A hierarchical latent variable model for data visualization

Visualization has proven to be a powerful and widely-applicable tool the analysis and interpretation of 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 sub-clusters 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 first on a toy data set, and then apply the algorithm to the visualization 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.

The stratigraphy of the Fishguard volcanic complex and associated sediments, SW Dyfed, Wales

This thesis describes the geology of a Lower Palaeozoic terrain, situated west of the town of Fishguard, SW Dyfed, Wales. The area is dominated by the Fishguard Volcanic Complex (Upper Llanvirn), and sediments that range in age from the Middle Cambrian to the Lower Llandeilo. The successions represent an insight into sedimentation and volcanism for c. 100 Ma. along the south-western margin of the Lower Palaeozoic Welsh Basin. The stratigraphy of the sedimentary sequence has been completely revised and the existing volcanostratigraphy modified. The observed complexity of the stratigraphy is primarily the consequence of Caldedonide deformation which resulted in large scale repetition. Fold-thrust tectonics dominates the structural style of the area. Caledonide trending (NE-SW) cross-faults complicate preexisting structures. Middle Cambrian (?) sedimentation is documented by shallow marine clastics and red shales deposited within tidal - subtidal environments. Upper Cambrian sedimentation was dominated by shallow marine `storm' and `fair weather' sedimentation within a muddy shelf environment. Shallow marine conglomerates and heterolithic intertidal siliciclastics mark the onset of Ordovician sedimentation during the lower Arenig transgression. Mid-Arenig sediments reflect deposits influenced by storm, fair-weather and wave related processes in various shallow marine environments, including; shoreface, inner shelf, shoaling bar, and deltaic. Graptolitic marine shales were deposited from the upper mid-Arenig through to the lower Llandeilo; during which time sediments accumulated by pelagic processes and fine grained turbidites. The varied nature of sedimentation reflects both localised change within the depositional system and the influence of larger regional eustatic events. Ordovician subaqueous volcanic activity produced thick accumulations of lavas, pyroclastics, hydroclastics, and hyaloclastics. The majority of volcanism was effusive in nature, erupted below the Pressure Compensation Level. Basaltic volcanism was characterised by pillowed lavas and tube networks, whilst sheet-flow lavas, pillow breccias and minor hyaloclastites developed locally. Silicic volcanism was dominated by rhyolitic clastics of various affinities, although coherent silicic obsidian lavas, sheet-flow lavas and pyroclastics developed. Hypabyssal intrusives of variable composition and habit occur throughout the volcanic successions. Low-grade regional metamorphism has variably affected the area, conditions of the prehnite-pumpellyite and greenschist facies having been attained. Numerous secondary phases developed in response to the conditions imposed, which collectively indicate that P-T conditions were of low-pressure facies series in the range P= 1.2-2.0 kbars and T= 230-350oC, under an elevated geothermal gradient of 40-45oC km-1. Polymineralic cataclastites associated with Caledonide deformation indicate that tectonism and metamorphism were in part contemporaneous.

Multi-level visualisation using Gaussian process latent variable models

Projection of a high-dimensional dataset onto a two-dimensional space is a useful tool to visualise structures and relationships in the dataset. However, a single two-dimensional visualisation may not display all the intrinsic structure. Therefore, hierarchical/multi-level visualisation methods have been used to extract more detailed understanding of the data. Here we propose a multi-level Gaussian process latent variable model (MLGPLVM). MLGPLVM works by segmenting data (with e.g. K-means, Gaussian mixture model or interactive clustering) in the visualisation space and then fitting a visualisation model to each subset. To measure the quality of multi-level visualisation (with respect to parent and child models), metrics such as trustworthiness, continuity, mean relative rank errors, visualisation distance distortion and the negative log-likelihood per point are used. We evaluate the MLGPLVM approach on the ‘Oil Flow’ dataset and a dataset of protein electrostatic potentials for the ‘Major Histocompatibility Complex (MHC) class I’ of humans. In both cases, visual observation and the quantitative quality measures have shown better visualisation at lower levels.