Mathematical modelling and optimisation of the formulation and manufacture of aggregate food products

In this PhD study, mathematical modelling and optimisation of granola production has been carried out. Granola is an aggregated food product used in breakfast cereals and cereal bars. It is a baked crispy food product typically incorporating oats, other cereals and nuts bound together with a binder, such as honey, water and oil, to form a structured unit aggregate. In this work, the design and operation of two parallel processes to produce aggregate granola products were incorporated: i) a high shear mixing granulation stage (in a designated granulator) followed by drying/toasting in an oven. ii) a continuous fluidised bed followed by drying/toasting in an oven. In addition, the particle breakage of granola during pneumatic conveying produced by both a high shear granulator (HSG) and fluidised bed granulator (FBG) process were examined. Products were pneumatically conveyed in a purpose built conveying rig designed to mimic product conveying and packaging. Three different conveying rig configurations were employed; a straight pipe, a rig consisting two 45° bends and one with 90° bend. It was observed that the least amount of breakage occurred in the straight pipe while the most breakage occurred at 90° bend pipe. Moreover, lower levels of breakage were observed in two 45° bend pipe than the 90° bend vi pipe configuration. In general, increasing the impact angle increases the degree of breakage. Additionally for the granules produced in the HSG, those produced at 300 rpm have the lowest breakage rates while the granules produced at 150 rpm have the highest breakage rates. This effect clearly the importance of shear history (during granule production) on breakage rates during subsequent processing. In terms of the FBG there was no single operating parameter that was deemed to have a significant effect on breakage during subsequent conveying. A population balance model was developed to analyse the particle breakage occurring during pneumatic conveying. The population balance equations that govern this breakage process are solved using discretization. The Markov chain method was used for the solution of PBEs for this process. This study found that increasing the air velocity (by increasing the air pressure to the rig), results in increased breakage among granola aggregates. Furthermore, the analysis carried out in this work provides that a greater degree of breakage of granola aggregates occur in line with an increase in bend angle.

On classical and other methods of discriminant analysis and estimation of log-odds

For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.

Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.

Computational studies of the transverse structure of AGN jets

Both the emission properties and the evolution of the radio jets of Active Galactic Nuclei are dependent on the magnetic (B) fields that thread them. A number of observations of AGN jets suggest that the B fields they carry have a significant helical component, at least on parsec scales. This thesis uses a model, first proposed by Laing and then developed by Papageorgiou, to explore how well the observed properties of AGN jets can be reproduced by assuming a helical B field with three parameters; pitch angle, viewing angle and degree of entanglement. This model has been applied to multifrequency Very Long Baseline Interferometry (VLBI) observations of the AGN jets of Markarian 501 and M87, making it possible to derive values for the helical pitch angle, the viewing angle and the degree of entanglement for these jets. Faraday rotation measurements are another important tool for investigating the B fields of AGN jets. A helical B field component should result in a systematic gradient in the observed Faraday rotation across the jet. Real observed radio images have finite resolution; typical beam sizes for cm-wavelength VLBI observations are often comparable to or larger than the intrinsic jet widths, raising questions about how well resolved a jet must be in the transverse direction in order to reliably detect transverse Faraday-rotation structure. This thesis presents results of Monte Carlo simulations of Faraday rotation images designed to directly investigate this question, together with a detailed investigation into the probabilities of observing spurious Faraday Rotation gradients as a result of random noise and finite resolution. These simulations clearly demonstrate the possibility of detecting transverse Faraday-rotation structures even when the intrinsic jet widths are appreciably smaller than the beam width.

Uncertainty visualization in 3D scalar data

One problem in most three-dimensional (3D) scalar data visualization techniques is that they often overlook to depict uncertainty that comes with the 3D scalar data and thus fail to faithfully present the 3D scalar data and have risks which may mislead users’ interpretations, conclusions or even decisions. Therefore this thesis focuses on the study of uncertainty visualization in 3D scalar data and we seek to create better uncertainty visualization techniques, as well as to find out the advantages/disadvantages of those state-of-the-art uncertainty visualization techniques. To do this, we address three specific hypotheses: (1) the proposed Texture uncertainty visualization technique enables users to better identify scalar/error data, and provides reduced visual overload and more appropriate brightness than four state-of-the-art uncertainty visualization techniques, as demonstrated using a perceptual effectiveness user study. (2) The proposed Linked Views and Interactive Specification (LVIS) uncertainty visualization technique enables users to better search max/min scalar and error data than four state-of-the-art uncertainty visualization techniques, as demonstrated using a perceptual effectiveness user study. (3) The proposed Probabilistic Query uncertainty visualization technique, in comparison to traditional Direct Volume Rendering (DVR) methods, enables radiologists/physicians to better identify possible alternative renderings relevant to a diagnosis and the classification probabilities associated to the materials appeared on these renderings; this leads to improved decision support for diagnosis, as demonstrated in the domain of medical imaging. For each hypothesis, we test it by following/implementing a unified framework that consists of three main steps: the first main step is uncertainty data modeling, which clearly defines and generates certainty types of uncertainty associated to given 3D scalar data. The second main step is uncertainty visualization, which transforms the 3D scalar data and their associated uncertainty generated from the first main step into two-dimensional (2D) images for insight, interpretation or communication. The third main step is evaluation, which transforms the 2D images generated from the second main step into quantitative scores according to specific user tasks, and statistically analyzes the scores. As a result, the quality of each uncertainty visualization technique is determined.