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.

Constructing localized non-linear projection manifolds in a principled way:hierarchical generative topographic mapping

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 (Bishop98a) in several directions: 1. We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. 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. 3. Using tools from differential geometry we derive expressions for local directionalcurvatures 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 andapply our system to two more complex 12- and 19-dimensional data sets.

A Clarke–Ledyaev Type Inequality for Certain Non–Convex Sets

We consider the question whether the assumption of convexity of the set involved in Clarke-Ledyaev inequality can be relaxed. In the case when the point is outside the convex hull of the set we show that Clarke-Ledyaev type inequality holds if and only if there is certain geometrical relation between the point and the set.

On Pareto Sets in Multi-Criteria Optimization

Здравко Д. Славов - В тази работа се разглеждат Паретовските решения в непрекъсната многокритериална оптимизация. Обсъжда се ролята на някои предположения, които влияят на характеристиките на Паретовските множества. Авторът се е опитал да премахне предположенията за вдлъбнатост на целевите функции и изпъкналост на допустимата област, които обикновено се използват в многокритериалната оптимизация. Резултатите са на базата на конструирането на ретракция от допустимата област върху Парето-оптималното множество.

Piecewise Convex Curves and their Integral Representation

2000 Mathematics Subject Classification: 52A10.

Stable sets in one-seller assignment games

We consider von Neumann -- Morgenstern stable sets in assignment games with one seller and many buyers. We prove that a set of imputations is a stable set if and only if it is the graph of a certain type of continuous and monotone function. This characterization enables us to interpret the standards of behavior encompassed by the various stable sets as possible outcomes of well-known auction procedures when groups of buyers may form bidder rings. We also show that the union of all stable sets can be described as the union of convex polytopes all of whose vertices are marginal contribution payoff vectors. Consequently, each stable set is contained in the Weber set. The Shapley value, however, typically falls outside the union of all stable sets.

New developments on VCA unmixing algorithm

Hyperspectral sensors are being developed for remote sensing applications. These sensors produce huge data volumes which require faster processing and analysis tools. Vertex component analysis (VCA) has become a very useful tool to unmix hyperspectral data. It has been successfully used to determine endmembers and unmix large hyperspectral data sets without the use of any a priori knowledge of the constituent spectra. Compared with other geometric-based approaches VCA is an efficient method from the computational point of view. In this paper we introduce new developments for VCA: 1) a new signal subspace identification method (HySime) is applied to infer the signal subspace where the data set live. This step also infers the number of endmembers present in the data set; 2) after the projection of the data set onto the signal subspace, the algorithm iteratively projects the data set onto several directions orthogonal to the subspace spanned by the endmembers already determined. The new endmember signature corresponds to these extreme of the projections. The capability of VCA to unmix large hyperspectral scenes (real or simulated), with low computational complexity, is also illustrated.

Tourism Destination Preferences: A Longitudinal Analysis of Consumer Decision Sets

There have only been a small number of applications of consumer decision set theory to holiday destination choice, and these studies have tended to rely on a single cross sectional snapshot of research participants’ stated preferences. Very little has been reported on the relationship between stated destination preferences and actual travel, or changes in decision set composition over time. The paper presents a rare longitudinal examination of destination decision sets, in the context of short break holidays by car in Queensland, Australia. Two questionnaires were administered, three months apart. The first identified destination preferences while the second examined actual travel and revisited destination preferences. In relation to the conference theme, there was very little change in consumer preferences towards the competitive set of destinations over the three month period. A key implication for the destination of interest, which, in an attempt to change market perceptions, launched a new brand campaign during the period of the project, is that a long term investment in a consistent brand message will be required to change market perceptions. The results go some way to support the proposition that the positioning of a destination into a consumer’s decision set represents a source of competitive advantage.

Generation of a 3D proximal femur shape from a single projection 2D radiographic image.

Summary Generalized Procrustes analysis and thin plate splines were employed to create an average 3D shape template of the proximal femur that was warped to the size and shape of a single 2D radiographic image of a subject. Mean absolute depth errors are comparable with previous approaches utilising multiple 2D input projections. Introduction Several approaches have been adopted to derive volumetric density (g cm-3) from a conventional 2D representation of areal bone mineral density (BMD, g cm-2). Such approaches have generally aimed at deriving an average depth across the areal projection rather than creating a formal 3D shape of the bone. Methods Generalized Procrustes analysis and thin plate splines were employed to create an average 3D shape template of the proximal femur that was subsequently warped to suit the size and shape of a single 2D radiographic image of a subject. CT scans of excised human femora, 18 and 24 scanned at pixel resolutions of 1.08 mm and 0.674 mm, respectively, were equally split into training (created 3D shape template) and test cohorts. Results The mean absolute depth errors of 3.4 mm and 1.73 mm, respectively, for the two CT pixel sizes are comparable with previous approaches based upon multiple 2D input projections. Conclusions This technique has the potential to derive volumetric density from BMD and to facilitate 3D finite element analysis for prediction of the mechanical integrity of the proximal femur. It may further be applied to other anatomical bone sites such as the distal radius and lumbar spine.