Parallel and Distributed Map-Reduce Models for External Clustering Validation Indexes

Procedures that evaluate the results of clustering algorithms are known as clustering validation (CV) indexes. There are several (CV) indexes usually classified into two broad classes namely external and internal clustering validation indexes depending on whether on ground truth or optimal clustering are known in advance or not respectively. Traditional cluster validation indexes are even impossible to perform especially when the size of the data set is very large. In this paper, we are interested in external validation of clustering large data sets. To solve the issue of CV in a big data context, we propose in this paper a parallel external clustering validation especially F-measure (MR_F-measure) model that is based on MapReduce. The experimental results reveal that MR_F-measure scales very well with increasing data set sizes.

This paper presents a comparison between external and internal cluster validity indices with a similar bounded index range. F-measure (FM) and Fowlkes-Mallows (FMI) of external validity indices, as well as Silhouette (SIL) of internal validity index, were chosen for this comparative analysis. Ten numerical data sets, namely Haberman, BUPA (liver disorder), Wisconsin Diagnostic Breast Cancer (WDBC), Iris, Seeds, Wine, User Knowledge, Cleveland, Segmentation, and Glass, were deployed to benchmark the clustering outcomes based on Fuzzy C-Mean (FCM) algorithm. Mean, minimum, and maximum scores were calculated to determine the similarities and differences among the indices. Pearson correlation was reported as well. As a result, the index scores displayed a slight difference between the external and internal validity indices. A moderate and positive correlation was noted between the external and internal validity index (r=. 66, r=.65 p<0.01) scores. This correlation signifies a similar graph pattern between the cluster validity indices. This comparative analysis revealed that the external and internal cluster validity indices with similar bounded index ranges and slightly different index scores generate a moderate and positive correlation with a similar graph pattern.

Ground truth bias in external cluster validity indices

&lt;p&gt;Differential Evolution (DE) is an algorithm for evolutionary optimization. Clustering problems have been solved by using DE based clustering methods but these methods may fail to find clusters hidden in subspaces of high dimensional datasets. Subspace and projected clustering methods have been proposed in literature to find subspace clusters that are present in subspaces of dataset. In this paper we propose VINAYAKA, a semi-supervised projected clustering method based on DE. In this method DE optimizes a hybrid cluster validation index. Subspace Clustering Quality Estimate index (SCQE index) is used for internal cluster validation and Gini index gain is used for external cluster validation in the proposed hybrid cluster validation index. Proposed method is applied on Wisconsin breast cancer dataset.&lt;/p&gt;

Gene selection from microarray gene expression datasets and clustering of samples into different groups are important data mining tasks for disease identification. Selection of more interpretable genes from the gene expression dataset is an essential data-preprocessing task, which helps to study on cancer diseases. Gene selection during sample clustering is inherently a difficult task as there is no obvious criterion to guide the search. Simultaneous gene selection and sample clustering is a two-way data analysis technique which has recently gained attention in research area. The traditional clustering techniques are unable to handle noisy data properly. So, effective clustering algorithms are more desirable which can deal with the relevant and noise free data. Therefore, target genes selection before sample clustering is essential and of course effective if both the tasks are done simultaneously. In this chapter, optimal gene subset is selected and sample clustering is performed simultaneously using Multi-Objective Genetic Algorithm (MOGA). Different versions of MOGA are employed to choose the optimal gene subset, where natural number of optimal clusters of samples is automatically obtained at the end of the process. Non-dominated sorting genetic algorithm (NSGA), Strength pareto evolutionary algorithm (SPEA) and its modified version SPEA2 are applied for the purpose. The methods use nonlinear hybrid uniform cellular automata for generating initial population, tournament selection strategy, two-point crossover operation, and a suitable jumping gene mutation mechanism to maintain diversity in the population. It uses mutual correlation coefficient; internal and external cluster validation indices as objective functions to find out the non-dominated solutions. To measure the cluster validation indices, clustering algorithm is applied on data subset associated to chromosomes in the population to find out different clusters. After the convergence of genetic algorithm, the best solution from the non-dominated solutions is identified that provides the important genes and categorizes the samples into clusters. The experimental results express the correctness of the proposed simultaneous gene selection and sample categorization method. The goodness of optimality of the clusters obtained using different genetic algorithms is expressed by comparing various cluster validation indices.

There are two notoriously hard problems in cluster analysis, estimating the number of clusters, and checking whether the population to be clustered is not actually homogeneous. Given a dataset, a clustering method and a cluster validation index, this paper proposes to set up null models that capture structural features of the data that cannot be interpreted as indicating clustering. Artificial datasets are sampled from the null model with parameters estimated from the original dataset. This can be used for testing the null hypothesis of a homogeneous population against a clustering alternative. It can also be used to calibrate the validation index for estimating the number of clusters, by taking into account the expected distribution of the index under the null model for any given number of clusters. The approach is illustrated by three examples, involving various different clustering techniques (partitioning around medoids, hierarchical methods, a Gaussian mixture model), validation indexes (average silhouette width, prediction strength and BIC), and issues such as mixed type data, temporal and spatial autocorrelation.

Purpose – The most commonly used approaches for cluster validation are based on indices but the majority of the existing cluster validity indices do not work well on data sets of different complexities. The purpose of this paper is to propose a new cluster validity index (ARSD index) that works well on all types of data sets. Design/methodology/approach – The authors introduce a new compactness measure that depicts the typical behaviour of a cluster where more points are located around the centre and lesser points towards the outer edge of the cluster. A novel penalty function is proposed for determining the distinctness measure of clusters. Random linear search-algorithm is employed to evaluate and compare the performance of the five commonly known validity indices and the proposed validity index. The values of the six indices are computed for all nc ranging from (nc min, nc max) to obtain the optimal number of clusters present in a data set. The data sets used in the experiments include shaped, Gaussian-like and real data sets. Findings – Through extensive experimental study, it is observed that the proposed validity index is found to be more consistent and reliable in indicating the correct number of clusters compared to other validity indices. This is experimentally demonstrated on 11 data sets where the proposed index has achieved better results. Originality/value – The originality of the research paper includes proposing a novel cluster validity index which is used to determine the optimal number of clusters present in data sets of different complexities.

Transmissions in the mmWave spectrum benefit from a-priori knowledge of radio channel propagation models. This paper is concerned with one important task that helps provide a more accurate channel model, namely, the clustering of all multipath components arriving at the receiver. Our work focuses on directive transmissions in urban outdoor scenarios and shows the importance of the correct estimation of the number of clusters for mmWave radio channels simulated with a software ray-tracer tool. We investigate the effectiveness of k-means and k-power-means clustering algorithms in predicting the number of clusters through the use of cluster validity indices (CVIs) and score fusion techniques. Our investigation shows that clustering is a difficult task because the optimal number of clusters is not always given by one or by a combination of more CVIs. However, using score fusion methods, we find the optimal partitioning for the k-means algorithm based on the power and time of arrival of the multipath rays or based on their angle of arrival. When the k-power-means algorithm is used, the power of each arriving ray is the most important clustering factor, making the dominant received paths pull the other ones around them, to form a cluster. Thus, the number of clusters is smaller and the decision based on CVIs or score fusion factors easier to be taken.

PDBI: A partitioning Davies-Bouldin index for clustering evaluation

The existence of large volumes of time series data in many applications has motivated data miners to investigate specialized methods for mining time series data. Clustering is a popular data mining method due to its powerful exploratory nature and its usefulness as a preprocessing step for other data mining techniques. This article develops two novel clustering algorithms for time series data that are extensions of a crisp c-shapes algorithm. The two new algorithms are heuristic derivatives of fuzzy c-means (FCM). Fuzzy c-Shapes plus (FCS+) replaces the inner product norm in the FCM model with a shape-based distance function. Fuzzy c-Shapes double plus (FCS++) uses the shape-based distance, and also replaces the FCM cluster centers with shape-extracted prototypes. Numerical experiments on 48 real time series data sets show that the two new algorithms outperform state-of-the-art shape-based clustering algorithms in terms of accuracy and efficiency. Four external cluster validity indices (the Rand index, Adjusted Rand Index, Variation of Information, and Normalized Mutual Information) are used to match candidate partitions generated by each of the studied algorithms. All four indices agree that for these finite waveform data sets, FCS++ gives a small improvement over FCS+, and in turn, FCS+ is better than the original crisp c-shapes method. Finally, we apply two tests of statistical significance to the three algorithms. The Wilcoxon and Friedman statistics both rank the three algorithms in exactly the same way as the four cluster validity indices.

Fuzzy C-Means (FCM) algorithm has been widely used in cluster analysis of gene expression data. It can converge rapidly and provide more information regarding relationships within the data thanks to the usage of fuzzy sets to represent the degrees of cluster membership for every data point. However, FCM has a shortcoming in that it requires a priori specification of cluster number and cluster validation. Most cluster validity indices are based on the data themselves and may not be applicable for gene expression data. In this paper, we propose a Bayesian method using Gene Ontology (GO) annotations for gene expression cluster validation. We show that our method outperforms popular validity indices on gene expression datasets.

Relational visual cluster validity (RVCV)

This paper introduces a novel hierarchical fuzzy algorithm for clustering categorical attributes, which consists of three basic design steps. It incorporates a potential-based clustering scheme with a cluster validity index into a framework that is based on the use of the weighted fuzzy c-modes. The novelty of the contribution lies in the following properties: (a) the potential-based clustering scheme reduces the dependence of the algorithm on initialization, (b) the weighted fuzzy c-modes provides flexibility in detecting the real data structure, and (c) the cluster validity index determines the appropriate number of clusters. The algorithm is applied to model (classify) cultural data related to a number of painters of the seventeenth century, where its performance is compared to the respective performance of an agglomerative hierarchical clustering algorithm.

Clustering is a basic requirement for most analyses on electricity load demand profile datasets. Different algorithms often generate varied clustering results. In absence of ground truth labels, several cluster validation indices (CVI) exist to compare these results and choose the best clustering. However, the issue subsists because these indices might also recommend distinct algorithms. This paper shows that the choice of appropriate CVI depends upon the characteristics of the concerned load demand profile data under study. It provides a comprehensive analysis of the different CVIs response upon changes in data characteristics. The paper finally recommends the indices that ought to be (or not to be) used for a dataset if certain data characteristics are emerging in the clustering results.

Several cluster validity measures have been proposed for evaluating clustering results. However, existing methods may not work well for the following two kinds of data sets. The first one is that the data set contains cluster groups with different densities. The second one is that some of the cluster groups are closely positioned. In this paper, we introduce a new cluster validity index. In this method, we define the index as the ratio between the squared total length of the data eigen-axes and the between-cluster separation. Compared with existing cluster validity indices, the proposed index produces more accurate results and is able to handle the two kinds of data sets mentioned above.