunn's indices that extend and improve the original measure in various ways. This paper presents online versions of two modified generalized Dunn's indices that can be used for the dynamic evaluation of an evolving (cluster) structure in streaming data. We argue that this method is a good way to monitor the ongoing performance of streaming clustering algorithms, and we illustrate several types of inferences that can be drawn from such indices. Streaming clustering algorithms are incremental, process incoming data points only once and then discard them, adapt as the data stream evolves, flag outliers, and most importantly, spawn new emerging structures. We compare the two new indices to the incremental Xie-Beni and Davies-Boudin indices, which to our knowledge offer the only comparable approach, with numerical examples on a variety of synthetic and real datasets.

Development of ‘speed ratio’ based level of service criteria on undivided urban streets in mixed traffic context

Developing countries are facing challenges in sustaining urban traffic congestion due to rapid urbanization. To manage urban traffic, a traffic engineer first needs to assess the current operational condition on urban roads. Level of service (LOS) is used to define the operational traffic condition within a traffic stream in terms of service quality that a facility is providing to its user. This paper proposes a novel approach to estimate ‘speed ratio’ by considering individual free-flow speed (FFS) of different vehicle categories. This study also provides a comparison between FFS estimated using the methods given in Highway Capacity Manual 2010 and Indian-Highway Capacity Manual 2018. LOS criteria were developed using five clustering technique: K-means, K-medoids, Clustering Large Applications, Fuzzy-C Means, and Hierarchical Agglomerative Clustering. Both internal and external cluster validation indices were used to find the optimal number of clusters and suitable clustering algorithms.

A consensus multi-view multi-objective gene selection approach for improved sample classification

BackgroundIn the field of computational biology, analyzing complex data helps to extract relevant biological information. Sample classification of gene expression data is one such popular bio-data analysis technique. However, the presence of a large number of irrelevant/redundant genes in expression data makes a sample classification algorithm working inefficiently. Feature selection is one such high-dimensionality reduction technique that helps to maximize the effectiveness of any sample classification algorithm. Recent advances in biotechnology have improved the biological data to include multi-modal or multiple views. Different ‘omics’ resources capture various equally important biological properties of entities. However, most of the existing feature selection methodologies are biased towards considering only one out of multiple biological resources. Consequently, some crucial aspects of available biological knowledge may get ignored, which could further improve feature selection efficiency.ResultsIn this present work, we have proposed a Consensus Multi-View Multi-objective Clustering-based feature selection algorithm called CMVMC. Three controlled genomic and proteomic resources like gene expression, Gene Ontology (GO), and protein-protein interaction network (PPIN) are utilized to build two independent views. The concept of multi-objective consensus clustering has been applied within our proposed gene selection method to satisfy both incorporated views. Gene expression data sets of Multiple tissues and Yeast from two different organisms (Homo Sapiens and Saccharomyces cerevisiae, respectively) are chosen for experimental purposes. As the end-product of CMVMC, a reduced set of relevant and non-redundant genes are found for each chosen data set. These genes finally participate in an effective sample classification.ConclusionsThe experimental study on chosen data sets shows that our proposed feature-selection method improves the sample classification accuracy and reduces the gene-space up to a significant level. In the case of Multiple Tissues data set, CMVMC reduces the number of genes (features) from 5565 to 41, with 92.73% of sample classification accuracy. For Yeast data set, the number of genes got reduced to 10 from 2884, with 95.84% sample classification accuracy. Two internal cluster validity indices - Silhouette and Davies-Bouldin (DB) and one external validity index Classification Accuracy (CA) are chosen for comparative study. Reported results are further validated through well-known biological significance test and visualization tool.

Simultaneous feature selection and clustering of micro-array and RNA-sequence gene expression data using multiobjective optimization

In this paper, we have devised a multiobjective optimization solution framework for solving the problem of gene expression data clustering in reduced feature space. Here clustering problem is viewed from two different aspects: clustering of genes in reduced sample space or clustering of samples in reduced gene space. Three objective functions: two internal cluster validity indices and the count on the number of features are optimized simultaneously by a popular multiobjective simulated annealing based approach, namely AMOSA. Here, point symmetry based distance is used for the assignment of gene data points to different clusters. Seven publicly available benchmark gene expression data sets are used for experimental purpose. Both aspects of clustering in reduced feature space is demonstrated. The proposed gene expression clustering technique outperforms the existing nine clustering techniques. Apart from this, also some statistical and biological significant tests have been carried out to show that the proposed FSC-MOO technique is more statistically and biologically enriched

A novel internal cluster validity index

A novel internal cluster validity index

A Refined 3-in-1 Fused Protein Similarity Measure: Application in Threshold-Free Hub Detection.

An exhaustive literature survey shows that finding protein/gene similarity is an important step towards solving widespread bioinformatics problems, such as predicting protein-protein interactions, analyzing Protein-Protein Interaction Networks (PPINs), gene prioritization, and disease gene/protein detection. In this article, we have proposed an improved 3-in-1 fused protein similarity measure called FuSim-II. It is built upon combining the weighted average of biological knowledge extracted from three potential genomic/ proteomic resources such as Gene Ontology (GO), PPIN, and protein sequence. Furthermore, we have shown the application of the proposed measure in detecting potential hub-proteins from a given PPIN. Aiming that, we have proposed a multi-objective clustering-based protein hub detection framework with FuSim-II working as the underlying proximity measure. The PPINs of H. Sapiens and M. Musculus organisms are chosen for experimental purposes. Unlike most of the existing hub-detection methods, the proposed technique does not require to follow any protein degree cut-off or threshold to define hubs. A thorough assessment of efficiency between proposed and existing eight protein similarity measures along with eight single/multi-objective clustering methods has been carried out. Internal cluster validity indices like Silhouette and Davies Bouldin (DB) are deployed to accomplish analytical study. Also, a comparative performance analysis between proposed and five existing hub-proteins detection algorithms is conducted through the enrichment of essentiality study. The reported results show the improved performance of FuSim-II over existing protein similarity measures in terms of identifying functionally related proteins as well as relevant hub-proteins. Supplementary material is available at http://csse.szu.edu.cn/staff/cuilz/eng/index.html.

Weighted Cluster Ensemble Based on Partition Relevance Analysis With Reduction Step

Over the last decade, the advent of the cluster ensemble framework has enabled more accurate and robust data analysis than traditional single clustering algorithms. The improved clustering of microarray data has had a particularly strong impact in the fields of genomics and medicine. However, when we bring several ensemble members together to form a consensus, low-quality data partitions can seriously compromise the final solution. One way to overcome this problem is the weighted cluster ensemble approach based on Partition Relevance Analysis (PRA), which uses internal cluster validity indices to evaluate and weight the ensemble members before the fusion. Unfortunately, the selection of appropriate validation indices for given data is far from trivial. In this paper, we propose an additional step in PRA that reduces the size of the committee of cluster validation indices. It does so by eliminating redundant and noisy indices using data dimensionality reduction methods. Our extension works in an unsupervised way, minimizing the amount of user intervention and required expert knowledge. We adapted three conventional consensus functions based on the principle of evidence accumulation to work with PRA weights. We demonstrate the advantages of the proposed reduction step of PRA based on extensive experiments with 25 gene expression and 15 non-genetic real-world datasets, where we compared 15 consensus functions. The source code is available at https://github.com/nejci/PRAr.

A Hierarchical Clustering algorithm based on Silhouette Index for cancer subtype discovery from genomic data

Identifying potential novel subtypes of cancers from genomic data requires techniques to estimate the number of natural clusters in the data. Determining the number of natural clusters in a dataset has been a challenging problem in Machine Learning. Employing an internal cluster validity index such as Silhouette Index together with a clustering algorithm has been a widely used technique for estimating the number of natural clusters, which has limitations. We propose a Hierarchical Agglomerative Clustering algorithm which automatically estimates the numbers of natural clusters and gives the associated clustering solutions along with dendrograms for visualizing the clustering structure. The algorithm has a Silhouette Index-based criterion for selecting the pair of clusters to merge, in the process of building the clustering hierarchy. The proposed algorithm could find decent estimates for the number of natural clusters, and the associated clustering solutions when applied to a collection of cancer gene expression datasets and general datasets. The proposed method showed better overall performance when compared to that of a set of prominent methods for estimating the number of natural clusters, which are used for cancer subtype discovery from genomic data. The proposed method is deterministic. It can be a better alternative to contemporary approaches for identifying potential novel subtypes of cancers from genomic data.

A kernel semi-supervised distance metric learning with relative distance: Integration with a MOO approach

Metric learning, which aims to determine an appropriate distance function to measure the similarity and dissimilarity between data points accurately, is one of the most popular methods to enhance the performance of many machine learning methods such as K-means clustering and K nearest neighbor classifier algorithms. These algorithms may not perform well because of the use of normal Euclidean distance function that ignores any statistical regularities that might be estimated from a large training set of labeled examples. In many real-world applications, the Euclidean distance may not be fit to capture the intrinsic similarity and dissimilarity between the data points. Compared to existing metric learning algorithms, which use large amount of labeled data in the form of must-link (ML) and cannot-link constraints as side information where the granularity of true clustering is unknown, our proposed approach uses few labeled data in the form of relative-distance constraints such as equality constraints, Ceq, and inequality constraints, Cneq. For satisfying such constraints, we need to project the initial Euclidean distance matrix by using Bregman projection on the convex subset of constraints in such a way that all the constraints are satisfied. Since Bregman projection is not orthogonal, means while satisfying the current constraint previously satisfied constraints may get unsatisfied, we need to select a proper subset of constraints for learning better distance function. The multi-objective framework is utilized for selecting a good subset of constraints which can help in getting the proper labeling of the data set. The selected subset of constraints is used for adjusting the initial kernel-matrix. K-means clustering technique is applied to the adjusted kernel matrix to label the data set. In order to evaluate the quality of obtained labeling, different external and internal cluster validity indices are deployed. The values of these indices are simultaneously optimized using the search capability of MOO with the aim of selecting the appropriate subset of constraints. The proposed approach is evaluated on UCI Human Activity Recognition using Smartphone Dataset v1.0 along with nine other popular data sets. Results show that our approach outperforms the state of the art semi-supervised metric learning algorithms with respect to different internal and external cluster validity indices.

Automatic Scientific Document Clustering Using Self-organized Multi-objective Differential Evolution

Document clustering is the partitioning of a given collection of documents into various K- groups based on some similarity/dissimilarity criterion. This task has applications in scope detection of journals/conferences, development of some automated peer-review support systems, topic-modeling, latest cognitive-inspired works on text summarization, and classification of documents based on semantics, etc. In the current paper, a cognitive-inspired multi-objective automatic document clustering technique is proposed which is a fusion of self-organizing map (SOM) and multi-objective differential evolution approach. The variable number of cluster centers are encoded in different solutions of the population to determine the number of clusters from a data set in an automated way. These solutions undergo various genetic operations during evolution. The concept of SOM is utilized in designing new genetic operators for the proposed clustering technique. In order to measure the goodness of a clustering solution, two cluster validity indices, Pakhira-Bandyopadhyay-Maulik index, and Silhouette index, are optimized simultaneously. The effectiveness of the proposed approach, namely self-organizing map based multi-objective document clustering technique (SMODoc_clust) is shown in automatic classification of some scientific articles and web-documents. Different representation schemas including tf, tf-idf and word-embedding are employed to convert articles in vector-forms. Comparative results with respect to internal cluster validity indices, namely, Dunn index and Davies-Bouldin index, are shown against several state-of-the-art clustering techniques including three multi-objective clustering techniques namely MOCK, VAMOSA, NSGA-II-Clust, single objective genetic algorithm (SOGA) based clustering technique, K-means, and single-linkage clustering. Results obtained clearly show that our approach is better than existing approaches. The validation of the obtained results is also shown using statistical significant t tests.

A novel internal validity index based on the cluster centre and the nearest neighbour cluster

It is crucial to evaluate the clustering quality in cluster analysis. In this paper, a new internal cluster validity index based on the cluster centre and the nearest neighbour cluster is designed according to the geometric distribution of objects. Moreover, a method for determining the optimal number of clusters is proposed. The new methodology can evaluate the clustering results produced by a certain clustering algorithm and determine the optimal number of clusters for a given dataset. Theoretical research and experimental results indicate the validity and good performance of the proposed index and method.

Exploring Multiobjective Optimization for Multiview Clustering

We present a new multiview clustering approach based on multiobjective optimization. In contrast to existing clustering algorithms based on multiobjective optimization, it is generally applicable to data represented by two or more views and does not require specifying the number of clusters a priori . The approach builds upon the search capability of a multiobjective simulated annealing based technique, AMOSA, as the underlying optimization technique. In the first version of the proposed approach, an internal cluster validity index is used to assess the quality of different partitionings obtained using different views. A new way of checking the compatibility of these different partitionings is also proposed and this is used as another objective function. A new encoding strategy and some new mutation operators are introduced. Finally, a new way of computing a consensus partitioning from multiple individual partitions obtained on multiple views is proposed. As a baseline and for comparison, two multiobjective based ensemble clustering techniques are proposed to combine the outputs of different simple clustering approaches. The efficacy of the proposed clustering methods is shown for partitioning several real-world datasets having multiple views. To show the practical usefulness of the method, we present results on web-search result clustering, where the task is to find a suitable partitioning of web snippets.

Approximating Dunn's Cluster Validity Indices for Partitions of Big Data.

Dunn's internal cluster validity index is used to assess partition quality and subsequently identify a "best" crisp partition of n objects. Computing Dunn's index (DI) for partitions of n p -dimensional feature vector data has quadratic time complexity O(pn2) , so its computation is impractical for very large values of n . This note presents six methods for approximating DI. Four methods are based on Maximin sampling, which identifies a skeleton of the full partition that contains some boundary points in each cluster. Two additional methods are presented that estimate boundary points associated with unsupervised training of one class support vector machines. Numerical examples compare approximations to DI based on all six methods. Four experiments on seven real and synthetic data sets support our assertion that computing approximations to DI with an incremental, neighborhood-based Maximin skeleton is both tractable and reliably accurate.

Improving the efficacy of clustering by using far enhanced clustering algorithm

There are several aspects on which research works are carried out on clustering. The prime focus is on finding the near optimal cluster centres and determining the best possible clusters. Hence, we have emphasised our work on finding a technique which contemplates on these facets in a way which is far more efficient than several novel approaches. In this paper, we have examined four varieties of clustering algorithms namely; K-Means, FEKM, ECM and proposed FECA implemented on varying data sets. We used few internal cluster validity indices like Dunn's index, Davies-Bouldin's index, Silhouette Coefficient, C index and Calinski index for quantitative evaluation of the clustering results obtained. The results obtained from simulation were compared, and as per our expectation it was found that, the quality of clustering produced by FECA is far more satisfactory than the others. Almost every value of validity indices used give encouraging results for FECA, implying good cluster formation. Further experiments support that the proposed algorithm also produces minimum quantisation error almost for all the data sets used.

Improving the efficacy of clustering by using far enhanced clustering algorithm

There are several aspects on which research works are carried out on clustering. The prime focus is on finding the near optimal cluster centres and determining the best possible clusters. Hence, we have emphasised our work on finding a technique which contemplates on these facets in a way which is far more efficient than several novel approaches. In this paper, we have examined four varieties of clustering algorithms namely; K-Means, FEKM, ECM and proposed FECA implemented on varying data sets. We used few internal cluster validity indices like Dunn's index, Davies-Bouldin's index, Silhouette Coefficient, C index and Calinski index for quantitative evaluation of the clustering results obtained. The results obtained from simulation were compared, and as per our expectation it was found that, the quality of clustering produced by FECA is far more satisfactory than the others. Almost every value of validity indices used give encouraging results for FECA, implying good cluster formation. Further experiments support that the proposed algorithm also produces minimum quantisation error almost for all the data sets used.