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<p><b>Abstract</b>—Visualization has proven to be a powerful and widely-applicable tool for the analysis and interpretation of multivariate data. Most visualization algorithms aim to find a projection from the data space down to a two-dimensional visualization space. However, for complex data sets living in a high-dimensional space, it is unlikely that a single two-dimensional projection can reveal all of the interesting structure. We therefore introduce a hierarchical visualization algorithm which allows the complete data set to be visualized at the top level, with clusters and subclusters of data points visualized at deeper levels. The algorithm is based on a hierarchical mixture of latent variable models, whose parameters are estimated using the expectation-maximization algorithm. We demonstrate the principle of the approach on a toy data set, and we then apply the algorithm to the visualization of a synthetic data set in 12 dimensions obtained from a simulation of multiphase flows in oil pipelines, and to data in 36 dimensions derived from satellite images. A Matlab software implementation of the algorithm is publicly available from the World Wide Web.</p>
Latent variables, data visualization, EM algorithm, hierarchical mixture model, density estimation, principal component analysis, factor analysis, maximum likelihood, clustering, statistics.

M. E. Tipping and C. M. Bishop, "A Hierarchical Latent Variable Model for Data Visualization," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 20, no. , pp. 281-293, 1998.
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