Splines and vector splines

The thrust of this report concerns spline theory and some of the background to spline theory and follows the development in (Wahba, 1991). We also review methods for determining hyper-parameters, such as the smoothing parameter, by Generalised Cross Validation. Splines have an advantage over Gaussian Process based procedures in that we can readily impose atmospherically sensible smoothness constraints and maintain computational efficiency. Vector splines enable us to penalise gradients of vorticity and divergence in wind fields. Two similar techniques are summarised and improvements based on robust error functions and restricted numbers of basis functions given. A final, brief discussion of the application of vector splines to the problem of scatterometer data assimilation highlights the problems of ambiguous solutions.

Gaussian processes:iterative sparse approximations

In recent years there has been an increased interest in applying non-parametric methods to real-world problems. Significant research has been devoted to Gaussian processes (GPs) due to their increased flexibility when compared with parametric models. These methods use Bayesian learning, which generally leads to analytically intractable posteriors. This thesis proposes a two-step solution to construct a probabilistic approximation to the posterior. In the first step we adapt the Bayesian online learning to GPs: the final approximation to the posterior is the result of propagating the first and second moments of intermediate posteriors obtained by combining a new example with the previous approximation. The propagation of em functional forms is solved by showing the existence of a parametrisation to posterior moments that uses combinations of the kernel function at the training points, transforming the Bayesian online learning of functions into a parametric formulation. The drawback is the prohibitive quadratic scaling of the number of parameters with the size of the data, making the method inapplicable to large datasets. The second step solves the problem of the exploding parameter size and makes GPs applicable to arbitrarily large datasets. The approximation is based on a measure of distance between two GPs, the KL-divergence between GPs. This second approximation is with a constrained GP in which only a small subset of the whole training dataset is used to represent the GP. This subset is called the em Basis Vector, or BV set and the resulting GP is a sparse approximation to the true posterior. As this sparsity is based on the KL-minimisation, it is probabilistic and independent of the way the posterior approximation from the first step is obtained. We combine the sparse approximation with an extension to the Bayesian online algorithm that allows multiple iterations for each input and thus approximating a batch solution. The resulting sparse learning algorithm is a generic one: for different problems we only change the likelihood. The algorithm is applied to a variety of problems and we examine its performance both on more classical regression and classification tasks and to the data-assimilation and a simple density estimation problems.

Phase equilibrium studies at normal pressures

A total pressure apparatus has been developed to measure vapour-liquid equilibrium data on binary mixtures at atmospheric and sub-atmospheric pressures. The method gives isothermal data which can be obtained rapidly. Only measurements of total pressure are made as a direct function of composition of synthetic liquid phase composition, the vapour phase composition being deduced through the Gibbs-Duhem relationship. The need to analyse either of the phases is eliminated. As such the errors introduced by sampling and analysis are removed. The essential requirements are that the pure components be degassed completely since any deficiency in degassing would introduce errors into the measured pressures. A similarly essential requirement was that the central apparatus would have to be absolutely leak-tight as any leakage of air either in or out of the apparatus would introduce erroneous pressure readings. The apparatus was commissioned by measuring the saturated vapour pressures of both degassed water and ethanol as a function of temperature. The pressure-temperature data on degassed water measured were directly compared with data in the literature, with good agreement. Similarly the pressure-temperature data were measured for ethanol, methanol and cyclohexane and where possible a direct comparison made with the literature data. Good agreement between the pure component data of this work and those available in the literature demonstrates firstly that a satisfactory degassing procedure has been achieved and that secondly the measurements of pressure-temperature are consistent for any one component; since this is true for a number of components, the measurements of both temperature and pressure are both self-consistent and of sufficient accuracy, with an observed compatibility between the precision/accuracy of the separate means of measuring pressure and temperature. The liquid mixtures studied were of ethanol-water, methanol-water and ethanol-cyclohexane. The total pressure was measured as the composition inside the equilibrium cell was varied at a set temperature. This gave P-T-x data sets for each mixture at a range of temperatures. A standard fitting-package from the literature was used to reduce the raw data to yield y-values to complete the x-y-P-T data sets. A consistency test could not be applied to the P-T-x data set as no y-values were obtained during the experimental measurements. In general satisfactory agreement was found between the data of this work and those available in the literature. For some runs discrepancies were observed, and further work recommended to eliminate the problems identified.

Parametric study of the behaviour of reinforced concrete columns in fire

The research is concerned with the application of the computer simulation technique to study the performance of reinforced concrete columns in a fire environment. The effect of three different concrete constitutive models incorporated in the computer simulation on the structural response of reinforced concrete columns exposed to fire is investigated. The material models differed mainly in respect to the formulation of the mechanical properties of concrete. The results from the simulation have clearly illustrated that a more realistic response of a reinforced concrete column exposed to fire is given by a constitutive model with transient creep or appropriate strain effect The assessment of the relative effect of the three concrete material models is considered from the analysis by adopting the approach of a parametric study, carried out using the results from a series of analyses on columns heated on three sides which produce substantial thermal gradients. Three different loading conditions were used on the column; axial loading and eccentric loading both to induce moments in the same sense and opposite sense to those induced by the thermal gradient. An axially loaded column heated on four sides was also considered. The computer modelling technique adopted separated the thermal and structural responses into two distinct computer programs. A finite element heat transfer analysis was used to determine the thermal response of the reinforced concrete columns when exposed to the ISO 834 furnace environment. The temperature distribution histories obtained were then used in conjunction with a structural response program. The effect of the occurrence of spalling on the structural behaviour of reinforced concrete column is also investigated. There is general recognition of the potential problems of spalling but no real investigation into what effect spalling has on the fire resistance of reinforced concrete members. In an attempt to address the situation, a method has been developed to model concrete columns exposed to fire which incorporates the effect of spalling. A total of 224 computer simulations were undertaken by varying the amounts of concrete lost during a specified period of exposure to fire. An array of six percentages of spalling were chosen for one range of simulation while a two stage progressive spalling regime was used for a second range. The quantification of the reduction in fire resistance of the columns against the amount of spalling, heating and loading patterns, and the time at which the concrete spalls appears to indicate that it is the amount of spalling which is the most significant variable in the reduction of fire resistance.

General and multiplicative non-parametric corporate performance models with interval ratio data

The increasing intensity of global competition has led organizations to utilize various types of performance measurement tools for improving the quality of their products and services. Data envelopment analysis (DEA) is a methodology for evaluating and measuring the relative efficiencies of a set of decision making units (DMUs) that use multiple inputs to produce multiple outputs. All the data in the conventional DEA with input and/or output ratios assumes the form of crisp numbers. However, the observed values of data in real-world problems are sometimes expressed as interval ratios. In this paper, we propose two new models: general and multiplicative non-parametric ratio models for DEA problems with interval data. The contributions of this paper are fourfold: (1) we consider input and output data expressed as interval ratios in DEA; (2) we address the gap in DEA literature for problems not suitable or difficult to model with crisp values; (3) we propose two new DEA models for evaluating the relative efficiencies of DMUs with interval ratios, and (4) we present a case study involving 20 banks with three interval ratios to demonstrate the applicability and efficacy of the proposed models where the traditional indicators are mostly financial ratios. © 2011 Elsevier Inc.

The role of pinning and instability in a class of non-equilibrium growth models

We study the dynamics of a growing crystalline facet where the growth mechanism is controlled by the geometry of the local curvature. A continuum model, in (2+1) dimensions, is developed in analogy with the Kardar-Parisi-Zhang (KPZ) model is considered for the purpose. Following standard coarse graining procedures, it is shown that in the large time, long distance limit, the continuum model predicts a curvature independent KPZ phase, thereby suppressing all explicit effects of curvature and local pinning in the system, in the "perturbative" limit. A direct numerical integration of this growth equation, in 1+1 dimensions, supports this observation below a critical parametric range, above which generic instabilities, in the form of isolated pillared structures lead to deviations from standard scaling behaviour. Possibilities of controlling this instability by introducing statistically "irrelevant" (in the sense of renormalisation groups) higher ordered nonlinearities have also been discussed.