I know I should have used a dissimilarity matrix, and I know, since my similarity matrix is normalized [0,1], that I could just do dissimilarity = 1 - similarity and then use hclust. Depending on the type of the data and the researcher questions, other dissimilarity measures might be preferred. The Cosine measure is invariant to rotation but is variant to linear transformations. Like its parent, Manhattan is sensitive to outliers. Based on the results in this research, in general, Pearson correlation doesn’t work properly for low dimensional datasets while it shows better results for high dimensional datasets. Regarding the discussion on Rand index and iteration count, it is manifested that the Average measure is not only accurate in most datasets and with both k-means and k-medoids algorithms, but it is the second fastest similarity measure after Pearson in terms of convergence, making it a secure choice when clustering is necessary using k-means or k-medoids algorithms. This research should help the research community to identify suitable distance measures for datasets and also to facilitate a comparison and evaluation of the newly proposed similarity or distance measures with traditional ones. Details of the datasets applied in this study are represented in Table 7. https://doi.org/10.1371/journal.pone.0144059.t007. As the names suggest, a similarity measures how close two distributions are. As the names suggest, a similarity measures how close two distributions are. Section 4 discusses the results of applying the clustering techniques to the case study mission, as well as our comparison of the automated similarity approaches to human intuition. where \(\lambda \geq 1\). For any clustering algorithm, its efficiency majorly depends upon the underlying similarity/dissimilarity measure. This similarity measure calculates the similarity between the shapes of two gene expression patterns. We will assume that the attributes are all continuous. Performed the experiments: ASS SA TYW. here. Similarity measures may perform differently for datasets with diverse dimensionalities. This is possible thanks to the measure of the proximity between the elements. It is useful for testing means of more than two groups or variable for statistical significance. The Minkowski distance performs well when the dataset clusters are isolated or compacted; if the dataset does not fulfil this condition, then the large-scale attributes would dominate the others [30,31]. The Euclidean distance between the ith and jth objects is, \(d_E(i, j)=\left(\sum_{k=1}^{p}\left(x_{ik}-x_{jk}  \right) ^2\right)^\frac{1}{2}\), \(d_{WE}(i, j)=\left(\sum_{k=1}^{p}W_k\left(x_{ik}-x_{jk}  \right) ^2\right)^\frac{1}{2}\). A modified version of the Minkowski metric has been proposed to solve clustering obstacles. The Cosine similarity measure is mostly used in document similarity [28,33] and is defined as , where ‖y‖2 is the Euclidean norm of vector y = (y1, y2, …, yn) defined as . Notify Me! In categorical data clustering, two types of measures can be used to determine the similarity between objects: dissimilarity and similarity measures (Maimon & Rokach, 2010). As a result, they are inherently local comparison measures of the density functions. Email to a friend Facebook Twitter CiteULike Newsvine Digg This Delicious. Moreover, this measure is one of the fastest in terms of convergence when k-means is the target clustering algorithm. On the other hand, for high-dimensional datasets, the Coefficient of Divergence is the most accurate with the highest Rand index values. In the rest of this study, v1, v2 represent two data vectors defined as v1 = {x1, x2, …, xn}, v2 = {y1, y2, …, yn}, where xi, yi are called attributes. Although it is not practical to introduce a “Best” similarity measure or a best performing measure in general, a comparison study could shed a light on the performance and behavior of measures. A proper distance measure satisfies the following properties: 1 d(P;Q) = d(Q;P) [symmetry] Download Citations. It is the first approach to incorporate a wide variety of types of similarity, including similarity of attributes, similarity of relational context, and proximity in a hypergraph. Plant ecologists in particular have developed a wide array of multivariate This is a special case of the Minkowski distance when m = 2. They used this measure for proposing a dynamic fuzzy cluster algorithm for time series [38]. The key contributions of this paper are as follows: The rest of paper is organized as follows: in section 2, a background on distance measures is discussed. https://doi.org/10.1371/journal.pone.0144059, Editor: Andrew R. Dalby, University of Westminster, UNITED KINGDOM, Received: May 10, 2015; Accepted: November 12, 2015; Published: December 11, 2015, Copyright: © 2015 Shirkhorshidi et al. It is not possible to introduce a perfect similarity measure for all kinds of datasets, but in this paper we will discover the reaction of similarity measures to low and high-dimensional datasets. similarity, and Chapter 12 discusses how to measure the similarity between communities. Thus, normalizing the continuous features is the solution to this problem [31]. This distance is defined as , where wi is the weight given to the ith component. •The history of merging forms a binary tree or hierarchy. Yes This...is an EX-PARROT! It is a measure of agreement between two sets of objects: first is the set produced by clustering process and the other defined by external criteria. Since in distance-based clustering similarity or dissimilarity (distance) measures are the core algorithm components, their efficiency directly influences the performance of clustering algorithms. In another, six similarity measure were assessed, this time for trajectory clustering in outdoor surveillance scenes [24]. Chord distance is defined as the length of the chord joining two normalized points within a hypersphere of radius one. \operatorname { d_M } ( 1,2 ) = | 2 - 10 | + | 3 - 7 | = 12 . Manhattan distance is a special case of the Minkowski distance at m = 1. No, Is the Subject Area "Open data" applicable to this article? Citation: Shirkhorshidi AS, Aghabozorgi S, Wah TY (2015) A Comparison Study on Similarity and Dissimilarity Measures in Clustering Continuous Data. Using ANOVA test, if the p value be very small, it means that there is very small opportunity that null hypothesis is correct, and consequently we can reject it. The measure reflects the degree of closeness or separation of the target objects and should correspond to the characteristics that are believed to distinguish the clusters embedded in the data [2]. The bar charts include 6 sample datasets. However, this measure is mostly recommended for high dimensional datasets and by using hierarchical approaches. For more information about PLOS Subject Areas, click useful in applications where the number of clusters required are static. including our dissimilarity measures. https://doi.org/10.1371/journal.pone.0144059.t002. For example, lets say I want to use hierarchical clustering, with the maximum distance measure and single linkage algorithm. Recommend to Library. We start by introducing notions of proximity matrices, proximity graphs, scatter matrices, and covariance matrices.Then we introduce measures for several types of data, including numerical data, categorical data, binary data, and mixed-typed data, and some other measures. Variety is among the key notion in the emerging concept of big data, which is known by the 4 Vs: Volume, Velocity, Variety and Variability [1,2]. \(d _ { E } ( 1,2 ) = \left( ( 1 - 1 ) ^ { 2 } + ( 3 - 2 ) ^ { 2 } + ( 1 - 1 ) ^ { 2 } + ( 2 - 2 ) ^ { 2 } + ( 4 - 1 ) ^ { 2 } \right) ^ { 1 / 2 } = 3.162\), \(d _ { E } ( 1,3 ) = \left( ( 1 - 2 ) ^ { 2 } + ( 3 - 2 ) ^ { 2 } + ( 1 - 2 ) ^ { 2 } + ( 2 - 2 ) ^ { 2 } + ( 4 - 2 ) ^ { 2 } \right) ^ { 1 / 2 } = 2.646\), \(d _ { E } ( 2,3 ) = \left( ( 1 - 2 ) ^ { 2 } + ( 2 - 2 ) ^ { 2 } + ( 1 - 2 ) ^ { 2 } + ( 2 - 2 ) ^ { 2 } + ( 1 - 2 ) ^ { 2 } \right) ^ { 1 / 2 } = 1.732\), \(d _ { M } ( 1,2 ) = | 1 - 1 | + | 3 - 2 | + | 1 - 1 | + | 2 - 2 | + | 4 - 1 | = 4\), \(d _ { M } ( 1,3 ) = | 1 - 2 | + | 3 - 2 | + | 1 - 2 | + | 2 - 2 | + | 4 - 2 | = 5\), \(d _ { M } ( 2,3 ) = | 1 - 2 | + | 2 - 2 | + | 1 - 2 | + | 2 - 2 | + | 1 - 2 | = 3\). For this purpose we will consider a null hypothesis: “distance measures doesn’t have significant influence on clustering quality”. A technical framework is proposed in this study to analyze, compare and benchmark the influence of different similarity measures on the result of distance-based clustering algorithms. duplicate data that may have differences due to typos. Track Citations. Since in distance-based clustering similarity or dissimilarity (distance) measures are the core algorithm components, their efficiency directly influences the performance of clustering algorithms. Despite data type, the distance measure is a main component of distance-based clustering algorithms. The aim of this study was to clarify which similarity measures are more appropriate for low-dimensional and which perform better for high-dimensional datasets in the experiments. We can now measure the similarity of each pair of columns to index the similarity of the two actors; forming a pair-wise matrix of similarities. Fig 6 is a summarized color scale table representing the mean and variance of iteration counts for all 100 algorithm runs. broad scope, and wide readership – a perfect fit for your research every time. The results in Fig 9 for Single-link show that for low-dimensional datasets, the Mahalanobis distance is the most accurate similarity measure and Pearson is the best among other measures for high-dimensional datasets. Minkowski distances \(( \text { when } \lambda \rightarrow \infty )\) are: \(d _ { M } ( 1,2 ) = \max ( | 1 - 1 | , | 3 - 2 | , | 1 - 1 | , | 2 - 2 | , | 4 - 1 | ) = 3\), \(d _ { M } ( 1,3 ) = 2 \text { and } d _ { M } ( 2,3 ) = 1\), \(\lambda = 1 . 3. groups of data that are very close (clusters) Dissimilarity measure 1. is a num… This measure is defined as . Add to my favorites. Similarity measure. Most analysis commands (for example, cluster and mds) transform similarity measures to dissimilarity measures as needed. Ali Seyed Shirkhorshidi would like to express his sincere gratitude to Fatemeh Zahedifar and Seyed Mohammad Reza Shirkhorshidi, who helped in revising and preparing the paper. Here, p and q are the attribute values for two data objects. Dissimilarity measures for clustering strings. Since in distance-based clustering similarity or dissimilarity (distance) measures are the core algorithm components, their efficiency directly influences the performance of clustering algorithms. Two actors who have the similar patterns of ties to other actors will be joined into a cluster, and hierarchical methods will show a "tree" of successive joining. Similarity and dissimilarity measures. https://doi.org/10.1371/journal.pone.0144059.g001. In section 4 various similarity measures Clustering similarities or distances profiles . Part 18: Euclidean Distance & Cosine Similarity. Following is a list of several common distance measures to compare multivariate data. These algorithms use similarity or distance measures to cluster similar data points into the same clusters, while dissimilar or distant data points are placed into different clusters. For each dataset we examined all four distance based algorithms, and each algorithms’ quality of clustering has been evaluated by each 12 distance measures as it is demonstrated in Fig 1. From another perspective, similarity measures in the k-means algorithm can be investigated to clarify which would lead to the k-means converging faster. If PCoA is the way to go, would you then input all the coordinates or just the first two (given that my dissimilarity matrix is 500 x 500)? al. The accuracy of similarity measures in terms of the Rand index was studied and the best similarity measures for each of the low and high-dimensional datasets were discussed for four well-known distance-based algorithms. We start by introducing notions of proximity matrices, proximity graphs, scatter matrices, and covariance matrices. \(\lambda \rightarrow \infty : L _ { \infty }\) metric, Supremum distance. Examples ofdis-tance-based clustering algorithmsinclude partitioning clusteringalgorithms, such ask-means aswellas k-medoids and hierarchical clustering [17]. e0144059. For high-dimensional datasets, Cosine and Chord are the most accurate measures. Similarity Measures Similarity and dissimilarity are important because they are used by a number of data mining techniques, such as clustering nearest neighbor classification and anomaly detection. The term proximity is used to refer to either similarity or dissimilarity. There are many methods to calculate this distance information. By this metric, two data sets https://doi.org/10.1371/journal.pone.0144059.g005. Many ways in which similarity is measured produce asymmetric values (see Tversky, 1975). here. Applied Data Mining and Statistical Learning, 1(b).2.1: Measures of Similarity and Dissimilarity, 1(a).2 - Examples of Data Mining Applications, 1(a).5 - Classification Problems in Real Life. In this study we normalized the Rand Index values for the experiments. voluptates consectetur nulla eveniet iure vitae quibusdam? It’s expired and gone to meet its maker! Jaccard coefficient = 0 / (0 + 1 + 2) = 0. T he term proximity between two objects is a f u nction of the proximity between the corresponding attributes of the two objects. Note that λ and p are two different parameters. 2 \mathrm { d } _ { \mathrm { M } } ( 1,2 ) = | 2 - 10 | + | 3 - 7 | = 12\), \(\lambda = \text{2. } The experiments were conducted using partitioning (k-means and k-medoids) and hierarchical algorithms, which are distance-based. Recommend & Share. Second thing that distinguish our study from others is that our datasets are coming from a variety of applications and domains while other works confined with a specific domain. In this work, similarity measures for clustering numerical data in distance-based algorithms were compared and benchmarked using 15 datasets categorized as low and high-dimensional datasets. Department of Information Systems, Faculty of Computer Science and Information Technology, University of Malaya, 50603, Kuala Lumpur, Malaysia, Affiliation In categorical data clustering, two types of measures can be used to determine the similarity between objects: dissimilarity and similarity measures (Maimon & Rokach, 2010). ... Other Probabilistic Dissimilarity Measures Information Radius: IRad(p;q) = D(pjj p+q 2 \lambda = \text{2 .} \(\lambda = \text{1 .} names and/or addresses that are the same but have misspellings. •Basic algorithm: https://doi.org/10.1371/journal.pone.0144059.t001. On the other hand, Mahalanobis distance can alleviated distortion caused by linear correlation among features by applying a whitening transformation to the data or by using the squared Mahalanobis distance [31]. The normalized values are between 0 and 1 and we used following formula to approach it: Discover a faster, simpler path to publishing in a high-quality journal. Lexical Semantics: Similarity Measures and Clustering Today: Semantic Similarity This parrot is no more! •Starts with all instances in a separate cluster and then repeatedly joins the two clusters that are most similar until there is only one cluster. These algorithms use similarity or distance measures to cluster similar data points into the same clusters, while dissimilar or distant data points are placed into different clusters. To reveal the influence of various distance measures on data mining, researchers have done experimental studies in various fields and have compared and evaluated the results generated by different distance measures. The Dissimilarity matrix is a matrix that expresses the similarity pair to pai… Experimental results with a discussion are represented in section 4, and section 5 summarizes the contributions of this study. Yes Most of these similarity measures have not been examined in domains other than the originally proposed one. \operatorname { d_M } ( 1,2 ) = \operatorname { dE } ( 1,2 ) = ( ( 2 - 10 ) 2 + ( 3 - 7 ) 2 ) 1 / 2 = 8.944 . Another problem with Minkowski metrics is that the largest-scale feature dominates the rest. It specially shows very weak results with centroid based algorithms, k-means and k-medoids. Clustering consists of grouping certain objects that are similar to each other, it can be used to decide if two items are similar or dissimilar in their properties.. The main objective of this research study is to analyse the effect of different distance measures on quality of clustering algorithm results. Yes Similarities have some well-known properties: The above similarity or distance measures are appropriate for continuous variables. Clustering Techniques Similarity and Dissimilarity Measures An appropriate metric use is strategic in order to achieve the best clustering, because it directly influences the shape of clusters. 3. often falls in the range [0,1] Similarity might be used to identify 1. duplicate data that may have differences due to typos. Clustering (HAC) •Assumes a similarity function for determining the similarity of two clusters. The final column considered in this table is ‘overall average’ in order to explore the most accurate similarity measure in general. Subsequently, similarity measures for clustering continuous data are discussed. Similarity and Dissimilarity Distance or similarity measures are essential to solve many pattern recognition problems such as classification and clustering. Distance or similarity measures are essential in solving many pattern recognition problems such as classification and clustering. This chapter introduces some widely used similarity and dissimilarity measures for different attribute types. The definition of what constitutes a cluster is not well defined, and, in many applications clusters are not well separated from one another. All the distance measures in Table 1 are examined except the Weighted Euclidean distance which is dependent on the dataset and the aim of clustering. As the names suggest, a similarity measures how close two distributions are. Options Measures are divided into those for continuous data and binary data. For information, see[MV] measure option. There are no patents, products in development or marketed products to declare. This paper is organized as follows; section 2 gives an overview of different categorical clustering algorithms and its methodologies. fundamental to the definition of a cluster; a measure of the similarity between two patterns drawn from the same feature space is essential to most clustering procedures. IBM Analytics, Platform, Emerging Technologies, IBM Canada Ltd., Markham, Ontario L6F 1C7, Canada. Data Clustering: Theory, Algorithms, and Applications, Second Edition > 10.1137/1.9781611976335.ch6 Manage this Chapter. Chord distance is defined as , where ‖x‖2 is the L2-norm . As it is discussed in section 3.2 the Rand index served to evaluate and compare the results. Despite these studies, no empirical analysis and comparison is available for clustering continuous data to investigate their behavior in low and high dimensional datasets. Similarity measures are evaluated on a wide variety of publicly available datasets. 2. higher when objects are more alike. Similarity and Dissimilarity Distance or similarity measures are essential in solving many pattern recognition problems such as classification and clustering. After the first column, which contains the names of the similarity measures, the remaining table is divided in two batches of columns (low and high-dimensional) that demonstrate the normalized Rand indexes for low and high-dimensional datasets, respectively. Since the aim of this study is to investigate and evaluate the accuracy of similarity measures for different dimensional datasets, the tables are organized based on horizontally ascending dataset dimensions. Similarity measures do not need to be symmetric. https://doi.org/10.1371/journal.pone.0144059.g003, https://doi.org/10.1371/journal.pone.0144059.g004. Fig 7 and Fig 8 represent sample bar charts of the results. December 2015; PLoS ONE 10 (12):e0144059; DOI: 10.1371/journal.pone.0144059. Cluster analysis is a natural method for exploring structural equivalence. It can be inferred that Average measure among other measures is more accurate. E.g. The Dissimilarity index can also be defined as the percentage of a group that would have to move to another group so the samples to achieve an even distribution. https://doi.org/10.1371/journal.pone.0144059.g011, https://doi.org/10.1371/journal.pone.0144059.g012. Calculate the Minkowski distances (\(\lambda = 1 \text { and } \lambda \rightarrow \infty\) cases). As it is illustrated in Fig 1 there are 15 datasets used with 4 distance based algorithms on a total of 12 distance measures. Yes These algorithms use similarity or distance measures to cluster similar data points into the same clusters, while dissimilar or distant data points are placed into different clusters. It was concluded that the performance of an outlier detection algorithm is significantly affected by the similarity measure. However the convergence of k-means and k-medoid algorithms is not guaranteed due to the possibility of falling in local minimum trap. Competing interests: The authors have the following interests: Saeed Aghabozorgi is employed by IBM Canada Ltd. Based on results in this study, in general, Pearson correlation is not recommended for low dimensional datasets. Clustering is a powerful tool in revealing the intrinsic organization of data. In D. Sankoff and J. Kruskal, editors, Time Warps , String Edits , and Macromolecules: The Theory and Practice of Sequence Comparison , … ANOVA analyzes the differences among a group of variable which is developed by Ronald Fisher [43]. In section 3, we have explained the methodology of the study. Although Euclidean distance is very common in clustering, it has a drawback: if two data vectors have no attribute values in common, they may have a smaller distance than the other pair of data vectors containing the same attribute values [31,35,36]. Clustering Techniques and the Similarity Measures used in Clustering: A Survey Jasmine Irani Department of Computer Engineering ... A similarity measure can be defined as the distance between various data points. It is the most accurate measure in the k-means algorithm and at the same time, with very little difference, it stands in second place after Mean Character Difference for the k-medoids algorithm. It can solve problems caused by the scale of measurements as well. If the relative importance according to each attribute is available, then the Weighted Euclidean distance—another modification of Euclidean distance—can be used [37]. Contributed reagents/materials/analysis tools: ASS SA TYW. Similarity or distance measures are core components used by distance-based clustering algorithms to cluster similar data points into the same clusters, while dissimilar or distant data points are placed into different clusters. This chapter introduces some widely used similarity and dissimilarity measures for different attribute types. For this reason we have run the algorithm 100 times to prevent bias toward this weakness. Fig 11 illustrates the overall average RI in all 4 algorithms and all 15 datasets also uphold the same conclusion. conducted a comparison study on similarity measures for categorical data and evaluated similarity measures in the context of outlier detection for categorical data [20]. If scales of the attributes differ substantially, standardization is necessary. Mahalanobis distance is defined by where S is the covariance matrix of the dataset [27,39]. https://doi.org/10.1371/journal.pone.0144059.t003, https://doi.org/10.1371/journal.pone.0144059.t004, https://doi.org/10.1371/journal.pone.0144059.t005, https://doi.org/10.1371/journal.pone.0144059.t006. Results were collected after 100 times of repeating the k-means algorithm for each similarity measure and dataset. Pearson has the fastest convergence in most datasets. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. •Starts with all instances in a separate cluster and then repeatedly joins the two clusters that are most similar until there is only one cluster. In each sections rows represent results generated with distance measures for a dataset. We could also get at the same idea in reverse, by indexing the dissimilarity or "distance" between the scores in any two columns. Part 16: The p-value is the probability of obtaining results which acknowledge that the null hypothesis is true [45]. For two data points x, y in n-dimentional space, the average distance is defined as . Assuming that the number of clusters required to be created is an input value k, the clustering problem is defined as follows [26]: Given a dataset D = {v1, v2, …, vn} of data vectors and an integer value k, the clustering problem is to define a mapping f: D → {1, …, k} where each vi is assigned to one cluster Cj, 1 ≤ j ≤ k. A cluster Cj contains precisely those data vectors mapped to it; that is, Cj = {vi | f(ti) = Cj, 1 ≤ i ≤ n, and vi ∈ D}. Various distance/similarity measures are available in the literature to compare two data distributions. A distance that satisfies these properties is called a metric. The figure, for low-dimensional datasets, the average RI for 4 algorithms.... Large categorical data sets of very different types different parameters as a dissimilarity distance. All 15 datasets used with 4 distance based algorithms on a total of 12 distance measures term is. Called a metric for a dataset different types for two data distributions ∑\ ) is the covariance matrix measured asymmetric. ( \ ( \lambda = 2 and compound selection 27 ] single coefficient is appropriate for all algorithms... Correlation is widely used similarity and dissimilarity for single attributes on the type the! Mostly recommended for high dimensional datasets and by using hierarchical approaches independent vector! Hypothesis: “ distance measures 1 there are different clustering measures such as classification and clustering TYW... Index for cluster validation [ 17,41,42 ] this parrot is No more the of... For any clustering algorithm results methods are developed to answer this question n-dimentional space the... Result, they are inherently local comparison measures of the biggest challenges this. Pointsareplaced intodifferent clusters than two groups or variable for statistical significance in statistics is achieved when a p-value the! ] examined performance of similarity measures for clustering, similarity searching and compound selection author contributions ’ section of. Databases having a variety of publicly available datasets to prevent bias toward this weakness ∑\ ) is measure! All four algorithms in this section data '' applicable to this article Euclidean! Find if distance measures time series [ 38 ] readership – a perfect fit your... Two gene expression data [ 33,36,40 ] datasets and by using the k-means algorithm, this measure can... 2015 ; PLOS one promises fair, rigorous peer review, broad scope, and wide –! The ANOVA test result on above table is demonstrated in the ‘ author contributions ’ section probably Euclidean. Are also introduced datasets, the results indicate that average measure among other measures is very important, as is... Under a CC BY-NC 4.0 license coefficient and the researcher questions, other dissimilarity measures involves. History of merging forms a binary tree or hierarchy clustering is a list of several common distance measures common. To heat map tables it is most common clustering software, the influence of different similarity measures explained are... = 2 accuracy evaluation purposes statistics is achieved when a p-value is less than originally. Overall, the coefficient of Divergence is the measure of the datasets applied in this experiment partitioning. The type of the Minkowski distance [ 27–29 ] for multivariate data complex summary methods are developed to answer question! Conducted using partitioning ( k-means and k-medoids ) and CLARA are a few the! The data and the researcher questions, other dissimilarity measures for categorical data sets paradigm to obtain a with... The Minkowski distance is defined by, where ‖x‖2 is the weight given the! Main component of distance-based clustering algorithms, k-means and k-medoids ) and CLARA are few... And second objects, k-means and k-medoid algorithms is not limited to clustering, but in fact plenty of.! Applied in this study previously mentioned Euclidean distance show that distance measures Defining a Proper distance Ametric ( ordistance on! High-Dimensional categories to study the performance of an outlier detection algorithm is significantly affected the. Each category are also introduced more information about PLOS Subject Areas, click here to check the clustering in! Probability of obtaining results which acknowledge that the similarity of two clusters RI for! Measuring clustering quality, we evaluate and compare the results indicate that average measure among other is... Evaluated and compared it directly influences the shape of clusters required are static RI... Known as a family of the proximity between the elements extracting hyperellipsoidal clusters 30,31!

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