Comparing soft clusters and partitions

2006

Cluster validity indexes aim at evaluating thc degree to which a partition obtain4 from a clustering algorithm approximates the real structure of w data set. Most of them reduce to the search of the right number of clusters. This paper presents mch a new validity index for fuzzy clustering bascd on the aggregation of the multing membership degrees with no additional informatian, e.g. the geornelrica! stmctum of the data. It exploits the tendency for a data point to helong to a unique cluster, Le, both the tendency to belong to one cluster and the tendency not to belong to the othem dusters. Clustering has been defined as the "art of finding groups (clusters) in data sets" [9]. The underlying idea is that dam pojnts from different clusters are as dissimilar as possible while objects belongjng to any of the dusters are similar. In such a framework, the label of a data point, i,e, the group to which it belongs is unknown. Clustering is then viewed as an instance of unsupervised classification and is one of the most important task in pattern recognition. A major challenge in cluster analysis is the validation of clusters resulting from clustering methods. The performance of a particular method clearly depends on the natural (groups-)structure of the data but also of the algorithm parameters, e.g. the shape of the clusters or their nurnber. Comparative studies of validity indexes, mainly for fuzzy clustering, can be; found in [4], 1141, [I 01. Two kinds of classical indexes are generally admitted: those that only use membership labels and those that exploit some geometrical infomation about the structure of the Carl Wlicat is with the Labomminire Irafonnrrrique-Image-!nrrmair~~t, Universit6 de I-a RochelTe, 17042 Ida Rtxhelle Cedex I , FRANCE (phone: $33 546 458 234; far: c33 546 4% 242; ernail: carl,frclicotQuniv-lr.fr), L a u~n t Mascarillid is with the Lahnrafnire fnjom1aidqu.e-I M~P-!nieracfirjn, UnivcnitE dc La Rochellc, 17042 La Rochclle Cedcx

Clustering (or cluster analysis) has been used widely in pattern recognition, image processing, and data analysis. It aims to organize a collection of data items into c clusters, such that items within a cluster are more similar to each other than they are items in the other clusters. The number of clusters c is the most important parameter, in the sense that the remaining parameters have less influence on the resulting partition. To determine the best number of classes several methods were made, and are called validity index. This paper presents a new validity index for fuzzy clustering called a Modified Partition Coefficient And Exponential Separation (MPCAES) index. The efficiency of the proposed MPCAES index is compared with several popular validity indexes. More information about these indexes is acquired in series of numerical comparisons and also real data Iris.

Lecture Notes in Computer Science, 2007

Clustering is one of the most well known types of unsupervised learning. Evaluating the quality of results and determining the number of clusters in data is an important issue. Most current validity indices only cover a subset of important aspects of clusters. Moreover, these indices are relevant only for data sets containing at least two clusters. In this paper, a new bounded index for cluster validity, called the score function (SF), is introduced. The score function is based on standard cluster properties. Several artificial and real-life data sets are used to evaluate the performance of the score function. The score function is tested against four existing validity indices. The index proposed in this paper is found to be always as good or better than these indices in the case of hyperspheroidal clusters. It is shown to work well on multidimensional data sets and is able to accommodate unique and sub-cluster cases.

IAEME PUBLICATION, 2020

In this research investigation, the authors have presented a new type of Cluster Weight Index. Firstly, the author considers parameters like Cardinality of a Cluster, Ratio Of Sum Of Squares Between to Sum Of Squares Within (for a Cluster), Scatter of a Cluster, Separation (Davies Bouldin like Index) of a Cluster, Dunn like Index for a Cluster, and coins new parameters like Property Adherence Factor of a Cluster, Mass Of a Cluster, Density of a Cluster and normalizes these to construct a new parameter in the range of 0-1. The authors then construct a new General formula that accounts for the behavior of all these parameters together. Finally, this general formula parameter is also normalized again to arrive at the Cluster Weight Index of a Cluster of concern

Quality of clustering is an important issue in application of clustering techniques. Most traditional cluster validity indices are geometry-based cluster quality measures. This work proposes a cluster validity index based on the decision-theoretic rough set model by considering various loss functions. Real time retail data show the usefulness of the proposed validity index for the evaluation of rough and crisp clustering. The measure is shown to help determine optimal number of clusters, as well as an important parameter called threshold in rough clustering. The experiments with a promotional campaign for the retail data illustrate the ability of the proposed measure to incorporate financial considerations in evaluating quality of a clustering scheme. This ability to deal with monetary values distinguishes the proposed decision-theoretic measure from other distance-based measures. Our proposed system validity index can also be efficient for evaluating other clustering algorithms such as fuzzy clustering.

2013

Clustering can be defined as the process of grouping physical or abstract objects into classes of similar objects. It’s an unsupervised learning problem of organizing unlabeled objects into natural groups in such a way objects in the same group is more similar than objects in the different groups. Conventional clustering algorithms cannot handle uncertainty that exists in the real life experience. Fuzzy clustering handles incompleteness, vagueness in the data set efficiently. The goodness of clustering is measured in terms of cluster validity indices where the results of clustering are validated repeatedly for different cluster partitions to give the maximum efficiency i.e. to determine the optimal number of clusters. Especially, fuzzy clustering has been widely applied in a variety of areas and fuzzy cluster validation plays a very important role in fuzzy clustering. Since then Fuzzy clustering has been evaluated using various cluster validity indices. But primary indices have used...

Computers, Materials & Continua, 2022

Unsupervised clustering and clustering validity are used as essential instruments of data analytics. Despite clustering being realized under uncertainty, validity indices do not deliver any quantitative evaluation of the uncertainties in the suggested partitionings. Also, validity measures may be biased towards the underlying clustering method. Moreover, neglecting a confidence requirement may result in over-partitioning. In the absence of an error estimate or a confidence parameter, probable clustering errors are forwarded to the later stages of the system. Whereas, having an uncertainty margin of the projected labeling can be very fruitful for many applications such as machine learning. Herein, the validity issue was approached through estimation of the uncertainty and a novel low complexity index proposed for fuzzy clustering. It involves only uni-dimensional membership weights, regardless of the data dimension, stipulates no specific distribution, and is independent of the underlying similarity measure. Inclusive tests and comparisons returned that it can reliably estimate the optimum number of partitions under different data distributions, besides behaving more robust to over partitioning. Also, in the comparative correlation analysis between true clustering error rates and some known internal validity indices, the suggested index exhibited the highest strong correlations. This relationship has been also proven stable through additional statistical acceptance tests. Thus the provided relative uncertainty measure can be used as a probable error estimate in the clustering as well. Besides, it is the only method known that can exclusively identify data points in dubiety and is adjustable according to the required confidence level.

Proceedings of the 26th Annual International Conference on Machine Learning, 2009

Information theoretic based measures form a fundamental class of similarity measures for comparing clusterings, beside the class of pair-counting based and set-matching based measures. In this paper, we discuss the necessity of correction for chance for information theoretic based measures for clusterings comparison. We observe that the baseline for such measures, i.e. average value between random partitions of a data set, does not take on a constant value, and tends to have larger variation when the ratio between the number of data points and the number of clusters is small. This effect is similar in some other non-information theoretic based measures such as the well-known Rand Index. Assuming a hypergeometric model of randomness, we derive the analytical formula for the expected mutual information value between a pair of clusterings, and then propose the adjusted version for several popular information theoretic based measures. Some examples are given to demonstrate the need and usefulness of the adjusted measures.

Pattern Recognition, 2013

The validation of the results obtained by clustering algorithms is a fundamental part of the clustering process. The most used approaches for cluster validation are based on internal cluster validity indices. Although many indices have been proposed, there is no recent extensive comparative study of their performance. In this paper we show the results of an experimental work that compares 30 cluster validity indices in many different environments with different characteristics. These results can serve as a guideline for selecting the most suitable index for each possible application and provide a deep insight into the performance differences between the currently available indices.

Validating a given clustering result is a very challenging task in real world. So for this purpose, several cluster validity indices have been developed in the literature. Cluster validity indices are divided into two main categories: external and internal. External cluster validity indices rely on some supervised information available and internal validity indices utilize the intrinsic structure of the data. In this paper a new external cluster validity index, MMI and its normalized version NMMI have been implemented based on Max-Min distance along data points and prior information using structure of data. A new probabilistic approach has been implemented to find the correct correspondence between the true and obtained clustering. Different possibilities for probabilistic approaches have been considered and tried to rectify their problems. Genetic K-means clustering algorithm (GAK-means) and single linkage clustering technique have been used as the underlying clustering techniques. Results of proposed index for classifying the true partitioning results have been shown for six artificial and two real-life data sets. GAK-means and single linkage clustering techniques are used as the underlying partitioning techniques with the number of clusters varied in a range. The MMI and NMMI index are then used to determine the appropriate number of clusters. Performance of MMI along with its two versions MMI old and MMI new along with its normalized version NMMI are compared with the existing external cluster validity indices, F-measure, purity, normalized mutual information (NMI), rand index (RI), adjusted rand index (ARI). Proposed MMI index works well for two class and multi class data sets.

ArXiv, 2020

We introduce a cluster evaluation technique called Tree Index. Our Tree Index algorithm aims at describing the structural information of the clustering rather than the quantitative format of cluster-quality indexes (where the representation power of clustering is some cumulative error similar to vector quantization). Our Tree Index is finding margins amongst clusters for easy learning without the complications of Minimum Description Length. Our Tree Index produces a decision tree from the clustered data set, using the cluster identifiers as labels. It combines the entropy of each leaf with their depth. Intuitively, a shorter tree with pure leaves generalizes the data well (the clusters are easy to learn because they are well separated). So, the labels are meaningful clusters. If the clustering algorithm does not separate well, trees learned from their results will be large and too detailed. We show that, on the clustering results (obtained by various techniques) on a brain dataset, ...

Intelligent Data Analysis, 2009

Clustering quality or validation indices allow the evaluation of the quality of clustering in order to support the selection of a specific partition or clustering structure in its natural unsupervised environment, where the real solution is unknown or not available. In this paper, we investigate the use of quality indices mostly based on the concepts of clusters' compactness and separation, for the evaluation of clustering results (partitions in particular). This work intends to offer a general perspective regarding the appropriate use of quality indices for the purpose of clustering evaluation. After presenting some commonly used indices, as well as indices recently proposed in the literature, key issues regarding the practical use of quality indices are addressed. A general methodological approach is presented which considers the identification of appropriate indices thresholds. This general approach is compared with the simple use of quality indices for evaluating a clustering solution.

2009

This paper presents a new approach to find the optimal number of clusters of a fuzzy partition. It is based on a fuzzy modeling approach which combines measures of clusters' separation and overlap. Theses measures are based on triangular norms and a discrete Sugeno integral. Results on artificial and real data sets prove its efficiency compared to indexes from the literature.

Research report RJ, 2001

In this paper we propose a measure of similarity/association between two partitions of a set of objects. Our motivation is the desire to use the measure to characterize the quality or accuracy of clustering algorithms by somehow comparing the clusters they produce with \ground truth" consisting of classes assigned to the patterns by manual means or some other means in whose veracity there is con dence. Such measures are referred to as \external". Our measure also allows clusterings with di erent numbers of clusters to be compared in a quantitative and principled way. Our evaluation scheme quantitatively measures how useful the cluster labels of the patterns are as predictors of their class labels. When all clusterings to be compared have the same number of clusters, the measure is equivalent to the mutual information between the cluster labels and the class labels. In cases where the numbers of clusters are di erent, however, it computes the reduction in the number of bits that would be required to encode (compress) the class labels if both the encoder and decoder have free access to the cluster labels. To achieve this encoding the estimated conditional probabilities of the class labels given the cluster labels must also be encoded. These estimated probabilities can be seen as a \model" for the class labels and their associated code length as a \model cost". In addition to de ning the measure we compare it to other commonly used external measures and demonstrate its superiority as judged by certain criteria.

Neural Processing Letters, 2006

Cluster validity has been widely used to evaluate the fitness of partitions produced by clustering algorithms. This paper presents a new validity, which is called the Vapnik–Chervonenkis-bound (VB) index, for data clustering. It is estimated based on the structural risk minimization (SRM) principle, which optimizes the bound simultaneously over both the distortion function (empirical risk) and the VC-dimension (model complexity). The smallest bound of the guaranteed risk achieved on some appropriate cluster number validates the best description of the data structure. We use the deterministic annealing (DA) algorithm as the underlying clustering technique to produce the partitions. Five numerical examples and two real data sets are used to illustrate the use of VB as a validity index. Its effectiveness is compared to several popular cluster-validity indexes. The results of comparative study show that the proposed VB index has high ability in producing a good cluster number estimate and in addition, it provides a new approach for cluster validity from the view of statistical learning theory.