Michael E. Tipping

A hierarchical latent variable model for data visualization

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.

Feed-forward neural networks and topographic mappings for exploratory data analysis

A recent novel approach to the visualisation and analysis of datasets, and one which is particularly applicable to those of a high dimension, is discussed in the context of real applications. A feed-forward neural network is utilised to effect a topographic, structure-preserving, dimension-reducing transformation of the data, with an additional facility to incorporate different degrees of associated subjective information. The properties of this transformation are illustrated on synthetic and real datasets, including the 1992 UK Research Assessment Exercise for funding in higher education. The method is compared and contrasted to established techniques for feature extraction, and related to topographic mappings, the Sammon projection and the statistical field of multidimensional scaling.

Shadow targets: A novel algorithm for topographic projections by radial basis functions

Abstract The archetypal neural network topographic paradigm, Kohonen’s self-organising map, has proven highly effective in many applications but nevertheless has significant disadvantages which can limit its utility. Alternative feed-forward neural network approaches, including a model called “ Neuro Scale ”, have recently been developed based on explicit distance-preservation criteria. Excellent generalisation properties have been observed for such models, and recent analysis indicates that such behaviour is relatively insensitive to model complexity. As such, it is important that the training of such networks is performed efficiently, as computation of error and gradients scales in the order of the square of the number of patterns to be mapped. We therefore detail and demonstrate a novel training algorithm for Neuro Scale which outperforms present approaches.