Bayesian analysis of the scatterometer wind retrieval inverse problem:Some new approaches

The retrieval of wind vectors from satellite scatterometer observations is a non-linear inverse problem. A common approach to solving inverse problems is to adopt a Bayesian framework and to infer the posterior distribution of the parameters of interest given the observations by using a likelihood model relating the observations to the parameters, and a prior distribution over the parameters. We show how Gaussian process priors can be used efficiently with a variety of likelihood models, using local forward (observation) models and direct inverse models for the scatterometer. We present an enhanced Markov chain Monte Carlo method to sample from the resulting multimodal posterior distribution. We go on to show how the computational complexity of the inference can be controlled by using a sparse, sequential Bayes algorithm for estimation with Gaussian processes. This helps to overcome the most serious barrier to the use of probabilistic, Gaussian process methods in remote sensing inverse problems, which is the prohibitively large size of the data sets. We contrast the sampling results with the approximations that are found by using the sparse, sequential Bayes algorithm.

Studies on the generalisation of Gaussian processes and Bayesian neural networks

The assessment of the reliability of systems which learn from data is a key issue to investigate thoroughly before the actual application of information processing techniques to real-world problems. Over the recent years Gaussian processes and Bayesian neural networks have come to the fore and in this thesis their generalisation capabilities are analysed from theoretical and empirical perspectives. Upper and lower bounds on the learning curve of Gaussian processes are investigated in order to estimate the amount of data required to guarantee a certain level of generalisation performance. In this thesis we analyse the effects on the bounds and the learning curve induced by the smoothness of stochastic processes described by four different covariance functions. We also explain the early, linearly-decreasing behaviour of the curves and we investigate the asymptotic behaviour of the upper bounds. The effect of the noise and the characteristic lengthscale of the stochastic process on the tightness of the bounds are also discussed. The analysis is supported by several numerical simulations. The generalisation error of a Gaussian process is affected by the dimension of the input vector and may be decreased by input-variable reduction techniques. In conventional approaches to Gaussian process regression, the positive definite matrix estimating the distance between input points is often taken diagonal. In this thesis we show that a general distance matrix is able to estimate the effective dimensionality of the regression problem as well as to discover the linear transformation from the manifest variables to the hidden-feature space, with a significant reduction of the input dimension. Numerical simulations confirm the significant superiority of the general distance matrix with respect to the diagonal one.In the thesis we also present an empirical investigation of the generalisation errors of neural networks trained by two Bayesian algorithms, the Markov Chain Monte Carlo method and the evidence framework; the neural networks have been trained on the task of labelling segmented outdoor images.

Bayesian data assimilation

This thesis addresses data assimilation, which typically refers to the estimation of the state of a physical system given a model and observations, and its application to short-term precipitation forecasting. A general introduction to data assimilation is given, both from a deterministic and' stochastic point of view. Data assimilation algorithms are reviewed, in the static case (when no dynamics are involved), then in the dynamic case. A double experiment on two non-linear models, the Lorenz 63 and the Lorenz 96 models, is run and the comparative performance of the methods is discussed in terms of quality of the assimilation, robustness "in the non-linear regime and computational time. Following the general review and analysis, data assimilation is discussed in the particular context of very short-term rainfall forecasting (nowcasting) using radar images. An extended Bayesian precipitation nowcasting model is introduced. The model is stochastic in nature and relies on the spatial decomposition of the rainfall field into rain "cells". Radar observations are assimilated using a Variational Bayesian method in which the true posterior distribution of the parameters is approximated by a more tractable distribution. The motion of the cells is captured by a 20 Gaussian process. The model is tested on two precipitation events, the first dominated by convective showers, the second by precipitation fronts. Several deterministic and probabilistic validation methods are applied and the model is shown to retain reasonable prediction skill at up to 3 hours lead time. Extensions to the model are discussed.

Precipitation studies on alloys from the system copper chromium zirconium

The precipitation reactions occurring in a series of copper-based alloys selected from the system copper-chromium-zirconium have been studied by resistometric and metallographic techniques. A survey of the factors influencing the development of copper-based alloys for high strength, high conductivity applications is followed by a more general review of contemporary materials, and illustrates that the most promising alloys are those containing chromium and zirconium. The few systematic attempts to study alloys from this system have been collated, discussed, and used as a basis for the selection of four alloy compositions viz:- Cu - 0.4% Cr Cu - 0.24. Zr Cu - 0. 3% Cr - 0.1% Zr Cu - 0.2% Cr - 0.2% Zr A description of the experimental techniques used to study the precipitation behaviour of these materials is preceeded by a discussion of the currently accepted theories relating to precipitate nucleation and growth. The experimental results are presented and discussed for each of the alloys independently, and are then treated jointly to obtain an overall assessment of the way in which the precipitation kinetics, metallography and mechanical properties vary with alloy composition and heat treatment. The metastable solid solution of copper-chromium is found to decompose by the rejection of chromium particles which maintain a coherent interface and a Kurdjumov-Sachs type crystallographic orientation relationship with the copper matrix. The addition of 0.1% zirconium to the alloy retards the rate of transformation by a factor of ten and modifies the dispersion characteristics of the precipitate without markedly altering the morphology. Further additions of zirconium lead to the growth of stacking faults during ageing, which provide favourable nucleation sites for the chromium precipitate. The partial dislocations bounding such stacking faults are also found to provide mobile heterogeneous nucleation sources for the precipitation reactions occurring in copper-zirconium.

Alpha helical trans-membrane proteins:enhanced prediction using a Bayesian approach

Membrane proteins, which constitute approximately 20% of most genomes, are poorly tractable targets for experimental structure determination, thus analysis by prediction and modelling makes an important contribution to their on-going study. Membrane proteins form two main classes: alpha helical and beta barrel trans-membrane proteins. By using a method based on Bayesian Networks, which provides a flexible and powerful framework for statistical inference, we addressed alpha-helical topology prediction. This method has accuracies of 77.4% for prokaryotic proteins and 61.4% for eukaryotic proteins. The method described here represents an important advance in the computational determination of membrane protein topology and offers a useful, and complementary, tool for the analysis of membrane proteins for a range of applications.

Beta barrel trans-membrane proteins:enhanced prediction using a Bayesian approach

Membrane proteins, which constitute approximately 20% of most genomes, form two main classes: alpha helical and beta barrel transmembrane proteins. Using methods based on Bayesian Networks, a powerful approach for statistical inference, we have sought to address beta-barrel topology prediction. The beta-barrel topology predictor reports individual strand accuracies of 88.6%. The method outlined here represents a potentially important advance in the computational determination of membrane protein topology.

Message-passing for inference and optimization of real variables on sparse graphs

The inference and optimization in sparse graphs with real variables is studied using methods of statistical mechanics. Efficient distributed algorithms for the resource allocation problem are devised. Numerical simulations show excellent performance and full agreement with the theoretical results. © Springer-Verlag Berlin Heidelberg 2006.

Recognition on Finite Set of Events: Bayesian Analysis of Generalization Ability and Classification Tree Pruning

The problem of recognition on finite set of events is considered. The generalization ability of classifiers for this problem is studied within the Bayesian approach. The method for non-uniform prior distribution specification on recognition tasks is suggested. It takes into account the assumed degree of intersection between classes. The results of the analysis are applied for pruning of classification trees.

Detecting Precipitation Climate Changes: An Approach Based on a Stochastic Daily Precipitation Model

2002 Mathematics Subject Classification: 62M10.

Statistical Inference for Processes Depending on Environments and Application in Regenerative Processes

We build the Conditional Least Squares Estimator of 0 based on the observation of a single trajectory of {Zk,Ck}k, and give conditions ensuring its strong consistency. The particular case of general linear models according to 0=( 0, 0) and among them, regenerative processes, are studied more particularly. In this frame, we may also prove the consistency of the estimator of 0 although it belongs to an asymptotic negligible part of the model, and the asymptotic law of the estimator may also be calculated.

On Supra-Bayesian Weighted Combination of Available Data Determined by Kerridge Inaccuracy and Entropy

2010 Mathematics Subject Classification: 94A17.

Bayesian Prediction of Weibull Distribution Based on Fixed and Random Sample Size

2000 Mathematics Subject Classification: 62E16, 65C05, 65C20.

Bayesian and Frequentist Two-Sample Predictions of the Inverse Weibull Model Based on Generalized Order Statistics

2000 Mathematics Subject Classification: 62E16,62F15, 62H12, 62M20.

Bayesian adaptive estimation of arbitrary points on a psychometric function

Bayesian adaptive methods have been extensively used in psychophysics to estimate the point at which performance on a task attains arbitrary percentage levels, although the statistical properties of these estimators have never been assessed. We used simulation techniques to determine the small-sample properties of Bayesian estimators of arbitrary performance points, specifically addressing the issues of bias and precision as a function of the target percentage level. The study covered three major types of psychophysical task (yes-no detection, 2AFC discrimination and 2AFC detection) and explored the entire range of target performance levels allowed for by each task. Other factors included in the study were the form and parameters of the actual psychometric function Psi, the form and parameters of the model function M assumed in the Bayesian method, and the location of Psi within the parameter space. Our results indicate that Bayesian adaptive methods render unbiased estimators of any arbitrary point on psi only when M=Psi, and otherwise they yield bias whose magnitude can be considerable as the target level moves away from the midpoint of the range of Psi. The standard error of the estimator also increases as the target level approaches extreme values whether or not M=Psi. Contrary to widespread belief, neither the performance level at which bias is null nor that at which standard error is minimal can be predicted by the sweat factor. A closed-form expression nevertheless gives a reasonable fit to data describing the dependence of standard error on number of trials and target level, which allows determination of the number of trials that must be administered to obtain estimates with prescribed precision.