A hierarchical latent variable model for data visualization
Christopher M. Bishop, Michael E. Tipping
Latent variables, data visualization, EM algorithm, hierarchical mixture model, density estimation, principal component analysis, factor analysis, maximum likelihood, clustering, statistics.
Visualisation has proven to be a powerful and widely-applicable tool for the analysis and interpretation of multi-variate data. Most visualisation algorithms aim to find a projection from the data space down to a two-dimensional visualisation 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 visualisation algorithm which allows the complete data set to be visualised at the top level, with clusters and sub-clusters of data points visualised 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 visualisation of a synthetic data set in 12 dimensions obtained from a simulation of multi-phase 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.
IEEE Transactions on Pattern Analysis and Machine Intelligence