Over the last decade, an increasing amount of data has become available for data analysts to understand. Datasets containing books, images, networks, or other types of data have been studied. A recent group of methods proposes to analyze samples in datasets based on a description of their shape. This group of methods is often referred to as Topological Data Analysis (TDA). In this thesis, an extension to the most commonly used TDA method, called Persistent Homology (PH), is proposed. PH only describes topological features, while this extension additionally allows for the description of geometric properties. The new information is obtained via the persistent Laplacian, a recently proposed operator that encodes the topological information of persistent homology in its kernel and geometric information in its non-zero spectrum. The persistent Laplacian contains a lot of information and extracting the relevant parts has not yet been standardized. In this thesis, a new operator, the persistent multiplicity operator, is proposed. The new operator summarizes the information of the persistent Laplacian such that it can easily be extracted and used to extend PH. This allows the many previously studied methods based on persistent homology to additionally describe geometric properties, as opposed to only topological features. For the multiplicity operator, the trace is analyzed and the features captured by it are discussed. Besides analyzing the sum of the eigenvalues, it is argued that individual eigenvalues could contain more information. However, these are deemed hard to understand. Therefore, an adjusted multiplicity operator is proposed that contains separately interpretable eigenvalues. Finally, the operators are used to classify handwritten digits from the MNIST dataset and to make statistical tests that can detect different generation processes of artificially made cross sections of crystalline structures.

Meta-analysis is a powerful method to combine the treatment effects of multiple independent studies answering a common research question. To determine a causal treatment effect in a meta-analysis it is important to look at the characteristics of the participants, like age or diseases, also known as covariates. In the case of experimental studies, participants are randomly allocated to treatment groups. As a result, the covariate is in expectation equally distributed between the treatment groups. In observational studies, no randomisation has occurred and thus, the covariate imbalance between treatment groups may be more profound. This makes it difficult to determine a causal effect of the treatment, since the covariate may affect the outcome. Therefore, it is important to balance these covariates between the treatment, especially for observational studies. The condition when this balance is present is called combinability.

In this thesis, the covariate imbalance between treatment groups is assessed using five multi-sample test statistics. These assessment methods are based on the comparison of the empirical cumulative distribution functions of the covariate between the meta-arms, which are the collections of the similar treatment groups. Then, a permutation test is used to determine whether the covariate imbalance is significant, as assessed by the multi-sample test statistics. This is done by computing a distribution of the multi-sample test statistics under the null hypothesis that there is no covariate imbalance between the meta-arms. If the observed multi-sample test statistics are significantly large, then combinability is not satisfied.

Subsequently, a balancing procedure is proposed to minimise the covariate imbalance if combinability is not satisfied. This balancing procedure works by discarding a selection of treatment groups from the meta-analysis, such that the multi-sample test statistics indicate that the covariate imbalance is no longer significant. The result is a more combinable set of treatment groups that can be used for the purposes of meta-analysis. Finally, a simulation study of the balancing procedure is done for three and four treatment groups. In these simulations, the treatment groups are simulated with different underlying distributions, such that in theory the covariate imbalance is significant. These simulations seem to indicate that the more treatment groups there are, the more groups need to be discarded before the covariate imbalance is no longer significant. This is explained by the fact that the initial covariate imbalance is larger if there are more treatment groups. On average, in the case of three treatment groups, nearly a fifth of groups needs to discarded, while in in the case of four treatment groups, roughly a third of groups needs to be discarded. Finally the use of five multi-sample test statistics in the balancing procedure result in a sizeable overlap of groups that are discarded.

The sizes of three-dimensional particles at a microscopic level reveal properties at a macroscopic level for many applications in materials science, but can be difficult to measure. This thesis builds on existing methods that estimate the size distribution of such particles of the same three-dimensional shape, using information obtained from their profiles in two-dimensional cross-sections. This is a well-known stereological problem. An existing method is explained, which yields a maximum likelihood estimator based on the two-dimensional sizes of observed profiles. This method is expanded, resulting in a new maximum likelihood estimator, which is based on the paired two-dimensional sizes and shapes of the observed profiles. The performance of both the existing and the new estimators is analysed in simulations, as well as in an application to real data obtained from a steel microstructure. By comparing the performances of both estimators, this thesis aims to answer whether or not the additional shape-related information improves the resulting estimator. Its conclusions are that the new estimator performs better on average than the existing estimator, but not in general. Moreover, the new estimator only performs better in applications when the observed information is accurate, which is not always the case.

This thesis presents localized methods for traffic prediction and analysis. The prediction method presents an extension of a state-of-the-art Graph Neural Network inspired by traffic flow characteristics on a local level. This inspiration from traffic flow characteristics consists of two parts. The first intuition is that state at some location and at time T in the future will be not influenced by information which is further than traveling time T away. The second intuition is that traffic information traveling with or against the stream of traffic behaves differently. The developed model leverages these intuitions to increase model prediction performance. Further, a modification is made which allows the model to be applied at an arbitrary location in a network, at the cost of performance. Alongside these model extensions, a novel method of visualizing a local traffic state is presented through constructing a novel traveltime diagram. This diagram can be used as a visual tool for analyzing traffic locally. Further, the traveltime diagram is designed to be summarized using Topological Data Analysis to a quantity called the Travel Lifetime which can represent traffic states ranging from extremely calm to imminent congestion to a congested state in a single number. The newly proposed Travel Lifetime is tested as an input to a Neural Network model for predicting traffic speed showing that its use as an input can improve model performance.

The Dutch mortgage market is highly competitive, with financial institutions constantly seeking ways to optimise their pricing strategies and market share. However, there is limited publicly available academic research on the prediction of company-level mortgage application volumes and the calculation of price elasticity, due in large part to the confidential nature of company-specific data. The main research question of this thesis is: which forecasting model offers the highest prediction accuracy for predicting company-level mortgage application volumes one week ahead? An important second goal was analysing the responsiveness of company-level market share to relative pricing changes compared to competitors across different categories of mortgages.

To address this, a novel two-part forecasting framework is introduced that separates the total market volume of mortgage applications from company-specific market share predictions. A newly proposed hybrid model combining predictions of ARIMAX, XGBoost, and Random Forest was found to outperform existing methods and the benchmark model, achieving the highest prediction accuracy with a Root Mean Squared Error (RMSE) of 1.9E+07 for total mortgage volume predictions. Market share predictions utilized Double Machine Learning (DML), which yielded an MSE of 1.9E-03 and effectively quantified price elasticity across grouped mortgage loans with similar characteristics. Results showed significant elasticity differences: non-ported mortgages and longer fixed-rate period loans were highly elastic, while ported loans remained inelastic. This suggests that the relative price position plays a crucial role in predicting market share, though external factors such as announced future interest rate
increases of competitors also exert substantial influence.

For company-level volume predictions, a weighted average approach using market share and total mortgage application volume predictions was identified as the optimal model with a Ratio squared, defined as the squared division of the RMSE by the mean of the company volumes, value of 0.59 across all segments and a value of 0.05 for the aggregated company volume. This highlights the large variability for different types of mortgages. The model significantly outperformed the benchmark values and an ARIMAX model directly trained to predict the company, further underlining the importance of splitting the forecasting task into a two-part framework that integrates market-wide and company-specific dynamics.

The results confirm the importance of using multiple models to separate volume and market share predictions, offering financial institutions enhanced strategic decision-making capabilities. In conclusion, this thesis makes a novel contribution to the mortgage industry by providing a company-specific forecasting model and insights into price elasticity across mortgage types.