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Age-period-cohort (APC) models are used to analyse a variety of different health and demographic related outcomes. When including all three temporal trends into one model, there arises the well-known identification problem due to the structural link between the temporal trends (given two, the third can be calculated). Previous methods to resolve this problem focus on defining a model based off identifiable quantities. Most of the literature focusses on the case when APC models are fit to data aggregated in equal intervals (age and period widths). This may be, in part, due to the added complication that arises when fitting APC models to data in unequal intervals which causes a cyclic pattern in any estimates of the temporal trends. We first show that when an APC model is fit to data in unequal intervals, the previously identifiable terms, used to define a model that resolves the problem created by the structural link, are no longer identifiable. Using penalised smoothing splines, we show how the novel inclusion of a penalty on the previously identifiable terms resolves any problems that arise when fitting an APC model to data aggregated in unequal intervals and conclude the necessity of a penalty to provide a suitable solution. We demonstrate the suitability of our proposed model using theoretical and empirical results. Subsequently, we highlight the key information that links our proposed method to a class of APC models that utilise smoothing priors in a Bayesian paradigm. We give a full consideration of the problems that arise when fitting APC models to unequal interval data and how smoothing priors can be used to resolve them. Using theoretical and empirical results, we show that the smoothing prior models are performing a similar penalisation to that of a penalised smoothing spline, which we had previously shown to be a suitable solution, concluding their suitability is due to this penalty. We conclude with a novel application of an APC model to under-five mortality rates (U5MR) in Kenya 2006 to 2014. Previous methods to model U5MR include temporal terms for age and period, but not cohort. We extend the APC model to be suitable for the application by including spatial, spatio-temporal and strata specific terms alongside the terms for age, period, and cohort. The results indicate the inclusion of cohort is important to producing smooth fitting subnational estimates, an important goal for U5MR modelling which suffers from unstable estimates due to sparse data.

This thesis discusses Markov Chain Monte Carlo (MCMC) sampling algorithms which make use of differential geometry to improve performance. In particular, new sampling algorithms are introduced which exploit the dual geometry of statistical models developed by Amari, Chentsov and other authors in the field of Information Geometry. Previous work on differential-geometric MCMC has only used Riemannian geometry, i.e. without the dual structures studied in Information Geometry. Background chapters of the thesis discuss the development of Information Geometry by formulating statistical models as manifolds of data-generating measures. We review how Bregman divergences on such manifolds give rise to the dually-flat structure of Information Geometry, with examples provided by exponential families. Furthermore, Bayes' theorem is recast in a co-ordinate free form. Two novel sampling algorithms are introduced, called Orthogonal Gibbs and Orthogonal Gradient, which make use of dually-flat geometry. The Orthogonal Gibbs algorithm samples primal variables conditionally on dual variables and vice versa; similarly Orthogonal Gradient moves along the primal geodesic whose tangent vector is proportional to the gradient of the target log-density, and samples recursively on an m-flat submanifold orthogonal to the e-flat geodesic. We show that both algorithms move in directions in parameter space which are orthogonal with respect to the Riemannian metric. Furthermore, we show that when applied to a multivariate Gaussian target distribution, the Orthogonal Gibbs and Orthogonal Gradient algorithm respectively reduce to the Gaussian Elimination and Conjugate Gradient solvers for linear systems with added stochastic terms in the updates to the solution. The latter finding reflects work by previous authors on the connection between MCMC and optimization, including that of Fox and Parker. We show that the Orthogonal Gibbs algorithm produces independent samples for multivariate Gaussian targets, and produce numerical results suggesting the analogous result for Orthogonal Gradient. Finally, we present numerical results for a non-Gaussian target distribution for which the novel algorithms have favourable effective sample sizes compared to the No U-Turn Sampler (NUTS).

Owing to a combination of the difficulty in detection and disproportionately large component strength reduction, barely visible impact damage (BVID) induced by low energy impacts is a major concern to composite component manufacturers in the aerospace industry. Component testing to address this issue is both costly and time consuming. This results in manufacturers oversizing components, which negates the weight saving benefits that composite components offer over traditional metal equivalents. Therefore, the aim of this study is to present simple statistical damage models built from empirical observations that can aid the component design process by replacing initial impact testing. To assess the applicability of the impact damage models in the design process they are applied as the input to a semi-numerical compression after impact strength reduction model, the results of which are then compared to experimentally obtained data. To produce an empirical impact model a large database of impact damage is required, unlike any seen in contemporary literature. To this end, an extensive composite impact damage database has been constructed from impact testing of over 40 laminates, resulting in over 1400 unique damage observations. Variations in material, ply count and layup are all considered to produce this diverse database. Data-driven generalised additive models (GAMs) are selected to model the database due to their predictive capabilities when compared to both least-squares regression and machine learning techniques. The resulting smooth models are shown to be accurate approximations of the database. When applied to published damage data they are capable of accurate predictions, even when required to extrapolate beyond the limits of the variables in the damage database. The smooth models are approximated by simple polynomial regression fits in an attempt to create a reduced predictive model that is less accurate but more simple. These simpler models produce adequate approximations of the mean of the database, however model the variance poorly. They are also shown to be incapable of predicting the published data, becoming unstable when required to extrapolate outside of the database. Improvements to the semi-analytical model first presented by Choudhry, Rhead, Nielsen & Butler in their 2019 paper 'A plate model for compressive strength prediction of delaminated composites' are presented. 17 layups, consisting of 16 and 24 ply HTS/977-2 and T800/M21 laminates, have been subjected to 4 impact energies and crushed. Assessment of the model updates on this dataset show that the model is improved, producing consistently more accurate predictions across the layups and thicknesses tested. The updated compression after impact (CAI) strength model is applied to the experimentally obtained CAI strength data using the measured impact damage, smooth models and simple polynomial models as damage inputs. The results of this show that the smooth damage model is an acceptable alternative to experimentally obtained damage data when used as the compression model input. Accurate strength predictions are shown for both the in-house experimental testing as well the published data for BVID impacts. The combination of the smooth damage model and semi-analytical CAI strength model is shown to produce quick and accurate component strength reduction predictions across a wide range of layups, impact energies and materials.

In this thesis we unify and find new, and recover all known, explicit local examples of extremal toric Kähler metrics and describe how to compactify them. To do so, we define explicitly a class of toric geometries of Sasaki type with toric Kähler quotients, both called 'separable' geometries, using 'factorization structures'. We conjecture factorization structures to be decomposable in which case we find their explicit description to be of Segre-Veronese type. A compatible factorization structure gives rise to 'separable coordinates' on the image of the momentum map of a given separable geometry. In such coordinates the extremality equation for separable Kähler geometries becomes a functional system of ODEs which, in our case, is a system obtained from a generalisation of the method for separation of variables for PDEs. We derive necessary conditions for its solutions and find a complete set of solutions in the case of the product Segre-Veronese factorization structure with a decomposable Sasaki structure. We use generalised equipoised condition for extremal affine functions to geometrically characterise some compactifications of such extremal metrics.

This thesis focusses on the cumulative hazard function as a tool for modelling time-to-event data, as opposed to the more common hazard or survival functions. By focussing on and providing a detailed discussion of the properties of these functions a new framework is explored for building complex models from, the relatively simple, cumulative hazards. Parametric families are thoroughly explored in this thesis by detailing types of parameters for time-to-event models. The discussion leads to the proposal of combination parametric families, which aim to provide flexible behaviour of the cumulative hazard function. A common issue in the analysis of time-to-event data is the presence of informative censoring. This thesis explores new models which are useful for dealing with this issue.