School of Mathematics and Statistics - Theses
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ItemThe dynamics of water bells( 2005-11)When a vertically aimed liquid jet impacts on the underside of a horizontal plate, it spreads radially to an abrupt point, and then suddenly falls of its own accord. The falling film may coalesce to form a water bell, and by changing the flow rate of the impinging jet, many beautiful shapes are attained. We present an original formulation for the critical radius where the fluid departs the plate. This solution agrees remarkably well with experiments. We also give an approximation for the evolving water bell shape under a changing flow regime.
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ItemAccounting for biological variation in digital gene expression experiments( 2009)This thesis discusses modeling, estimation and testing methods for 'digital gene expression' (DGE) data developed in the Walter and Eliza Hall Institute's Bioinformatics Division and implemented in the software package 'edgeR'. The race to complete the sequence of the entire human genome catalyzed the development of ultra high-throughput DNA sequencing technologies. Over the last decade, these technologies have proven to be extremely valuable for genomics, their original application, but have recently been shown to offer a powerful approach to investigating gene expression. Sequencing methods generate DGE data, that is, `counts' of the number of times a particular gene is detected in an RNA sample, a discrete measure of expression level. The number of counts for a gene gives an excellent indication of the true expression level of that gene in the biological sample. Assessing which genes are differentially expressed between experimental groups nevertheless remains a difficult problem, given the small sample sizes typical of biological experiments and the multiple-testing issues that arise when trying to assess differential expression for tens of thousands of genes simultaneously. Poisson models are a natural and popular choice for modeling DGE data, but it has been shown that biological replication of samples introduces greater variability into the data than can be accounted for using the Poisson model. The negative binomial (NB) model offers greater flexibility in handling overdispersion relative to the Poisson model, and represents a very promising approach for accounting for biological variation in DGE experiments. Accurate modeling of the variance is vital for the assessment of differential expression, but the standard methods for estimating the dispersion parameter for the NB model are not satisfactory, especially in the small-sample context of DGE experiments. Conditional maximum likelihood estimation of the dispersion parameter proves superior for DGE data. The typically small sample sizes in DGE experiments make the usual asymptotic tests inadmissible for testing hypotheses for differential expression. We discuss exact testing procedures for assessing differential expression in DGE experiments, including a novel exact test for the NB model. Experience from the analysis of microarray data has taught us that inference can be improved substantially by sharing information across all genes. This knowledge has led to the development of a common parameter model and empirical Bayes approaches to modeling DGE data. Case studies on real DGE data show the success of these methods in practice.