Cluster validity measures for data with tolerance

Given the complexity of real-life event logs, several trace clustering techniques have been proposed to partition an event log into subsets with a lower degree of variation. In general, these techniques assume that the number of clusters is known in advance. However, this will rarely be the case in practice. Therefore, this paper presents approaches to determine the appropriate number of clusters in a trace clustering context. In order to fulfil the objective of identifying the most appropriate number of trace clusters, two approaches built on similarity are proposed: a stability- and a separation-based method. The stability-based method iteratively calculates the similarity between clustered versions of perturbed and unperturbed event logs. Alternatively, an approach based on between-cluster dissimilarity, or separation, is proposed. Regarding practical validation, both approaches are tested on multiple real-life datasets to investigate the complementarity of the different components. Our results suggest that both methods are successful in identifying an appropriate number of trace clusters.

Identification of the correct number of clusters is an important consideration in clustering where several cluster validity indexes, primarily utilizing the Euclidean distance, have been used in the literature. The property of symmetry is observed in most clustering solutions. In this paper, the symmetry versions of nine cluster validity indexes, namely, Davies-Bouldin index, Dunn index, generalized Dunn index, point symmetry (PS) index, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</i> index, Xie-Beni index, FS index, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> index, and SV index, are proposed. It is empirically established that incorporation of the property of symmetry significantly improves the capabilities of these indexes in identifying the appropriate number of clusters. A recently developed PS-based genetic clustering technique, GAPS clustering, is used as the underlying partitioning algorithm. Results on six artificially generated and five real-life datasets show that symmetry-distance-based <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</i> index performs the best as compared to all the other eight indexes.

Determining number of clusters and prototype locations via multi-scale clustering

Unsupervised classification needs no prior knowledge of interested region and has less human interference that is suitable for processing messes of digital images acquired from earth observation satellites. A typical sequence for unsupervised classification might begin with the analyst specifying minimum and maximum numbers of categories to be generated by the classification algorithm. Usually, a larger cluster number is assigned in advance for the initial clustering. Then, several classified clusters are merged into a larger group so that the number of clusters can be reduced. The selection of an appropriate cluster number is up to the analyst's experience. This paper illustrates the convergence of cluster-center optimization and the optimal cluster number varying with the dimension in Minkowski space. One of the simplest and most popular is Euclidean distance while Minkowski distance is measured in two dimensions. The experiment was executed on several digital images by a classifier employing Genetic Algorithm as a searching tool and Davies-Bouldin index as a cluster validity measure. The results show that a smaller dimension, e.g. one-, two-, and infinity dimensions, in Minkowski space derives less clusters, while the assignment of a larger dimension results in a larger cluster number with a more detailed segmentation.

Although clustering is an unsupervised learning approach, most clustering algorithms require setting several parameters (such as the number of clusters, minimum density or distance threshold) in advance to work properly. In this paper, we eliminate the necessity of setting the minimum cluster size parameter of the Randomized Gravitational Clustering algorithm proposed by Gomez et al. Basically, the minimum cluster size is estimated using a heuristic that takes in consideration the functional relation between the number of clusters and the clusters with at least a given number of points. Then a data point’s region of action (region of the space assigned to a point) is defined and a cluster refinement process is proposed in order to merge overlapping clusters. Our experimental results show that the proposed algorithm is able to deal with noise, while finding an appropriate number of clusters without requiring a manual setting of the minimum cluster size.

Clustering is one of the relevant data mining tasks, which aims to process data sets in an effective way. This paper introduces a new clustering heuristic combining the E-transitive heuristic adapted to quantitative data and the k-means algorithm with the goal of ensuring the optimal number of clusters and the suitable initial cluster centres for k-means. The suggested heuris-tic, called PFK-means, is a parameter-free clustering algorithm since it does not require the prior initialization of the number of clusters. Thus, it generates progressively the initial cluster centres until the appropriate number of clusters is automatically detected. Moreover, this paper exposes a thorough comparison between the PFK-means heuristic, its diverse variants, the E-Transitive heuristic for clustering quantitative data and the traditional k-means in terms of the sum of squared errors and accuracy using different data sets. The experiments results reveal that, in general, the proposed heuristic and its variants provide the appropriate number of clusters for different real-world data sets and give good clusters quality related to the traditional k-means. Furthermore, the experiments conducted on synthetic data sets report the performance of this heuristic in terms of processing time.

Within the scope of cluster analysis of variables, the selection of the appropriate number of clusters is of paramount interest. The strategy of determination of the appropriate number of clusters adopted herein is based on a hypothesis testing approach. It consists in testing whether the variation of a partition quality criterion between two consecutive partitions is far removed from the expected variation under the null-hypothesis stipulating a lack of structure. Three hypothesis testing strategies are detailed and compared in the scope of clustering of variables: Gap, Weighted Gap and a statistic associated with CLV methodology. Finally, an illustration is presented based on data from a preference study.

Clustering is a fundamental and widely used method for grouping similar records in one cluster and dissimilar records in the different cluster. In cluster analysis, a major problem is to determine the appropriate number of cluster in advance. It is difficult for a user (data miner) to estimate the appropriate number of clusters in advance. Another limitation of a well-known clustering technique called K-means is that it gets stuck at local optima. In order to overcome these limitations Genetic Algorithm (GA) based clustering techniques have been proposed in the 1990s. Since then many researchers have developed several evolutionary algorithm based clustering techniques, including GA and applied in various fields. This paper presents an up-to-date review of some major GA-based clustering techniques for the last twenty (20) years. A total of 45 ranked (i.e. based on citation reports and JCR/CORE rank) GA-based clustering approaches are reviewed, which are uses for real-life applications such as real-life data sets, highway construction projects, a Gas Company, cellular networks and satellite image segmentations. Almost two third of the techniques do not require any user to define the number of clusters. Finally, a thorough discussion and emerging research directions are presented.

Data clustering describes a set of frequently employed techniques in exploratory data analysis to extract "natural" group structure in data. Such groupings need to be validated to separate the signal in the data from spurious structure. In this context, finding an appropriate number of clusters is a particularly important model selection question. We introduce a measure of cluster stability to assess the validity of a cluster model. This stability measure quantifies the reproducibility of clustering solutions on a second sample, and it can be interpreted as a classification risk with regard to class labels produced by a clustering algorithm. The preferred number of clusters is determined by minimizing this classification risk as a function of the number of clusters. Convincing results are achieved on simulated as well as gene expression data sets. Comparisons to other methods demonstrate the competitive performance of our method and its suitability as a general validation tool for clustering solutions in real-world problems.

GX-Means: A model-based divide and merge algorithm for geospatial image clustering

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.

We propose a new principal component analysis (PCA) for interval-valued data by using a covariance involving a fuzzy classification structure based on dissimilarity in higher dimensional space in which objects exist. The covariance for interval-valued data is obtained adaptively by evaluating the validity of the fuzzy classification structure based on the selection of an appropriate number of clusters. In order to select an appropriate number of clusters, we propose an alignment criterion to evaluate the obtained classification structure and prove the concentration of the criterion around the expected value with respect to variation of similarity among clusters. The merit of this PCA is to consider not only the projection of objects to a lower dimensional space, but also the dissimilarity of objects in a higher dimensional space by using a weighted covariance matrix. The weight is estimated as the degree of contribution for the fuzzy classification structure based on dissimilarity of objects in the higher dimensional space. A numerical example of interval-valued data consisting of human based subjective decisions shows a better performance when compared with a result of an ordinary PCA.

This study proposes a novel algorithm to cluster for point data with significant improvements. First, each point is assigned to a circle with an appropriate radius, from which a new measure for their similarity is proposed. This distance is then used to determine the appropriate number of clusters. Once the number of clusters is determined, the steps to establish the fuzzy relationship of each element with the clusters using the proposed distance are detailed. The proposed algorithm is thoroughly detailed in each implementation step, numerical examples are illustrated, and it can be executed using an established MATLAB procedure. A key motivation for this algorithm is its application to image data, where characteristics are represented as points for recognition. Applications on various image datasets with different characteristics demonstrate that the proposed algorithm is stable and yields good results, outperforming many popular algorithms as well as recently proposed algorithms.

This paper presents a novel density based agglomerative clustering algorithm named TOBAE which is a parameter-less algorithm and automatically filters noise. It finds the appropriate number of clusters while giving a competitive running time. TOBAE works by tracking the cumulative density distribution of the data points on a grid and only requires the original data set as input. The clustering problem is solved by automatically finding the optimal density threshold for the clusters. It is applicable to any N-dimensional data set which makes it highly relevant for real world scenarios. The algorithm outperforms state of the art clustering algorithms by the additional feature of automatic noise filtration around clusters. The concept behind the algorithm is explained using the analogy of puddles ('tobae'), which the algorithm is inspired from. This paper provides a detailed algorithm for TOBAE along with the complexity analysis for both time and space. We show experimental results against known data sets and show how TOBAE competes with the best algorithms in the field while providing its own set of advantages.

Spatial clustering analysis is an important spatial data mining technique. It divides objects into clusters according to their similarities in both location and attribute aspects. It plays an essential role in density distribution identification, hot-spot detection, and trend discovery. Spatial clustering algorithms in the Euclidean space are relatively mature, while those in the network space are less well researched. This study aimed to present a well-known clustering algorithm, named density-based spatial clustering of applications with noise (DBSCAN), to network space and proposed a new clustering algorithm named network space DBSCAN (NS-DBSCAN). Basically, the NS-DBSCAN algorithm used a strategy similar to the DBSCAN algorithm. Furthermore, it provided a new technique for visualizing the density distribution and indicating the intrinsic clustering structure. Tested by the points of interest (POI) in Hanyang district, Wuhan, China, the NS-DBSCAN algorithm was able to accurately detect the high-density regions. The NS-DBSCAN algorithm was compared with the classical hierarchical clustering algorithm and the recently proposed density-based clustering algorithm with network-constraint Delaunay triangulation (NC_DT) in terms of their effectiveness. The hierarchical clustering algorithm was effective only when the cluster number was well specified, otherwise it might separate a natural cluster into several parts. The NC_DT method excessively gathered most objects into a huge cluster. Quantitative evaluation using four indicators, including the silhouette, the R-squared index, the Davis–Bouldin index, and the clustering scheme quality index, indicated that the NS-DBSCAN algorithm was superior to the hierarchical clustering and NC_DT algorithms.