In many physical, statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and found 'lines and planes of closest fit to system of points'. The famous k-means algorithm solves the approximation problem too, but by finite sets instead of lines and planes. This chapter gives a brief practical introduction into the methods of construction of general principal objects, i.e. objects embedded in the 'middle' of the multidimensional data set. As a basis, the unifying framework of mean squared distance approximation of finite datasets is selected. Principal graphs and manifolds are constructed as generalisations of principal components and k-means principal points. For this purpose, the family of expectation/maximisation algorithms with nearest generalisations is presented. Construction of principal graphs with controlled complexity is based on the graph grammar approach.

Machine-part cell formation is used in cellular manufacturing in order to process a large variety, quality, lower work in process levels, reducing manufacturing lead-time and customer response time while retaining flexibility for new products. This paper presents a new and novel approach for obtaining machine cells and part families. In the cellular manufacturing the fundamental problem is the formation of part families and machine cells. The present paper deals with the Self Organising Map (SOM) method an unsupervised learning algorithm in Artificial Intelligence, and has been used as a visually decipherable clustering tool of machine-part cell formation. The objective of the paper is to cluster the binary machine-part matrix through visually decipherable cluster of SOM color-coding and labelling via the SOM map nodes in such a way that the part families are processed in that machine cells. The Umatrix, component plane, principal component projection, scatter plot and histogram of SOM have been reported in the present work for the successful visualization of the machine-part cell formation. Computational result with the proposed algorithm on a set of group technology problems available in the literature is also presented. The proposed SOM approach produced solutions with a grouping efficacy that is at least as good as any results earlier reported in the literature and improved the grouping efficacy for 70% of the problems and found immensely useful to both industry practitioners and researchers.

This paper describes a graph clustering algorithm that aims to minimize the normalized cut criterion and has a model order selection procedure. The performance of the proposed algorithm is comparable to spectral approaches in terms of minimizing normalized cut. However, unlike spectral approaches, the proposed algorithm scales to graphs with millions of nodes and edges. The algorithm consists of three components that are processed sequentially: a greedy agglomerative hierarchical clustering procedure, model order selection, and a local refinement. For a graph of n nodes and O(n) edges, the computational complexity of the algorithm is O(n log^2 n), a major improvement over the O(n^3) complexity of spectral methods. Experiments are performed on real and synthetic networks to demonstrate the scalability of the proposed approach, the effectiveness of the model order selection procedure, and the performance of the proposed algorithm in terms of minimizing the normalized cut metric.

Agglomerative hierarchical clustering has been the dominant approach to constructing embedded classification schemes. It is our aim to direct the reader's attention to practical algorithms and methods - both efficient (from the computational and storage points of view) and effective (from the application point of view). It is often helpful to distinguish between method, involving a compactness criterion and the target structure of a 2-way tree representing the partial order on subsets of the power set; as opposed to an implementation, which relates to the detail of the algorithm used. As with many other multivariate techniques, the objects to be classified have numerical measurements on a set of variables or attributes. Hence, the analysis is carried out on the rows of an array or matrix.

Notwithstanding the popularity of conventional clustering algorithms such as K-means and probabilistic clustering, their clustering results are sensitive to the presence of outliers in the data. Even a few outliers can compromise the ability of these algorithms to identify meaningful hidden structures rendering their outcome unreliable. This paper develops robust clustering algorithms that not only aim to cluster the data, but also to identify the outliers. The novel approaches rely on the infrequent presence of outliers in the data which translates to sparsity in a judiciously chosen domain. Capitalizing on the sparsity in the outlier domain, outlier-aware robust K-means and probabilistic clustering approaches are proposed. Their novelty lies on identifying outliers while effecting sparsity in the outlier domain through carefully chosen regularization. A block coordinate descent approach is developed to obtain iterative algorithms with convergence guarantees and small excess computational complexity with respect to their non-robust counterparts. Kernelized versions of the robust clustering algorithms are also developed to efficiently handle high-dimensional data, identify nonlinearly separable clusters, or even cluster objects that are not represented by vectors. Numerical tests on both synthetic and real datasets validate the performance and applicability of the novel algorithms.

If X is endowed with a metric, then this metric can be mapped onto an ultrametric. In practice, endowing X with a metric can be relaxed to a dissimilarity. An often used mapping from metric to ultrametric is by means of an agglomerative hierarchical clustering algorithm. A succession of n 1 pairwise merge steps take place by making use of the closest pair of singletons and/or clusters at each step. Here n is the number of observations, i.e. the cardinality of set X. Closeness between singletons is furnished by whatever distance or dissimilarity is in use. For closeness between singleton or non-singleton clusters, we need to define an inter-cluster distance or dissimilarity. This can be defined with reference to the cluster compactness or other property that we wish to optimize at each step of the algorithm. Since agglomerative hierarchical clustering requires consideration of pairwise dissimilarities at each stage it can be shown that even in the case of the most efficient algorithms, e.g.

Clustering in high-dimensional spaces is nowadays a recurrent problem in many scientific domains but remains a difficult task from both the clustering accuracy and the result understanding points of view. This paper presents a discriminative latent mixture (DLM) model which fits the data in a latent orthonormal discriminative subspace with an intrinsic dimension lower than the dimension of the original space. By constraining model parameters within and between groups, a family of 12 parsimonious DLM models is exhibited which allows to fit onto various situations. An estimation algorithm, called the Fisher-EM algorithm, is also proposed for estimating both the mixture parameters and the discriminative subspace. Experiments on simulated and real datasets show that the proposed approach performs better than existing clustering methods while providing a useful representation of the clustered data. The method is as well applied to the clustering of mass spectrometry data.

Many web-search queries serve as the beginning of an exploration of an unknown space of information, rather than looking for a specific web page. To answer such queries effec- tively, the search engine should attempt to organize the space of relevant information in a way that facilitates exploration. We describe the Aspector system that computes aspects for a given query. Each aspect is a set of search queries that together represent a distinct information need relevant to the original search query. To serve as an effective means to explore the space, Aspector computes aspects that are orthogonal to each other and to have high combined coverage. Aspector combines two sources of information to compute aspects. We discover candidate aspects by analyzing query logs, and cluster them to eliminate redundancies. We then use a mass-collaboration knowledge base (e.g., Wikipedia) to compute candidate aspects for queries that occur less frequently and to group together aspects that are likely to be semantically related. We present a user study that indicates that the aspects we compute are rated favorably against three competing alternatives related searches proposed by Google, cluster labels assigned by the Clusty search engine, and navigational searches proposed by Bing.

We consider a collection of prediction experiments, which are clustered in the sense that groups of experiments ex- hibit similar relationship between the predictor and response variables. The experiment clusters as well as the regres- sion relationships are unknown. The regression relation- ships define the experiment clusters, and in general, the predictor and response variables may not exhibit any clus- tering. We call this prediction problem clustered regres- sion with unknown clusters (CRUC) and in this paper we focus on linear regression. We study and compare several methods for CRUC, demonstrate their applicability to the Yahoo Learning-to-rank Challenge (YLRC) dataset, and in- vestigate an associated mathematical model. CRUC is at the crossroads of many prior works and we study several prediction algorithms with diverse origins: an adaptation of the expectation-maximization algorithm, an approach in- spired by K-means clustering, the singular value threshold- ing approach to matrix rank minimization under quadratic constraints, an adaptation of the Curds and Whey method in multiple regression, and a local regression (LoR) scheme reminiscent of neighborhood methods in collaborative filter- ing. Based on empirical evaluation on the YLRC dataset as well as simulated data, we identify the LoR method as a good practical choice: it yields best or near-best prediction performance at a reasonable computational load, and it is less sensitive to the choice of the algorithm parameter. We also provide some analysis of the LoR method for an asso- ciated mathematical model, which sheds light on optimal parameter choice and prediction performance.

We present the extention and application of a new unsupervised statistical learning technique--the Partition Decoupling Method--to gene expression data. Because it has the ability to reveal non-linear and non-convex geometries present in the data, the PDM is an improvement over typical gene expression analysis algorithms, permitting a multi-gene analysis that can reveal phenotypic differences even when the individual genes do not exhibit differential expression. Here, we apply the PDM to publicly-available gene expression data sets, and demonstrate that we are able to identify cell types and treatments with higher accuracy than is obtained through other approaches. By applying it in a pathway-by-pathway fashion, we demonstrate how the PDM may be used to find sets of mechanistically-related genes that discriminate phenotypes.