Random field models and priors on wind

This report seeks to make concrete some of the ideas we have been discussing about sensible priors for winds over the ocean. In particular, random field models are reviewed, as are permissible covariance functions. The criteria which these covariance functions must satisfy in order that vorticity and divergence exist and are continuous are defined. The use of Helmholtz theorem is discussed, and possible choices for the covariances are suggested.

An upper bound on the Bayesian error bars for generalized linear regression

In the Bayesian framework, predictions for a regression problem are expressed in terms of a distribution of output values. The mode of this distribution corresponds to the most probable output, while the uncertainty associated with the predictions can conveniently be expressed in terms of error bars. In this paper we consider the evaluation of error bars in the context of the class of generalized linear regression models. We provide insights into the dependence of the error bars on the location of the data points and we derive an upper bound on the true error bars in terms of the contributions from individual data points which are themselves easily evaluated.

Guiding local regression using visualisation

Solving many scientific problems requires effective regression and/or classification models for large high-dimensional datasets. Experts from these problem domains (e.g. biologists, chemists, financial analysts) have insights into the domain which can be helpful in developing powerful models but they need a modelling framework that helps them to use these insights. Data visualisation is an effective technique for presenting data and requiring feedback from the experts. A single global regression model can rarely capture the full behavioural variability of a huge multi-dimensional dataset. Instead, local regression models, each focused on a separate area of input space, often work better since the behaviour of different areas may vary. Classical local models such as Mixture of Experts segment the input space automatically, which is not always effective and it also lacks involvement of the domain experts to guide a meaningful segmentation of the input space. In this paper we addresses this issue by allowing domain experts to interactively segment the input space using data visualisation. The segmentation output obtained is then further used to develop effective local regression models.

Learning curves for Gaussian processes models: fluctuations and universality

Based on a statistical mechanics approach, we develop a method for approximately computing average case learning curves and their sample fluctuations for Gaussian process regression models. We give examples for the Wiener process and show that universal relations (that are independent of the input distribution) between error measures can be derived.

An investigation of the retail marketing mix and its effectiveness in small shops

The recent history of small shop and independent retailing has been one of decline. The most desirable form of assistance is the provision of information which will increase the efficiency model of marketing mix effeciveness which may be applied in small scale retailing. A further aim is to enhance theoretical development in the marketing field. Recent changes in retailing have affected location, product range, pricing and promotion practices. Although a large number of variables representing aspects of the marketing mix may be identified, it is not possible, on the basis of currently available information, to quantify or rank them according to their effect on sales performance. In designing a suitable study a major issue is that of access to a suitable representative sample of small retailers. The publish nature of the retail activities involved facilitates the use of a novel observation approach to data collection. A cross-sectional survey research design was used focussing on a clustered random sample of greengrocers and gent's fashion outfitters in the West Midlands. Linear multiple regression was the main analytical technique. Powerful regression models were evolved for both types of retailing. For greengrocers the major influences on trade are pedestrian traffic and shelf display space. For gent's outfitters they are centrality-to-other shopping, advertising and shelf display space. The models may be utilised by retailers to determine the relative strength of marketing mix variables. The level of precision is not sufficient to permit cost benefit analysis. Comparison of the findings for the two distinct kinds of business studied suggests an overall model of marketing mix effectiveness might be based on frequency of purchase, homogeneity of the shopping environment, elasticity of demand and bulk characteristics of the good sold by a shop.

Epidemiology of myopia in a United Kingdom urban child population

The Aston Eye Study (AES) was instigated in October 2005 to determine the distribution of refractive error and associated ocular biometry in a sample of UK urban school children. The AES is the first study to compare outcome measures separately in White, South Asian and Black children. Children were selected from two age groups (Year 2 children aged 6/7 years, Year8 children aged 12/13 years of age) using random cluster sampling of schools in Birmingham, West Midlands UK. To date, the AES has examined 598 children (302 Year 2,296 Year 8). Using open-field cycloplegic autorefraction, the overall prevalence of myopia (=-0.50D SER in either eye) determined was 19.6%, with a higher prevalence in older (29.4%) compared to younger (9.9%) children (p<0.001). Using multiple logistic regression models, the risk of myopia was higher in Year 8 South Asian compared to White children and higher in children attending grammar schools relative to comprehensive schools. In addition, the prevalence of uncorrected ametropia was found to be high (Year 8: 12.84%, Year 2: 15.23%), which will be of concern to bodies responsible for the implementation of school vision screening strategies. Biometric data using non-contact partial coherence interferometry revealed a contributory effect of axial length (AL) and central corneal radius (CR) on myopic refraction, resulting in a strong coefficient of determination of the AL/CR ratio on refractive error. Ocular biometric measures did not vary significantly as a function of ethnicity, suggesting a greater miscorrelation of components in susceptible ethnic groups to account for their higher myopia prevalence. Corneal radius was found to be steeper in myopes in both age groups, but was found to flatten with increasing axial length. Due to the inextricable link between myopia and axial elongation, the paradoxical finding of the cornea demands further longitudinal investigation, particularly in relation to myopia onset. Questionnaire analysis revealed a history of myopia in parents and siblings to be significantly associated with myopia in Year 8 children, with a dose-dependent rise in the odds ratio of myopia evident with increasing number of myopic parents. By classifying socioeconomic status (SES) using Index of Multiple Deprivation values, it was found that Year 8 children from moderately deprived backgrounds were more at risk of myopia compared with children located at both extremities of the deprivation spectrum. However, the main effect of SES weakened following multivariate analysis, with South Asian ethnicity and grammar schooling remaining associated with Year 8 myopia after adjustment.

Guiding local regression using visualisation

Solving many scientific problems requires effective regression and/or classification models for large high-dimensional datasets. Experts from these problem domains (e.g. biologists, chemists, financial analysts) have insights into the domain which can be helpful in developing powerful models but they need a modelling framework that helps them to use these insights. Data visualisation is an effective technique for presenting data and requiring feedback from the experts. A single global regression model can rarely capture the full behavioural variability of a huge multi-dimensional dataset. Instead, local regression models, each focused on a separate area of input space, often work better since the behaviour of different areas may vary. Classical local models such as Mixture of Experts segment the input space automatically, which is not always effective and it also lacks involvement of the domain experts to guide a meaningful segmentation of the input space. In this paper we addresses this issue by allowing domain experts to interactively segment the input space using data visualisation. The segmentation output obtained is then further used to develop effective local regression models.

Optimal design for correlated processes with input-dependent noise

Optimal design for parameter estimation in Gaussian process regression models with input-dependent noise is examined. The motivation stems from the area of computer experiments, where computationally demanding simulators are approximated using Gaussian process emulators to act as statistical surrogates. In the case of stochastic simulators, which produce a random output for a given set of model inputs, repeated evaluations are useful, supporting the use of replicate observations in the experimental design. The findings are also applicable to the wider context of experimental design for Gaussian process regression and kriging. Designs are proposed with the aim of minimising the variance of the Gaussian process parameter estimates. A heteroscedastic Gaussian process model is presented which allows for an experimental design technique based on an extension of Fisher information to heteroscedastic models. It is empirically shown that the error of the approximation of the parameter variance by the inverse of the Fisher information is reduced as the number of replicated points is increased. Through a series of simulation experiments on both synthetic data and a systems biology stochastic simulator, optimal designs with replicate observations are shown to outperform space-filling designs both with and without replicate observations. Guidance is provided on best practice for optimal experimental design for stochastic response models. © 2013 Elsevier Inc. All rights reserved.