This thesis focuses on optimising the mortgage acceptance process at Achmea Bank, based on improved probability of default (PD) modelling. The objective is to improve the existing Advanced Internal Ratings-Based (A-IRB) model, originally designed for calculating capital requirements, and better tailor it to the credit acceptance mechanism. This research identifies areas where adjustments are necessary, including feature selection optimisation, the use of a more suitable target variable, and the exploration of multivariate isotonic regression as a non-parametric model for better estimating the nonlinear interactions among features and the target variable.

Earlier on, the thesis introduces an adapted dataset to mimic the setting of a mortgage acceptance process. This is where only the first observation of each facility in the dataset is used, unlike the original A-IRB model, which worked with more than one observation per facility.

The research employs logistic regression to model the probability of default, with a focus on the feature selection process that involves both univariate and multivariate analysis. Comparison among different models with target defaulting in 12 months and 24 months reveals that changing the target variable and improving the feature selection process results in better model performance.

The second focus of this thesis is isotonic multivariate regression, giving greater flexibility by fitting a nonlinear relationship between the risk drivers and target, but with the constraint of monotonicity. The minimum Redundancy Maximum Relevance (mRMR) algorithm is used for the selection of features to reduce computational time.

Comparison across models reveals that the isotonic regression model, with greater recall and more accurate default detection, has a higher false-positive rate. Logistic regression with a 24-month target, on the other hand, strikes a better trade-off between precision and recall, leading to fewer false alarms and a lower number of flagged cases.

Overall, the thesis demonstrates that logistic regression and isotonic regression models both provide valuable information to Achmea Bank's mortgage acceptance. While logistic regression with a 24-month goal is an appropriate balance between recall and precision, multivariate isotonic regression can provide recall-improvement potential at the cost of precision. Subsequent studies should focus on reducing false positives within the isotonic regression model, exploring rejection inference procedures to adjust for potential sample bias, and examining other forms of classification procedures that can more effectively handle class imbalance.

This thesis examines rainfall event characteristics in the Netherlands over a 26-year period (1998–2023) using radar-derived precipitation data. Extreme precipitation is a major contributor to flooding, which impacts human life, infrastructure, and ecosystems. A life-cycle-based tracking approach is employed to analyse rainfall events in terms of spatial extent, duration, and intensity, using 5-minute precipitation data at a 2.4 km resolution.

The study reveals significant increases in the spatial extent and duration of rainfall events, with these trends persisting across most seasons. Intensity trends are more complex: while shorter-duration events show decreasing intensity, a closer look at smaller-scale shorter-duration higher-intensity events indicates an increase in intensity. No significant changes in the shape parameter of extreme event distributions are detected over the study period.

The analysis also finds strong correlations between event duration and area, and between intensity and both duration and dew point temperature, with these relationships evolving over time. This work provides valuable insights into the dynamics of rainfall events in the Netherlands and lays a foundation for future research on refining intensity metrics, distinguishing rainfall types, and improving flood risk assessments.

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.

Maskeringstechnieken kunnen nuttig zijn bij auteurherkenning om auteurherkenningsmethoden minder onderwerpafhankelijk te maken. Echter, als er teveel gemaskeerd wordt gaat er relevante informatie verloren waardoor deauteurherkenningsmethodejuist minder goed werkt. Hierin moeteenzorgvuldige afweging gemaakt worden. Het doel van dit onderzoek is om een aanbeveling te doen wat betreft het optimale aantal te maskeren woorden voor verschillende datasets. Ook worden er verschillende maskeringstechnieken bij verschil lende classificatiemethoden vergeleken. Er wordt gekeken naar maskering met behulp van een al gemene woordenlijst (COCA-woordenlijst) en maskering met behulp van een eigen frequentielijst per dataset. Hierbij worden twee classificatiemethoden gebruikt: support vector machines en logistis che regressie. Voor drie verschillende datasets: tweets, literaire teksten en brieven wordt gekeken welke manier van maskeren en welke classificatiemethode het beste werkt. Dit wordt gedaan door de prestaties te vergelijken. Ook wordt er daarnaast, vanuit de literatuur, gekeken naar de verschillen in onderwerprobuustheid en op basis daarvan wordt, gecombineerd met de informatie wat betreft de prestaties, een aanbeveling gedaan wat betreft het optimale aantal te maskeren woorden.

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.

Models governed by systems of ordinary differential equations (ODEs) often produce complex and unpredictable behaviors. To address this, we can use dimension reduction techniques, which simplify these models, allowing for the retention of specific behaviors while greatly decreasing the cost of numerical solutions and, in some cases, enabling analytical derivations of sufficient conditions for the existence of nonzero fixed points of the model’s ODEs. This thesis reviews the state-of-the-art reduction theories and extends the established proofs by Wu et al., by providing necessary assumptions, lemmas, and a novel proof of an important proposition used in their work. We additionally verify and confirm Wu et al.’s findings and predictions for a cooperative version of the Cowan-Wilson model, which describes a population of neurons’ firing activity. We derived a one-dimensional reduction, inspired by Laurence et al., for a generalized Cowan-Wilson model, which we introduced in this thesis. Unlike the original, this generalized model can produce oscillatory behavior without external stimulus. A valuable finding is that the method of reduction does not depend on the specific form of the Cowan-Wilson function, allowing it to be applied to a broader class of nonlinear dynamics. Our reduction is able to predict system behavior effectively, given the network yields a unique reduction. However, the reduction parameter was not unique in approximately half of the networks studied, which saw a 66% increase in average error, suggesting these networks are inherently multi-dimensional. This opens the door for future research into the existence of a multi-dimensional reduction framework that could mitigate this discrepancy.

The domination game is a two-player game played on a graph. The players have two roles, Dominator and Staller. Dominator wants to finish the game as soon as possible, while Staller tries to postpone the end of the game for as long as possible. The two players alternate turns and each turn they pick a vertex, and color the vertex itself and its neighbors. The players should always color at least one previously uncolored vertex. When all vertices are colored the game is finished. Calculating bounds on the total number of moves under optimal play for all graphs with n vertices is a hard open problem. In this research, the domination game on a specific family of graphs is analyzed, the king graphs. These are modeled after how kings move on a chessboard and provide an interesting graph with a lot of structure, but nontrivial optimal winning strategies.
For small boards, we found the optimal move counts with computer search for up to 9 x 9 boards. For general n, lower and upper bounds on the number of moves were found. For an n x n king graph, the number of moves under optimal play, gamma_g(G_n), satisfies
2/10n^2 - O(n) <= gamma_g(G_n) <= 2/9 n^2 + O(n).

In this thesis models of the real numbers will be constructed using a set-theoretic approach. The mathematical foundation we will assume is the first-order theory with equality known as Zermelo-Fraenkel set theory ZF. From the axioms of ZF, the necessary notions for the construction will be introduced as definitional extensions to the language of ZF. We will show how functions can be defined as relations, which in turn are defined as subsets of Cartesian products. Using this preliminary work we will first construct the natural numbers. We will see how our intuition of using the natural numbers for counting gives rise to the Dedekind-Peano axioms PA. Defining the natural numbers ℕ as the smallest set satisfying the requirement in the statement of the Axiom of Infinity, we find that they are indeed a model of PA. By using our knowledge of what properties operations relating to the natural numbers should satisfy, we induce algebraic structures, the most basal of which being the ordered semiring. We will see how the successor function and induction are intimately tied to the predecessor function and well-ordering. This structure will be used to prove the uniqueness of ℕ; it is the unique well-ordered commutative semiring in which every nonzero element has a predecessor. Next the integers ℤ are defined as equivalence classes of pairs of the natural numbers. We observe how the integers have additive inverses, which allows for a simple characterisation of them as the unique ordered commutative ring whose positive elements are well-ordered. Using the integers, we will define the rational numbers ℚ as fractions represented by equivalence classes of pairs of integers. An important result will be that the rational numbers are the smallest ordered field, in that every ordered field has a subfield isomorphic to ℚ. After this, we will turn to constructing the real numbers. Three models will be constructed: the Dedekind reals ℝ_D, the Cantor reals ℝ_C and the Schanuel reals ℝ_S. The first two constructions both induce a fundamental property of the real numbers: completeness. The third construction is a lesser known construction. We will compare the three constructions and prove they are indeed equivalent. That is, we will prove that any ordered field is Dedekind-complete if and only if it is Archimedean and Cauchy-complete. This property of the real numbers will be used to prove that they are the unique ordered Dedekind-complete field. Lastly some directions for further research will be discussed.

In the study of evolutionary biology, there exists a method called the “cherry picking algorithm” that produces the instructions needed to create a network that shows how different species are related.
This report explores what happens when the algorithm starts with a wrong choice, or a “suboptimal cherry” for the first step of the algorithm, and how this affects the accuracy of the algorithm. Imagine you are trying to build a family tree for different species, but you start with a mistake. This research looks at how such initial mistakes can impact the accuracy of the entire family tree. We conducted this study
using simulations of the algorithm that deliberately make an initial mistake, and afterwards continue as the algorithm would normally. The study found that starting with a wrong step in the algorithm usually makes the performance of the algorithm worse. Specifically, it lead to an average optimal performance decrease of 34,8% for networks relating a smaller number of species, and 11.3% for networks relating a larger number of species. Interestingly, the larger the number of species we are attempting to relate in the network produced by our algorithm, the less severe the impact of the initial mistake.
We concluded that making an initial mistake negatively effects the average performance of the algorithm, and the extent of the effect varies with the number of species we are trying to relate in our network.

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.

Quantum algorithms have shown much potential in solving complex problems more efficiently than our current classical algorithms, particularly in search problems with a large and complex search space. When leveraging the principles of superposition and interference, quantum walk based algorithms can lead to a faster convergence on a target state. The thesis begins with a theoretical framework for the study of classical and quantum walks on graphs. The usage in search algorithms is discussed together with the effect of a modified Hamiltonian that introduces a bias toward the target state. This is analyzed using perturbation theory showing a decrease of the ground state energy. Ultimately, this thesis focuses on the optimization of quantum random walk search algorithms using resetting techniques. Simulations of quantum random walks on simple graph structures, such as path and cycle graphs are used to provide concrete examples. Results show that resetting not only improves the probability of finding the target state, but also has a faster convergence. Furthermore, decoherence effects are shown to be reduced when using resetting techniques.

This thesis aims to enhance the understanding of quasicrystal formation by integrating global geometry optimization and atomic simulations using the Oscillating Pair Potential (OPP). Quasicrystals, with non-periodic long-range order and forbidden rotational symmetries, have captivated scientists since their discovery. This research combines the use of OPP and geometry optimization techniques to investigate the energetically stable structures of quasicrystalline materials.

The core of the thesis revolves around the implementation of the aforementioned computational methods using the HOOMD-blue Python package. Simulations explore clusters with varying numbers of atoms under different potentials, comparing the stable structures formed. The results provide valuable insights into the formation of icosahedral quasicrystals.

This study contributes to the understanding of quasicrystal formation, unlocking new possibilities for material design and technological advancements.

This Bachelor thesis provides an analysis of Runge-Kutta methods using Butcher tableaus. Runge-Kutta method are numerical methods used for approximating initial value problems. A Runge-Kutta method can be classified as either an explicit or an implicit method. A special kind of implicit methods are diagonally implicit methods. The type of method can be recognised by the Butcher tableau.

Using the entries of the Butcher tableau, one can compute the amplification factor of a Runge-Kutta method. The amplification factor can then be used to compute the order of the local truncation error and the stability region. Examples of these computations are given for seven methods. Furthermore, this thesis provides an algorithm to perform time steps for each of the three types of Runge-Kutta methods. Finally, in order to analyse the global truncation error of the seven methods, the algorithm to perform time steps is used with different step sizes.

The Leslie model considers age-specific birth and survival rates to describe how a population size and age distribution changes over time. This thesis investigates the long-term dynamics of the Leslie model for population growth, utilizing mathematical theorems such as Perron-Frobenius, Doeblin’s theorem, and Branching processes.

The Perron-Frobenius theorem guarantees the existence of a dominant eigenvalue. This dominant eigenvalue and its corresponding eigenvector represent the long-term behaviour of a population; The dominant eigenvalue indicates the long-term population growth, and the corresponding eigenvector indicates the long-term age distribution. Furthermore, the Perron-Frobenius theorem implies that a population asymptotically reaches a stable age distribution that is independent of its initial age structure. Once this stable age distribution is reached, the population continues to grow exponentially, exhibiting Malthusian behaviour. Doeblin’s theorem, although not directly applicable to the complete Leslie model, provides valuable insights into the long-term behaviour of Markov chains. As Doeblin‘s theorem can not be applied to the complete Leslie model, the reproduction process of the Leslie model is formulated as a Branching process. Introducing the Leslie model as a Branching process allows for the consideration of demographic stochasticity. Simulations reveal that for larger populations, the Branching process closely mirrors the Leslie model, while disparities become more pronounced in smaller populations. These results illustrate the impact of probabilistic factors in population dynamics, as well as the strength of the Leslie model for larger populations.

Thanks to increasing computational resources, the grid sizes of global climate and weather prediction models are decreasing to scales at which subgrid processes, especially the evolution of clouds, can become spatially too variable for traditional deterministic parameterization approaches. We propose a stochastic parameterization in which clouds are represented by a Markov-Chain (MC) lattice model. The state of each lattice site is characterized by cloud top height (CTH) and cloud optical depth (COD). Based on these two parameters, clouds can be categorized into different cloud types (cumulus, stratus, cirrus, deep convection, etc.). In contrast to previous work, where the transition probabilities between different cloud types were calculated, we determine the transition probability densities in the continuous CTH-COD domain. We moreover explore the influence of neighbouring lattice sites on these transition densities to research whether it is possible to simulate cloud organization structures, without the use of extra external large scale variables or forcings. We estimate the parameters of the distributions using GOES-16 satellite images. Using these estimates, we perform simulations of the continuous state space Markov Chain over the lattice. This results in 2D spatial cloud evolutions, which spatial organization we compare to the original images.
The simulations did not show any spatial structures as were observed in the original images. All the simulations resulted in a granulated image of different cloud types after only a couple of hours.
A simulation from the MC model describing transitions from a finite number of cloud types converges to the observed occurrence frequencies, which is property of a MC. When describing the MC in a continuous state space the simulations still converged to the observed cloud state frequencies. However, introducing spatial dependence this property was not a given anymore.
Since we were unable to reproduce the organizational structure from previous work using the GOES-16 satellite images, we cannot conclude whether it is possible to simulate cloud organization using the MC lattice model. A better understanding of these type of models was achieved. Neighbouring sides could diverge too fast in the performed models. We believe adding a stronger coupling between the cloud state evolution of neighbouring cells can improve the spatial models. Therefore one method to simulate correlated variables was tried to implement on the CTH evolution. This showed promising early results for further research. Also a better correction for horizontal advection on the original images could be performed to observe a specific cloud over time, which could improve the observed data quality.

In this thesis, we will be studying Lie groups and their connection to certain orthogonal polynomials. We will look into the classical Krawtchouk, Meixner and Laguerre polynomials, and the multivariate Krawtchouk and Meixner polynomials as defined by Iliev. Using representations of the Lie groups SU(2) and SU(1,1), it will be shown that the three classical polynomials can be described as matrix coefficients of the representations. Using this connection of the polynomials to Lie groups, we derive various properties of the polynomials from the unitarity of the representation and the associated Lie algebra representation. Next, the representations are generalised to higher dimensional spaces. Doing so, a new connection is shown between the Lie group SU(d+1) and the multivariate Krawtchouk polynomials, extending the known theory for the univariate polynomials. Another new result that will be established is the connection between the multivariate Meixner polynomials and Lie theory. This will be done by defining a representation of SU(1,d) in the Bergman space of the d-dimensional unit ball. Similar as for the univariate polynomials, we will derive the orthogonality, recurrence relations and difference equations from the associated Lie theory.

De Wilson- en Racahpolynomen zijn hypergeometrische orthogonale polynomen die helemaal bovenaan staan in het Askey-schema. Deze polynomen zijn de meest algemene hypergeometrische orthogonale polynomen in één variabele en generaliseren de andere hypergeometrische orthogonale polynomen in het Askey-schema.

In deze scriptie wordt ingegaan op twee specifieke eigenschappen van de Wilson- en Racahpolynomen: de drieterms recurrente betrekking en de orthogonaliteitsrelatie. Deze eigenschappen worden met analytische en algebraïsche methoden bestudeerd.

Bij de analytische methode wordt eerst algemene theorie van hypergeometrische functies en orthogonale polynomen bestudeerd. Er worden identiteiten, transformaties en aaneengesloten relaties voor hypergeometrische functies afgeleid. Hiermee kan de drieterms recurrente betrekking van de Wilson- en Racahpolynomen worden afgeleid. Met behulp van de residuenstelling van Cauchy en de theorie van hypergeometrische functies kan de orthogonaliteitsrelatie van beide polynomen worden verkregen.

Bij de algebraïsche methode wordt de Racah-Wilsonalgebra bestudeerd. De Racah-Wilsonalgebra voldoet aan een zogenaamde laddereigenschap waarmee een keten van eigenvectoren kan worden geconstrueerd. Hiermee is het mogelijk om een eindig dimensionale irreducibele representatie te krijgen. Door een inproduct te definiëren op de bases van eigenvectoren van de generatoren van Racah-Wilsonalgebra, kan met behulp van overlapcoëfficiënten een drieterms recurrente betrekking worden afgeleid. Door vervolgens enkele transformaties toe te passen, kan worden aangetoond dat deze drieterms recurrente betrekking overeenkomt met de drieterms recurrente betrekking van de Racahpolynomen. Ten slotte laten we met dit gekozen inproduct zien dat de Racahpolynomen orthogonale polynomen zijn.

In this thesis a novel model is proposed to solve the Robust Tail Assignment problem. The Robust Tail Assignment problem aims to assign aircraft to flights, while minimize expected costs of operating a flight schedule, including expected delay costs. This problem is difficult, because delays can propagate between successive flights in the schedule, creating dependencies between flights assigned to the same aircraft.

Using probability distributions of delay for every individual flight, as well as expected costs associated with delaying flights, the expected delay costs of a full flight schedule can be estimated. The workings of a simulator are described, which can be used to evaluate the total expected costs of solution schedules for the Robust Tail Assignment problem.

To be able to incorporate expected delay costs in a mathematical model, the construction of a multi-commodity flow network is described, which uses departure and arrival states for flight rotations, corresponding to discrete amounts of delay. The amount of flow through edges of this network represents the probability of these states transitioning into other states. By activating and deactivating edges, based on the assignment of aircraft to rotations, this network can be used in a model to approximate the total expected delay costs of a model solution.

The proposed robust flow model uses such a state network in a MIP model, that can be solved using an iterative solver to find good solutions to the Robust Tail Assignment problem. Delay costs are imposed on edges in the network, to quantify the expected delay costs. In the model, the network size is reduced by only considering connections between rotations that have high probabilities of propagating delay. This reduces the accuracy of the model, but shortens the run-time of the optimization process significantly.

Several experiments are done to test the run-time and performance of the robust flow model. The model proved hard to solve to optimality, but is able to find good solutions, if the model parameters are well tuned. Recommendations are given for using the model, as well as future research directions.

In this study, we investigate the usage of generative adversarial networks for modelling a collection of sounds. The proposed method incites an interpretation of musical sound synthesis based on audio collections rather than synthesizer component controls. This promises the generation of arbitrarily complex sounds without the restrictions of traditional synthesizer components. Furthermore,
the method promises to introduce non-linear interpolations within abritrarily varied collections of sounds. These two elements motivate a new approach in creating musical instruments. Here, we introduce a proof of principle method with qualifications and quantifactions of the results. First, we cover the imagelike audio signal representation and neural network architectures that compose a trainable system capable of producing audio signals. Despite some artifacts, the trained system is able to produce structural similarities in the spectral information compared to the training data set. Furthermore, we introduce a metric to quantitatively compare signal characteristics between two sets of signals. The difference between characteritics appears to decline throughout the training of the system.