In this thesis, we consider random sums with heavy-tailed increments. By the term random sum, we mean a sum of random variables where the number of summands is also random. Our interest is to analyse the tail behaviour of random sums and to construct an efficient method to calculate quantiles. For the sake of efficiency, we simulate rare-events (tail-events) using a Markov chain Monte Carlo (MCMC) method. The asymptotic behaviour of sum and the maximum of heavy-tailed random sums is identical. Therefore we compare random sum and maximum value for various distributions, to investigate from which point one can use the asymptotic approximation. Furthermore, we propose a new method to estimate quantiles and the estimator is shown to be efficient.

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Hyperspectral image analysis works with image data where each pixel contains hundreds of wavelengths acquired from spectral measurements. It is a growing field of research in the sciences and industries because it can distinguish visually similar objects. While many machine-learning methods work well for analysing regular images, little is known about how they perform on hyperspectral data. Standard methods for quantifying and classifying hyperspectral data include the chemometric methods PLS, PLS-DA and SIMCA. They provide rapid computations along with intuitive modelling and diagnostic tools, but cannot capture more complex data. I benchmarked the chemometric methods against machine learning methods from Microsoft's ML.NET library on six classification and two quantification problems. The ML.NET methods proved to be good complements to the chemometric methods. In particular, the decision tree methods provided accurate classification and quantification while the maximum entropy classification methods balanced between accuracy and computational time the best. While the remaining ML.NET methods performed equally well or better than the chemometric methods, finding their use requires testing on data sets with a wider range of properties. The best ML.NET methods are suitable for analysing more complex hyperspectral images by capturing nonlinearities disregarded by standard image analysis.

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Motivated by the study of the hypoxia problem in cancerous tissues, we propose a system of coupled partial differential equations defined on a heterogeneous, periodically perforated  domain describing the flux of oxygen from blood vessels towards the tissue and the corresponding oxygen diffusion within the tissue. Using heuristics based on dimensional analysis, we rephrase the initially parabolic problem as a semi-linear elliptic transmission problem. Focusing on the elliptic case, we are able to define a microscopic $\varepsilon$-dependent problem that is the starting point of our mathematical analysis; here $\varepsilon$ is linked to the scale of heterogeneity.   We study the well-posedness of the microscopic problem as well as the passage to the periodic homogenization limit. Additionally, we derive the strong formulation of the two-scale macroscopic limit problem. Finally, we prove a corrector estimate. This specific ingredient allows us to estimate, in an {\em a priori} way,  the discrepancy between solutions to the microscopic and, respectively, macroscopic problem. Our working techniques include energy-type estimates, fixed-point type iterations, monotonicity arguments, as well as the two-scale convergence tool.

In early drug discovery and the study of the effects of new chemical compounds on cancer cells, the change in cell shape over time provides vital information about cell health. Live-cell image analysis systems can be used to extract cell-shape describing parameters of individual cells during exposure to new drugs. Multivariate statistical analysis is then applied to understand cell morphology and the correlation between various shape descriptors. Principal component analysis integrated with histogram distribution analysis is a method to compress and summarize important cellular data features without loss of information about the individual cell shapes. A workflow for this kind of analysis is being developed at Sartorius and aims to aid in the biological interpretation of different experimental results. However, methods for exploring the time dimension in the experiments are not yet fully explored, and a temporal view of the data would increase understanding of the change in cell morphology metrics over time. In this study, we implement the workflow to a data set generated from the microscope IncuCyte and investigate a possible continuation of time-series analysis on the data. The results demonstrate how we can use principal component analysis in two steps together with histogram distributions of different experimental conditions to study cell shapes over time. Scores and loadings from the analysis are used as new observations representing the original data, and the evolution of score-value can be backtracked to cell morphology metrics changing in time. The results show a comprehensive way of studying how cells from all experimental conditions relate to each other during the course of an experiment.

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