In modern data analysis, high-dimensional datasets where the number of features far exceeds the number of observations are increasingly common. In such settings, identifying sparse informative features is a critical challenge. This thesis investigates the comparative performance of random forests and two penalized linear regression techniques, Ridge and Lasso regression, in scenarios with few informative features. Synthetic datasets were generated to simulate both linear and non-linear relationships between features and the target variable. The models were evaluated using mean squared error (MSE) to measure prediction accuracy and two custom metrics assessing the ability to identify informative features among the top-ranked variables. Additionally, computational efficiency was assessed.
In linear settings, Lasso consistently outperformed the other models in both prediction and feature selection, particularly at the lower informative ratios. Ridge regression demonstrated reasonable feature selection accuracy in linear settings but underperformed in prediction, likely due to a limited hyperparameter grid. In non-linear settings, random forests were most effective at identifying informative features, though the prediction performance of Lasso remained competitive and slightly outperformed random forests under high sparsity. Training time analysis further showed that the penalized linear models were substantially faster than randomforests. These findings underscore that no single model is universally optimal, as performance varies depending on datastructure, sparsity, and the modelling objective. They also demonstrate that improved feature selection does not necessarily guarantee better prediction in complex, high-dimensional settings. These results have practical implications for domains like genomics, where balancing accuracy, interpretability, and computational efficiency is critical.

Reconstructing high-resolution wind fields from sparse, low-resolution observations is a critical yet ill-posed problem in meteorological modeling. Classical approaches, such as Computational Fluid Dynamics (CFD), are often too computationally intensive to meet the demands of real- time or large-scale industrial applications. Meanwhile, conventional data-driven methods like Convolutional Neural Networks (CNNs) tend to produce overly smoothed outputs and struggle to recover fine-scale structures, especially under severe data sparsity.
This thesis explores the use of diffusion-based generative models for super-resolution in wind field reconstruction. A progressive SR3 (Super-Resolution via Repeated Refinement) frame- work is developed, combining a multi-stage architecture with stochastic denoising processes to gradually reconstruct high-resolution outputs. Extensive experiments demonstrate that the progressive SR3 consistently outperforms CNN-based baselines in terms of reconstruction accur- acy, perceptual quality, and robustness. Furthermore, a joint training strategy improves both performance and computational efficiency by enabling end-to-end optimization across stages.
The findings support the use of probabilistic diffusion models for meteorological super-resolution tasks and emphasize the effectiveness of progressive refinement in handling large upscaling factors. This approach provides a promising pathway for enhancing data-driven post-processing in atmospheric modeling.

Solution methods and computational efficiency of stochastic linear programs have been extensively studied over the years. Despite this, their practical applications remain limited due to persistent computational challenges and the lack of a user-friendly interface for modeling and solving these problems. This thesis revisits those computational challenges and explores the usability of stochastic linear programs in the field of energy system modeling. This is done by implementing two solution methods for solving two-stage stochastic linear programs: the Extensive Form method and the L-Shaped method, and applying both methods on existing input problems. To additionally test the performance on an energy system model, the existing large-scale energy model Oemof-B3 is extended to a stochastic linear program, and the solution methods are applied on this problem too. Since the stochastic linear programming framework can be easily applied, and both methods successfully produce solutions for various input problems, we conclude stochastic linear programming holds significant potential as a modeling tool. However, computational challenges remain, particularly when applying the stochastic programming framework to the energy model Oemof-B3. Additionally, the lack of user-friendly tools and readily available solvers for solving stochastic linear programs limits the practical applicability of these methods.

This thesis investigates the problem of time-varying function optimization. In particular, we study techniques to minimize the cumulative regret when optimizing a time-varying function in the Gaussian process setting. First, we introduce the problem and present a literature review of the current methods and results. Following this, we we propose enhancements to existing algorithms, demonstrating improved regret bounds. We discuss the applications of these algorithms and where they can provide a benefit compared to existing methods. With these applications in mind we introduce two new temporal models for time-varying functions and their associated algorithms. We test their performance in order to validate their effectiveness and potential benefits.

This thesis aims to introduce the reader to the mathematical principles underlying X-ray computed tomography. It begins with the derivation of both filtered and unfiltered back-projection reconstruction techniques. Following this, the thesis delves into two algorithms based on algebraic reconstruction techniques (ART): the Simultaneous Algebraic Reconstruction Technique (SART) and the Discrete Algebraic Reconstruction Technique (DART). This includes the derivation and implementation of both methods and the analysis and comparison of theDART algorithmto the aforementioned methods, all using simulated data. These ART methods approach the reconstruction problem by iteratively optimizing a large system of linear equations. In addition, DART leverages prior knowledge on grey values to steer the final reconstruction towards one that only contains these pre-determined grey values. Based on the experiments using simulated data it is shown that DART effectively deals with both errors in the approximation of the grey values as well as with noisy projection data. Furthermore, it is found that DART computes relatively accurate reconstructions using only a small number of projections or projections from a small angular range.

With the rise of zero-shot synthetic image generation models, such as Stability.ai's Stable Diffusion, OpenAI's DALLE or Google's Imagen, the need for powerful tools to detect synthetic generated images has never been higher. In this thesis we contribute to this goal by considering wavelet-based approaches for synthetic image detection.

We will introduce multi-level discrete wavelet transform, which to the best of our knowledge has never been considered for this goal prior to this work. A similar approach that has been considered for the goal of synthetic image detection, is the multi-level wavelet packet transform used by Wolter et al. We will show that not only is our proposed approach more efficient and easier interpretable, it also performs better in a number of experimental settings and therefore forms a suitable addition to the toolset for the detection of synthetic images.

Moreover, we will try and generalize performance of our used classifiers to out-of-dataset samples and see that our used classifier in general does not allow for such generalization. Finally, we will discuss the challenges of this work and offer interesting directions for further research.

This thesis explores the application of Benford's Law to wavelet coefficients derived from the Discrete Wavelet Transform (DWT) of images, aiming to provide a novel method for image differentiation. The study focuses on the DWT, specifically utilizing Haar and Daubechies wavelets, to decompose an image into approximation and detail coefficients. Benford's Law, predicting the frequency distribution of leading digits in natural datasets, is applied to the detail coefficients. The research investigates whether this approach can differentiate natural images from other genres, such as paintings or cartoons. The thesis provides a comprehensive understanding of wavelets, the DWT, and signal decomposition, followed by an introduction to Benford's Law and its applications. The final part involves applying Benford’s Law to the DWT coefficients of different image genres, analyzing the results, and discussing further research.

This thesis aims at introducing the reader to the mathematical concepts behind the imaging technique X-ray tomography, commonly known as a CT scan. It includes the derivation of the filtered and unfiltered backprojection reconstruction methods for noise-free data. It was concluded that for noisy data, X-ray tomography is an ill-posed, and therefore unstable, inverse problem that needs regularization in order to produce adequate reconstructions. The methods of truncated singular value decomposition regularization and Tikhonov regularization are derived, analyzed and compared using simulated as well as real-life data. It was found that both methods can produce stable reconstructions, but that Tikhonov regularization is less sensitive to parameter choice.

Background Dynamic SPECT scanning provides a non-invasive way to image the time-dependent distribution of radio-labelled tracers inside living tissue. Beside human medicine, dynamic SPECT also finds its applications in pre-clinical research on small animals. In pre-clinical research, multi-pinhole collimators are used to enable high-resolution sub-millimeter imaging. Conventional iterative reconstruction methods, such as Maximum Likelihood Expectation Maximisation (MLEM) perform poorly in reconstructing the noisy and low-count scans in dynamic SPECT. This limits the temporal resolution that can be achieved.
Method The reconstruction of noisy, low-count time-frames can be aided by incorporating information from earlier and later time-frames. Wang and Qi (2015) published the paper 'PET Image Reconstruction Using Kernel Method', with a proposed method entitled Kernelised Expectation Maximisation (KEM) for dynamic PET, a method that uses principles from Machine Learning, such as Support Vector Machines and the 'kernel trick' to incorporate prior information in the reconstruction algorithm. This method is highly adjustable due to a number of input parameters of the method. In this paper, KEM is implemented for dynamic multi-pinhole SPECT. The effects of the KEM parameters are explored in computer simulations. Two different dynamic phantoms are used, one of the striata in a mouse brain which were adapted from a paper by Vastenhouw et al. (2007) and one of the hepatobiliary system adapted from Vaissier et al. (2012). The results of KEM are benchmarked against conventional MLEM with a Gaussian post-filter. 
Results In high-count simulations, the MLEM reconstructions had a lower mean-squared-error than the KEM image, while the signal-to-noise ratio of KEM was better than MLEM. The images produced after 200 iterations were indistinguishable, however. In the low-count regime, KEM was shown to be more resistant to noise than MLEM. Varying the input parameters of KEM gave rise to differences in performance, such as (over-)smoothing effects and a different level of noise-suppresion in the reconstructed image. 
Conclusions In the simulations used in this paper, KEM was shown to outperform conventional MLEM with a Gaussian post-filter from low-count projections. The optimal input parameters of the KEM algorithm, however, need to be found ad hoc by searching the parameter space. Further research should look into finding a set of rules or guidelines for finding the optimal parameters.

Cyo-EM is a powerful technique in biophysics to model. By recording tens of thousands of these images and using computational methods to form a 3D density map. The interpretability of these cryo-EM maps decreases in high-resolution by using map sharpening to increase these areas of low contrast locally. It is possible to locally sharpen the cryo-EM map. While global sharpening methods would result in oversharpening of some areas should be and undersharpening in others, using a local sharpening method can adjust for a variation in contrast within a map. This resulted in the development of LocScale. Which is a local sharpening method using a (pseudo) atomic model. The main drawback of using a local sharpening method which would require an atomic model is the fact that it is impractical in cases where there is no model available as well as requiring atomic model refinement for sharpening, which takes a lot of resources. While other machine learning methods are able to be used to sharpen EM-maps. When decomposing the sharpened maps in Fourier space, it often becomes clear that not only the amplitudes but also the phases, instead of just the amplitudes of the EM-map have been changed. Resulting in hallucinations appearing, and ML sharpening methods are often implemented in such a way that the entire method is housed inside a black box, making it challenging to understand why certain parts of the macromolecule are sharpened. Using a machine learning-based approach to predicting local radial profiles it is possible to limit the use of machine learning to only one step, the step for determining the radial profiles, compared to the model-based method, LocScale. And use all other steps which would normally be associated with LocScale, vastly reducing the use of ML. Since the radial profiles describe the amplitudes of the EM-map it is possible to change these while leaving the phases unchanged. Reducing the chance of hallucinations appearing to zero.In this thesis, it is shown that it is possible to produce a machine learning method (named iLocScale) to effectively sharpen EM maps by predicting radial profiles. This resulted in a method which is almost as sharp as their model-based (target) counterpart (LocScale) in the modelled region and vastly outperforms LocScale in the unmodelled regions for sharpness. While also offering a large improvement in correlation coefficient for local resolution and B-factor between the iLocScale sharpened maps (0.3416) and unsharpened maps (0.0465). Moreover, iLocScale vastly outperforms LocScale in unmodeled regions. It also takes approximately 75% less time.

In this thesis we develop a Bayesian approach to graph contrastive learning and propose a new uncertainty measure based on the disagreement in likelihood due to different positive samples. Moreover, we extend contrastive learning to simplicial complexes and show that it can be used to generate high-quality representations of edge flow data.

This research consists of two applications of image processing, namely, image compression and image denoising. Image compression aims to reduce the size of an image without losing too many features. This is often used to store a large number of images such as fingerprints. Denoising is a technique for removing noise from an image while preserving as many of the edges and other detailed features as possible. This research studies the use of different discrete wavelets and the Dual Tree Complex Wavelets in the image compression and denoising process. The wavelet transform decomposes the original image into approximation and detail coefficients, where the approximation coefficients are calculated by averaging and the detail coefficients by taking differences. The wavelet transform is also invertible so that the image can be reconstructed again using the approximation and detail coefficients. The compression and denoising method consists of three steps: decomposition, thresholding and reconstruction. The difference between compression and denoising lies in the threshold part. For image compression, a percentage of the detail coefficient is chosen as the threshold. The wavelets db6, sym5, coif3, bior4.4 and rbio1.5 are chosen to compress the images. The images are tested with compression rates ranging from 5:1 to 43:1. Based on the Structural Similarity Index Measurement (SSIM), the bior4.4 wavelet performs best. For denoising, the threshold is optimised to obtain a denoised image. The discrete wavelets used are db4, coif3, bior2.8. Of these, the bior2.8 wavelet performs the best on images used in this research. Therefore, the bior2.8 wavelet is compared with the Dual Tree Complex Wavelet (DTCW). Based on the Peak Signal-toNoise-Ratio (PSNR), the denoised image using the DTCW performs better than the bior2.8 wavelet. Overal, wavelets are a powerful tool in image processing. The different wavelets each have their own characteristics. The choice of the optimal wavelet depends on the application and cannot be generalised.

When a second tumor arises in the contralateral breast in a patient with a previous or synchronous breast cancer, it is of clinical importance to determine if this tumor is a new unrelated tumor or a metastasis, i.e. clone, of the primary tumor. A new, unrelated tumor may be treated similarly as the first one since treatment was successful, while a distant metastasis demands a change of therapy and has a more adverse prognosis. In clinic, a second tumor is generally regarded as a new primary. If there is clinical suspicion that the second tumor may be a metastasis, clinico-pathological characteristics of the two tumors are used assess the clonality status. Clinico-pathological characteristics, however, are not reliable predictors to determine if a second tumor is a metastasis. Recent studies have investigated tumor clonality using techniques from molecular genetics. These models appear to perform well, but have several drawbacks.

In this thesis a more advanced classification model is being developed that can detect tumor clonality based on SNP array data. For this, two segmentation algorithms, ASCAT and OncoSNP, and two comparison methods, Log LR and adapted SI, have been incorporated. For each tumor, the segmentation algorithms construct a copy number profile based on the SNP array data. Given the copy number profiles, the comparison methods compute a p-value which reflects the probability that a pair is of clonal origin. Both comparison methods are permutation methods which test the null hypothesis of independence against the alternative hypothesis assuming clonality. The proposed model consists of a decision tree which assigns each pair to one of six categories depending on the significance of the four resulting p-values.

The model has been tested on 23 fresh frozen pairs by means of expert judgment. The results were promising: the four pairs which were unanimously labeled as clonal by the experts were also regarded as such by the model. No independent pairs were assigned as clonal by the model. Moreover, the decision tree showed to have a higher sensitivity than the clinical assessments as the latter only managed to detect two out of four clonal pairs. A discordance between the clinico-pathological judgments and decision tree results was found for three out of 18 pairs for which both assessments were available.

The model appears to be suitable in practice, but is not yet applicable as a stand-alone model. There were two ambiguous pairs which were labeled as independent by the model but for which the experts had varying opinions about the clonality status. Until the ambiguous pairs can be reliably categorized, it is advised to take into account both the model results and clinical assessments when determining tumor clonality. Finally, the performance of the model remains to be tested on FFPE pairs.

In this report, our goal is to find a way to get some information such as the mean out of high dimensional densities. If we want to calculate the mean we need to calculate integrals, which are difficult to do for high dimensional densities. We cannot use the analytical or classical (deterministic) numerical rules for high dimensional problems for which we want to calculate the mean. These methods take a lot of computational time. To solve this problem we introduced the Markov Chain Monte Carlo (MCMC) method, the method samples from the distribution, and with these samples, we can approximate the mean. Then we explain the theory behind these methods and how we can use it. Then we introduced one MCMC method, in particular, the Metropolis-Hastings Algorithm. We explain how this method works and the theory behind it. From this, we see that the method is very easy to implement and can be used to approximate the mean. Then we approximate the mean for some examples using this method.

The random graph is a mathematical model simulating common daily cases, such as ranking and social networks. Generally, the connection between different users in the network is established through preference, and this phenomenon leads to a power-law behaviour of the degree sequence of the random graph. Other than studying this feature, the thesis also focuses on the decay of degree sequence itself and the height of the random graph models. Researches are conducted on four kinds of random graph models, including two fully studied modes as benchmarks, the Preferential Attachment Model (PAM) and Random Recursive Tree (RRT), and another common model, the Bianconi-Barabási Model (BBM). Previous researches show that the degree sequence of PAM follows a power-law with coefficient -3, and the degree sequence of RRT has an exponential tail in contrast. In BBM, each node is assigned a fitness value from a certain distribution, once the node is generated, and researches show that the degree sequence of BBM follows a power-law with coefficient depending on the fitness distribution but close to the one of PAM. Then, the thesis aims to find a random graph model with fitness interpolating between RRT and PAM. Thus, a new Recursive Fitness Model (RFM) is developed by using a recursive formula to generate fitness values instead of the fixed distribution in BBM. By simulation we conjecture that the degree sequences of RFMs still follow a power-law with coefficient close to that of PAM. The simulation further shows an interesting phenomenon: a special linear height increment of a particular RFM, the Plus-1 model, appears, in contrast to other RFM models. Finally, the thesis invents a new Preferential Attachment Inverse Model (PAMinv) by using the expectation of degree sequence of PAM. Degree sequence of PAMinv shows a different behaviour from PAM and RRT, and it could be the tool to interpolate RRT from PAM.