# Survey of Clustering Algorithms

Data analysis plays an indispensable role for understanding various phenomena. Cluster analysis, primitive exploration with little or no prior knowledge, consists of research developed across a wide variety of communities. The diversity, on one hand, equips us with many tools. On the other hand, the profusion of options causes confusion. We survey clustering algorithms for data sets appearing in statistics, ** Modern heuristic techniques for combinatorial problems. Advanced topics in computer science **

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, and machine learning, and illustrate their applications in some benchmark data sets, the traveling salesman problem, and bioinformatics, a new ﬁeld attracting intensive efforts. Several tightly related topics, proximity measure, and cluster validation, are also discussed.

WE ARE living in a world full of data. Every day, people encounter a large amount of information and store or represent it as data, for further analysis and management. One of the vital means in dealing with these data is to classify or group them into a set of categories or clusters. Actually, as one of the most primitive activities of human beings [14], classi- ﬁcation plays an important and indispensable role in the long history of human development. In order to learn a new object or understand a new phenomenon, people always try to seek the features that can describe it, and further compare it with other known objects or phenomena, based on the similarity or dissimilarity, generalized as proximity, according to some certain standards or rules. “Basically, classiﬁcation systems are either supervised or unsupervised, depending on whether they assign new inputs to one of a ﬁnite number of discrete supervised classes or unsupervised categories, respectively [38], [60], [75]. In supervised classiﬁcation, the mapping from a set of input data vectors ( , where is the input space dimensionality), to a ﬁnite set of discrete class labels ( , where is the total number of class types), is modeled in terms of some mathematical function , where is a vector of adjustable parameters. The values of these parameters are determined (optimized) by an inductive learning algorithm (also termed inducer), whose aim is to minimize an empirical risk functional (related to an inductive principle) on a ﬁnite data set of input–output examples, , where is the ﬁnite cardinality of the available representative data set [38], . When the inducer reaches convergence or terminates, an induced classiﬁer is generated [167]. In unsupervised classiﬁcation, called clustering or exploratory data analysis, no labeled data are available [88], [150]. The goal of clustering is to separate a ﬁnite unlabeled data set into a ﬁnite and discrete set of “natural,” hidden data structures, rather than provide an accurate characterization of unobserved samples generated from the same probability distribution [23], [60]. This can make the task of clustering fall outside of the framework of unsupervised predictive learning problems, such as vector quantization [60] (see Section II-C), probability density function estimation [38] (see Section II-D), [60], and entropy maximization [99]. It is noteworthy that clustering differs from multidimensional scaling (perceptual maps), whose goal is to depict all the evaluated objects in a way that minimizes the topographical distortion while using as few dimensions as possible. Also note that, in practice, many (predictive) vector quantizers are also used for (nonpredictive) clustering analysis [60]. Nonpredictive clustering is a subjective process in nature, which precludes an absolute judgment as to the relative efﬁ- cacy of all clustering techniques [23], [152]. As pointed out by Backer and Jain [17], “in cluster analysis a group of objects is split up into a number of more or less homogeneous subgroups on the basis of an often subjectively chosen measure of similarity (i.e., chosen subjectively based on its ability to create “interesting” clusters), such that the similarity between objects within a subgroup is larger than the similarity between objects belonging to different subgroups Clustering algorithms partition data into a certain number of clusters (groups, subsets, or categories). There is no universally agreed upon deﬁnition [88]. Most researchers describe a cluster by considering the internal homogeneity and the external separation , i.e., patterns in the same cluster should be similar to each other, while patterns in different clusters should not. Both the similarity and the dissimilarity should be examinable in a clear and meaningful way. Here, we give some simple mathematical descriptions of several types of clustering, based on the descriptions in

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