Clustering is the unsupervised classification of patterns (observations, data items, or feature vectors) into groups (clusters). The clustering problem has been addressed in many contexts and by researchers in many disciplines; this reflects its broad appeal and usefulness as one of the steps in exploratory data analysis. However, clustering is a difficult problem combinatorially, and differences in assumptions and contexts in different communities has made the transfer of useful generic concepts and methodologies slow to occur. This paper presents an overview of pattern clustering methods from a statistical pattern recognition perspective, with a goal of providing useful advice and references to fundamental concepts accessible to the broad community of clustering practitioners. We present a taxonomy of clustering techniques, and identify cross-cutting themes and recent advances. We also describe some important applications of clustering algorithms such as image segmentation, object recognition, and information retrieval.

Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mid-eighties in an attempt to stochastically solve problems of this generic class. During the past decade, a variety of multiobjective EA (MOEA) techniques have been proposed and applied to many scientific and engineering applications. Our discussion's intent is to rigorously define multiobjective optimization problems and certain related concepts, present an MOEA classification scheme, and evaluate the variety of contemporary MOEAs. Current MOEA theoretical developments are evaluated; specific topics addressed include fitness functions, Pareto ranking, niching, fitness sharing, mating restriction, and secondary populations. Since the development and application of MOEAs is a dynamic and rapidly growing activity, we focus on key analytical insights based upon critical MOEA evaluation of c...

The self-organizing map (SOM) is an excellent tool in exploratory phase of data mining. It projects input space on prototypes of a low-dimensional regular grid that can be effectively utilized to visualize and explore properties of the data. When the number of SOM units is large, to facilitate quantitative analysis of the map and the data, similar units need to be grouped, i.e., clustered. In this paper, different approaches to clustering of the SOM are considered. In particular, the use of hierarchical agglomerative clustering and partitive clustering using-means are investigated. The two-stage procedure---first using SOM to produce the prototypes that are then clustered in the second stage---is found to perform well when compared with direct clustering of the data and to reduce the computation time.

Finding structures in vast multidimensional data sets, be they measurement data, statistics, or textual documents, is difficult and time-consuming. Interesting, novel relations between the data items may be hidden in the data. The selforganizing map (SOM) algorithm of Kohonen can be used to aid the exploration: the structures in the data sets can be illustrated on special map displays. In this work, the methodology of using SOMs for exploratory data analysis or data mining is reviewed and developed further. The properties of the maps are compared with the properties of related methods intended for visualizing highdimensional multivariate data sets. In a set of case studies the SOM algorithm is applied to analyzing electroencephalograms, to illustrating structures of the standard of living in the world, and to organizing full-text document collections. Measures are proposed for evaluating the quality of different types of maps in representing a given data set, and for measuring the robu...

Network models are widely used to represent relational information among interacting units. In studies of social networks, recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors. We develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved "social space." Inference for the social space is developed within a maximum likelihood and Bayesian framework, and Markov chain Monte Carlo procedures are proposed for making inference on latent positions and the effects of observed covariates. We present analyses of three standard datasets from the social networks literature, and compare the method to an alternative stochastic blockmodeling approach. In addition to improving upon model fit, our method provides a visual and interpretable model-based spatial representation of social relationships, and improves upon existing methods by allowing the statistical uncertainty in the social space to be quantified and graphically represented.

With Islands of Music we present a system which facilitates exploration of music libraries without requiring manual genre classification. Given pieces of music in raw audio format we estimate their perceived sound similarities based on psychoacoustic models. Subsequently, the pieces are organized on a 2-dimensional map so that similar pieces are located close to each other. A visualization using a metaphor of geographic maps provides an intuitive interface where islands resemble genres or styles of music. We demonstrate the approach using a collection of 359 pieces of music.

AbstractÐWe present a novel technique for texture mapping on arbitrary surfaces with minimal distortions by preserving the local and global structure of the texture. The recent introduction of the fast marching method on triangulated surfaces made it possible to compute a geodesic distance map from a given surface point in O…n lg n † operations, where n is the number of triangles that represent the surface. We use this method to design a surface flattening approach based on multidimensional scaling �MDS). MDS is a family of methods that map a set of points into a finite dimensional flat �Euclidean) domain, where the only given data is the corresponding distances between every pair of points. The MDS mapping yields minimal changes of the distances between the corresponding points. We then solve an ªinverseº problem and map a flat texture patch onto the curved surface while preserving the structure of the texture. Index TermsÐTexture mapping, multidimensional scaling, fast marching method, Geodesic distance, Euclidean distance. æ 1

Summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure. In this paper we provide an overview of some existing techniques for discovering such embeddings. We then introduce a novel probabilistic interpretation of principal component analysis (PCA) that we term dual probabilistic PCA (DPPCA). The DPPCA model has the additional advantage that the linear mappings from the embedded space can easily be nonlinearised through Gaussian processes. We refer to this model as a Gaussian process latent variable model (GP-LVM). Through analysis of the GP-LVM objective function, we relate the model to popular spectral techniques such as kernel PCA and multidimensional scaling. We then review a practical algorithm for GP-LVMs in the context of large data sets and develop it to also handle discrete valued data and missing attributes. We demonstrate the model on a range of real-world and artificially generated data sets.

The availability of large music collections calls for ways to efficiently access and explore them. We present a new approach which combines descriptors derived from audio analysis with meta-information to create different views of a collection. Such views can have a focus on timbre, rhythm, artist, style or other aspects of music. For each view the pieces of music are organized on a map in such a way that similar pieces are located close to each other. The maps are visualized using an Islands of Music metaphor where islands represent groups of similar pieces. The maps are linked to each other using a new technique to align self-organizing maps. The user is able to browse the collection and explore different aspects by gradually changing focus from one view to another. We demonstrate our approach on a small collection using a meta-information-based view and two views generated from audio analysis, namely, beat periodicity as an aspect of rhythm and spectral information as an aspect of timbre.

The Self-Organizing Map (SOM) is an efficient tool for visualization of multidimensional numerical data. In this paper, an overview and categorization of both old and new methods for the visualization of SOM is presented. The purpose is to give an idea of what kind of information can be acquired from different presentations and how the SOM can best be utilized in exploratory data visualization. Most of the presented methods can also be applied in the more general case of first making a vector quantization (e.g. k-means) and then a vector projection (e.g. Sammon's mapping).