Hierarchical GTM: Constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the parent visualization plot which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 19-dimensional data sets.

Hierarchical GTM: constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping (GTM). bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the ancestor visualization plots which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 18-dimensional data sets.

A principled approach to interactive hierarchical non-linear visualization of high-dimensional data

Hierarchical visualization systems are desirable because a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex high-dimensional data sets. We extend an existing locally linear hierarchical visualization system PhiVis [1] in several directions: bf(1) we allow for em non-linear projection manifolds (the basic building block is the Generative Topographic Mapping -- GTM), bf(2) we introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree, bf(3) we describe folding patterns of low-dimensional projection manifold in high-dimensional data space by computing and visualizing the manifold's local directional curvatures. Quantities such as magnification factors [3] and directional curvatures are helpful for understanding the layout of the nonlinear projection manifold in the data space and for further refinement of the hierarchical visualization plot. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. We demonstrate the visualization system principle of the approach on a complex 12-dimensional data set and mention possible applications in the pharmaceutical industry.

Semi-supervised learning of hierarchical latent trait models for data visualisation

An interactive hierarchical Generative Topographic Mapping (HGTM) ¸iteH_GTM has been developed to visualise complex data sets. In this paper, we build a more general visualisation system by extending the HGTM visualisation system in 3 directions: bf (1) We generalize HGTM to noise models from the exponential family of distributions. The basic building block is the Latent Trait Model (LTM) developed in ¸iteKaban_pami. bf (2) We give the user a choice of initializing the child plots of the current plot in either em interactive, or em automatic mode. In the interactive mode the user interactively selects ``regions of interest'' as in ¸iteH_GTM, whereas in the automatic mode an unsupervised minimum message length (MML)-driven construction of a mixture of LTMs is employed. bf (3) We derive general formulas for magnification factors in latent trait models. Magnification factors are a useful tool to improve our understanding of the visualisation plots, since they can highlight the boundaries between data clusters. The unsupervised construction is particularly useful when high-level plots are covered with dense clusters of highly overlapping data projections, making it difficult to use the interactive mode. Such a situation often arises when visualizing large data sets. We illustrate our approach on a toy example and apply our system to three more complex real data sets.

A basis function approach to Bayesian inference in diffusion processes

In this paper, we present a framework for Bayesian inference in continuous-time diffusion processes. The new method is directly related to the recently proposed variational Gaussian Process approximation (VGPA) approach to Bayesian smoothing of partially observed diffusions. By adopting a basis function expansion (BF-VGPA), both the time-dependent control parameters of the approximate GP process and its moment equations are projected onto a lower-dimensional subspace. This allows us both to reduce the computational complexity and to eliminate the time discretisation used in the previous algorithm. The new algorithm is tested on an Ornstein-Uhlenbeck process. Our preliminary results show that BF-VGPA algorithm provides a reasonably accurate state estimation using a small number of basis functions.

Gold mineralisation in the Caledonides of the British Isles with special reference to the Dolgellau Gold-belt, North Wales and the Southern Uplands, Scotland. Available in 2 volumes.

Two aspects of gold mineralisation in the Caledonides of the British Isles have been investigated: gold-telluride mineralisation at Clogau Mine, North Wales; and placer gold mineralisation in the Southern Uplands, Scotland. The primary ore assemblage at Clogau Mine is pyrite, arsenopyrite, cobaltite, pyrrhotine, chalcopyrite, galena, tellurbismuth, tetradymite, altaite, hessite, native gold, wehrlite, hedleyite, native bismuth, bismuthunite and various sulphosalts. The generalised paragenesis is early Fe, Co, Cu, As and S species, and later minerals of Pb, Bi, Ag, Au, Te, Sb. Electron probe micro-analysis (EPMA) of complex telluride-sulphide intergrowths suggests that these intergrowths formed by co-crystallisation/replacement processes and not exsolution. Minor element chemical variation, in the sulphides and tellurides, indicates that antimony and cadmium are preferentially partitioned into telluride minerals. Mineral stability diagrams suggest that during gold deposition log bf aTe2 was between -7.9 and -9.7 and log bf aS2 between -12.4 and -13.8. Co-existing mineral assemblages indicate that the final stages of telluride mineralisation were between c. 250 - 275oC. It is suggested that the high-grade telluride ore shoot was the result of remobilisation of Au, Bi, Ag and Te from low grade mineralisation elsewhere within the vein system, and that gold deposition was brought about by destabilisation of gold chloride complexes by interaction with graphite, sulphides and tellurbismuth. Scanning electron microscopy of planer gold grains from the Southern Uplands, Scotland, indicates that detailed studies on the morphology of placer gold can be used to elucidate the history of gold in the placer environment. In total 18 different morphological characteristics were identified. These were divided on an empirical basis, using the relative degree of mechanical attrition, into proximal and distal characteristics. One morphological characteristic (a porous/spongy surface at high magnification) is considered to be chemical in origin and represent the growth of `new' gold in the placer environment. The geographical distribution of morphological characteristics has been examined and suggests that proximal placer gold is spatially associated with the Loch Doon, Cairsphairn and Fleet granitoids. Quantitative EPMA of the placer gold reveals two compositional populations of placer gold. Examination of the geographical distribution of fineness suggests a loose spatial association between granitoids and low fineness placer gold. Also identified was chemically heterogeneous placer gold. EPMA studies of these heterogeneities allowed estimation of annealing history limits, which suggest that the heterogeneities formed between 150 and 235oC. It is concluded, on the basis of relationships between morphology and composition, that there are two types of placer gold in the Southern Uplands: (i) placer gold which is directly inherited from a hypogene source probably spatially associated with granitoids; and (ii) placer gold that has formed during supergene processes.