Authorship attribution is the task of determining the unknown author of a text. In forensic authorship attribution, the likelihood that a suspect has written a specific text of unknown origin is computed based on reference texts from both the suspect and a background population. The current method used at the Netherlands Forensic Institute contains a manual and a computational part. In this thesis, we attempted to improve the computational part of this process. We study this problem from three directions.

Firstly, the performance of state-of-the-art computational authorship attribution methods was assessed on Dutch, forensically relevant corpora. The compared methods were support vector machines combined with masking, using either word or character n-grams as features, BERT-based models using a mean pooling strategy to handle long texts and the baseline, which consists of a logistic regression model with the 100 most frequent Dutch words as features. We notice similar performance differences between state-of-the-art methods as in the literature. The best-performing method was a support vector machine without masking using character n-grams as features. In comparison, both the baseline and BERT-based models perform worse on our corpora.

Secondly, a score-based likelihood ratio system was created to modify the computational authorship attribution methods for usage in forensics. This method is based on kernel density estimators and uses cross-calibration to handle the small number of training and calibration texts of the suspect. For most methods, the performance is in line with the previous performances outside the likelihood ratio system, except for the BERT-based methods, which significantly underperform when part of a likelihood ratio system. This is likely caused by the combination of cross-calibration and the randomness in finetuning BERT models.

Additionally, authorship attribution methods should be topic-robust, such that their attribution is not biased by the topic of a text. We introduced two new metrics to measure the topic-robustness of authorship attribution methods, ‘topic impact’ and ‘conversation impact’. These metrics can only be used on specific types of corpora, the topic impact can be computed on topic-controlled corpora and the conversation impact can be computed on conversational corpora. To study whether these metrics both measured the topic-robustness of authorship attribution methods for their respective corpus type, we computed the correlation between the results of the metrics for varying authorship attribution methods.
We found a correlation of 0.68. As a result, we cannot conclude that the conversation impact is a perfect metric to measure the topic-robustness of methods using conversational corpora, but it does give a good indication of large differences between methods.

Using this new metric, we found that our best-performing methods suffered from a high conversation impact and, as a result, might be more likely to have a low topic-robustness. If more of the infrequent words were masked, the conversation impact decreased, but so did the performance. A trade-off between high performance and high topic-robustness must be made when a model is chosen for real forensic case work. The conversation impact metric we proposed can help quantify these effects on forensically relevant corpora and therefore assist in making better choices.

The introduction of data-based modeling in football (soccer) in the last decade has led to the creation of models that describe player performance through key performance indicators (KPIs). However, relying solely on historical and current KPI values is insufficient for scouting departments, as predicting future values could significantly enhance transfer decision-making. This research aimed to identify the optimal model for forecasting the development of player performance KPIs over the next year, focusing on explainability, uncertainty quantification, and predictive performance.
To achieve this, we implemented linear models, tree-based models, and time- series-based kNN models to forecast two specific KPIs one year in the future: SciSkill, which measures the general quality of a player, and Estimated Transfer Value, representing the player’s monetary value. Tree-based models showed the best predictive performance. The random forest in particular emerged as the best due to its explainable predictions, uncertainty quantification method based on bagging, and good predictive performance. In the Sciskill case study, the random forest model achieved low loss values, especially for young players. For the Estimated Transfer Value, the random forest model demonstrated the best predictive performance on the general set of players, and specifically on the subset of players valued at over €10 million.
Our findings suggest that tree-based models, particularly the random forest, are well-suited for predicting the future development of football player perfor- mance KPIs. Although it is important to monitor the predictive performance using the most recent data, the insights and the resulting models of this research can enhance scouting decisions via both data-informed and data-based decision- making. Finally, this research paves the way to study the influence of time series information or contextual information on player performance metrics.

The Kalman filter is a recursive algorithm that estimates the state of a dynamic system subject to measurement and model noise. If all noise terms affecting the system are white Gaussian noise with known mean and variance, and all noise terms are independent of each other, then the Kalman filter is the optimal estimator for the state variable. When measurements are collected from multiple sources, the covariance between these sources should be known or the sources should be independent to ensure that the estimate made by the Kalman filter is optimal. When the covariance between dependent measurement sources is not known, various methods exist which provide a solution to this problem. This thesis discusses two methods: the H∞ filter and covariance intersection...

In deze scriptie wordt een methode ontwikkeld om de viscositeit en zelfdiffusiviteit van moleculen te schatten door middel van moleculaire simulaties. De simulaties worden uitgevoerd in kubusvormige volumes met verschillende ribbelengtes $L$, waarbij de computationele kosten en de nauwkeurigheid van de metingen afhankelijk zijn van de kubusgrootte. Metingen in kleinere kubussen vereisen minder rekenkracht maar leveren onnauwkeurige resultaten op, terwijl metingen in grotere kubussen preciezere schattingen geven tegen hogere computationele kosten.

De relatie tussen de werkelijke zelfdiffusiviteit $D^\infty_{\text{self}}$ en de gemeten zelfdiffusiviteit $D^{\text{MS}}_{\text{self}}$ wordt beschreven met de formule:

$$D^\infty_{\text{self}} = D^{\text{MS}}_{\text{self}} + \frac{\xi k_B T}{6\pi \eta L}$$

waarbij $L$ de ribbelengte is en $\eta$ de viscositeit. Dit leidt tot een lineair regressiemodel waarbij $D^{\text{MS}}_{\text{self}}$ de afhankelijke variabele is en $\frac{1}{L}$ de onafhankelijke variabele is. De schatting van de intercept en de helling moet nauwkeurig worden bepaald.

Er is gebruik gemaakt van gewogen regressie om heteroscedasticiteit in de fouten van de simulaties aan te pakken. Metingen in grotere kubussen tellen zwaarder mee voor de regressie omdat ze nauwkeuriger zijn, terwijl metingen in kleinere kubussen minder meetellen. De variantiefunctie $\sigma(x)$ en de kostenfunctie $f(x)$ spelen een cruciale rol bij het bepalen van de optimale ribbelengtes en gewichten. In het bijzonder is aangetoond dat voor hogere ordes van de variantiefunctie (bijv. $\sigma(x) = x^3$) de voorkeur wordt gegeven aan grotere ribbelengtes.

Daarnaast zijn de simultane schattingen van zowel de viscositeit als de zelfdiffusiviteit verbeterd door gebruik te maken van de covariantiematrix van de regressie. De afhankelijkheid tussen de intercept en helling wordt gebruikt voor een verkleining van het betrouwbaarheidsgebied met 85\% ten opzichte van een traditionele benadering zonder covariantie. Dit is geïllustreerd met behulp van betrouwbaarheidsgebieden, waaruit bleek dat het meenemen van de covariantie bijdraagt aan het reduceren van de onzekerheid in de schattingen.

The whitening transformation transforms a random matrix into a whitened matrix with expectation 0 and covariance matrix I. By removing the first and second order statistical structures, higher order structures can be looked at for better classification. This is why Stage Gate 11 B.V. has employed whitening in the preprocessing of their hyperspectral data. The aim of this work is to gain insight into the whitening transformation and how it influences hyperspectral data.
To gain this insight, synthetic data was created and used to make synthetic scans. The signal-to-noise ratio of a target spectrum was calculated, and Monte Carlo simulations were used to reveal hidden patterns in the data. In case of a high contrast scenario, multi-area whitening was employed and the cosine similarity between the target spectrum and its signature was determined. It was observed that the shape and intensity of the whitened target spectrum differs, depending on if pixels were used as observations or wavelengths. However, both are subject to the ‘bleeding’ effect. Further, it was found that if the number of pixels in the scan is greater than the number of spectral bands (548), then the signal-to-noise ratio becomes better as the number of whitened pixels in the scan increases. In case of a high contrast scenario, multi-area whitening guarantees the uniformity of the spectra, resulting in a higher
cosine similarity between the target spectrum and its signature. But as multi-area whitening uses a smaller
number of pixels in the scan, it cannot be concluded if multi-area whitening is better than global whitening, as it is not known how the increase in cosine similarity and the decrease in signal-to-noise ratio relate to the classification process. Finally, it is concluded that when working with real and unknown data, using pixels as
observations is much more feasible.

Graph data is widely used in various applications, driving the rapid development of graph-based machine learning methods. However, traditional algorithms tailored for graphs have constraints in capturing intricate node relationships and higher-order patterns. Recent insights from prior research have shed light on comparing different graph neural network architectures. This work introduced higher-order neural networks capable of grasping complex graph patterns. Nevertheless, these models encounter scalability issues that hinder their application to real-world datasets. This thesis builds on this foundation by theoretically assessing the various neural architectures proposed in those studies. Moreover, we present novel models aiming to find a middle ground between capturing higher-order patterns and maintaining scalability. Our objective is to enhance the modeling capabilities of graph-based algorithms and address existing limitations. Additionally, we implemented our models on benchmark datasets to gauge their performance. The outcomes confirm that our models achieve notably improved generalization compared to conventional graph neural networks. Furthermore, our models exhibit substantial scalability enhancements when contrasted with other higher-order graph neural networks. This research contributes to graph machine learning by offering more efficient and scalable methods for capturing higher-order patterns in graph data.

Looking for making an investment, one objective could be to find a portfolio where the Sharpe ratio for in the future, known as the out-of-sample Sharpe ratio, is maximized. Since future data is not avail-able, the Sharpe ratio needs to be predicted using historical data, the in-sample data. This is often done using the Sharpe Ratio Information Criterion, which determines the bias for the in-sample Sharpe ratio to es-timate the out-of-sample Sharpe ratio. However, this approach assumes that the covariance matrix is known. In portfolio management, the covariance matrix is typically unknown and can only be estimated. This project will use the bootstrap method to estimate the out-of-sample Sharpe ratio using the estimated co-variance matrix and analogous methods used for the Akaike Information Criterion. By eliminating the assumption of a known covariance matrix, this method becomes more applicable. Simulations will also be done with a known covariance matrix, demonstrating that the bootstrap method is an effective approach for estimating the out-of-sample Sharpe ratio. We then look at some extensions for the bootstrap method and finally we will apply the bootstrap method to stocks in the Dutch and American stock markets, showing that the in-sample Sharpe ratio is often overly optimistic compared to the out-of-sample Sharpe ratio. We reached our goal that we found an effective way to estimate the out-of-sample Sharpe ratio without the assumption that the covariance matrix is known, resulting this method becomes much more suitable for predicting the Sharpe ratio in the future.
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There is a raising demand for player statistics in the world of football. With the developments over the last years in wearable sensors, Human Activity Recognition (HAR) based on wearable IMU sensors can be used to tackle this problem. This thesis builds upon an earlier research done for this topic, where an end-to-end pipeline based on deep learning was created that is able to be trained and used for football activity recognition. The goal is to test and improve said pipeline. This was done by adding change of directions (COD’s) to the classifiable activities. Furthermore, run velocities were build as a spectrum with several categories depending on the speed. A combination of convolutional and recurrent layers resulted in test accuracies up to 88.9%.
Afterwards, the pipeline was used to evaluate larger datasets containing football drill and a football physiotherapy training. For this a sliding window evaluation procedure was proposed. These evaluations gave promising results. Many actions and football related activities could be recognized, however many smaller, shorter actions were missed. This can be seen as lack in trainingdata. In this data, little activities with the ball were present. Hence the deep learning models could not be trained accordingly. Later, it was researched
if additional training of activities with ball increased the evaluation. This was indeed confirmed, since the evaluations showed more detailed and realistic results. Including even more additional trainingdata, could result in the pipeline performing reliably in real-life football scenario’s.

In this report a new approach to (forensic) handwriting analysis is presented; score-based likelihood ratio (SLR) systems are employed and their quality of performance is studied. These systems compare elements of handwriting based on their characteristics and give an insight into the degree of uncertainty of the statement that two writings have the same writer. They can be used in forensic and fraud investigations.

As a response to the dry summer of 2018, Witteveen+Bos developed a model for water demand prediction to improve insight into water demands. Validation by water board "Hunze en Aas" has revealed the predictive power of the irrigation model to be very limited. For this thesis project, we developed a methodology for the detection of irrigation of crop parcels based on the radar vegetation index (RVI) derived from remote SAR images. This methodology can be used to improve the existing irrigation model.

To achieve this, we developed a novel model to describe the evolution of a vegetation index (such as RVI) during the growth season. Unlike existing models, the model presented in this thesis includes the effect of precipitation deficit, both as a temporary inhibitor of a vegetation index, and as a long-term influence on the crop growth. The model is non-linear in many of its model parameters. Therefore, heuristic calibration methods are unavoidable. We show that the standard calibration methods non-linear least squares and differential evolution are outperformed by a hybrid of both methods that we specifically designed for this application.

After calibrating the model to time series of 1167 potato parcels in the north-east of the Netherlands, we investigate different ways to cluster the model parameters. We propose explanations for three important clusterings through their RVI time series (speculative) environmental factors. Comparison with information on irrigated parcels for the years 2018-2020 reveals a statistically significant correlation between some of the clusters and irrigation. However, the variation in irrigation rate never exceeded a factor two. Therefore, no accurate classifier can be built based on these clusters.

We recommend two important ways to improve the current implementation. Firstly, the baseline RVI is consistently overestimated, resulting in mostly negative normalized RVI. Because of this, the model cannot properly describe precipitation deficit-driven fluctuations in the RVI. These fluctuations are an important part of system behaviour, so improving the estimation of the baseline RVI should be the first priority for future research.

Secondly, the exact irrigation dates of a set of parcels will be very useful. Comparing these dates to the corresponding RVI time series will make it possible to uncover features of the RVI evolution that are indicators of irrigation. The model parameterization can then be tuned to optimize sensitivity to these features.

The problem of calibrating time-inhomogeneous exponential Lévy models with finite jump activity based on market prices of plain vanilla options is studied. Belomestny and Reiß introduced an estimation procedure for calibration in the homogeneous case with one maturity. The open-ended question that will be addressed is if we can extend this model to use all listed plain vanilla options with multiple maturities. This opens a way to use all available data to create a time-inhomogeneous model with time-dependent Lévy triplets between subsequent intervals based on the maturities.

We establish via an adapted Lévy-Khintchine representation an estimation procedure on the explicit inversion of the option price formulas in the spectral domain with a cut-off scheme for the regularisation of high frequencies. The estimation procedure will be shown to be well-defined and the parameters to be normally distributed asymptotically.

Practical implications imply that using the asymptotic variances of the normal distributions leads to insufficient confidence intervals for non-asymptotical cases. We, therefore, construct confidence intervals using an approximated finite sample variance.

Monte Carlo simulations are implemented in the computational software R to evaluate the stability and accuracy of the estimation procedure. Furthermore, the finite sample confidence intervals are assessed with coverage probabilities.

In the end, the estimation procedure is evaluated by calibrating to market data of plain vanilla S\&P500 options, which contain numerous maturities. The estimated parameters with confidence intervals between maturities support time-dependency and the constructed time-inhomogeneous exponential L{\'e}vy models appear favorable in contrast to the time-homogeneous model.

In research there is often a need to choose between multiple competing models. Two popular criteria for model selection are the AIC and BIC. The AIC excels in estimating the best model for the unknown data generating process. The BIC on the other hand is consistent in finding the true model. It is clear that for model selection these two information criterion give answers to different selection criteria. The question that arises is whether it is possible to construct a model selection criterion which combines the strengths of both AIC and BIC. In this study we will show that it is impossible to construct a model selection criterion which shares the above mentioned two strenghts by revisiting the proof of \cite{yang2005can} : That is, any consistent model selection criterion must be sub-optimal in the minimax convergence rate for regression estimation compared to the AIC.

Currently, speaker recognition research is mainly based on phonetics and speech signal processing. This research addresses speaker recognition from a new perspective, analysing the transcription of a fragment of speech with text analysis methods. Since text analysis is based on the transcription text only, it can be assumed independent from current automatic speaker recognition software. Hence, it would contribute significantly to the overall evidential value. The analysis is based on the frequencies of non-content, highly frequent words. We study whether information about the identity of the speaker is contained in the transcription of spoken text. The value of evidence is quantified using a score-based likelihood ratio. The score-based approach is chosen because in most forensic cases, there is not enough data from the suspect or of the disputed speech fragment available to model a robust feature-based likelihood ratio. Different methods to model the system from feature vector over score to likelihood ratio have been compared. As a baseline, a distance based method is used, where the score is the distance between the feature vectors. To improve upon this baseline, machine learning algorithms are implemented. The results from SVM and XGBoost are explored. As a third method a feature-based likelihood ratio is calculated and used as a score instead of as a direct likelihood ratio. With this method, both similarity and typicality are taken into account. The model is trained and tested on the FRIDA data set from the Netherlands Forensic Institute, consisting of Dutch conversations from a homogeneous group of 250 individuals. The performance of the likelihood ratio system is evaluated through computing the cost log-likelihood-ratio (Cllr), which is a measure for the accuracy and quality of the likelihood ratios, and the accuracy (A) of the likelihood ratios solely. The performance is also evaluated by inspecting the Tippett, empirical cross-entropy and pool-adjacent-violators plots. Different values for parameters used in the calculation of the likelihood ratios are investigated: the length of the sample, the number of frequent words (number of features) and the number of samples needed to train the model. The distance method showed a strong baseline, with good performance for large sample lengths. The SVM method outperformed the distance method for all parameter settings, with a peak performance of A=0.94 and Cllr=0.24. The XGBoost method showed promising results for smaller samples lengths, but a too large amount of data is needed to obtain good performance for larger sample lengths. The LR score method showed moderate results, but no improvements due to the necessity to estimate high-dimensional distributions. This thesis shows that information about the identity of the speaker is contained in transcriptions of speech. The complete process from data to likelihood ratio is constructed, where the likelihood ratio quantifies the evidential value of a transcribed speech fragment.

In de statistiek zijn er verschillende methodes voor het uitvoeren van model selectie. Het verschil in deze methodes komt voort uit het verschil in stromingen. Voor niet-geneste model selectie zijn de meest ganbare stromingen de Bayes Factor en de likelihood ratio. D. M. Ommen en C. P. Saunders presenteerden theoretische resultaten voor de relatie tussen de Bayes Factor en de likelihood ratio, waardoor de resultaten van beide paradigma's met elkaar kunnen worden vergeleken [5]. De Bayes Factor wordt uitgedrukt in de verwachting van likelihoodratio functie met betrekking tot de posterior verdeling van de parameters. In de bewijzen van deze theoriën ontbraken een aantal belangrijke aspecten. Om deze resultaten volledig te kunnen bewijzen, wordt in dit verslag aangetoond dat eerdere theoretische resultaten moeten worden uitgebreid met extra aannames of volledig moeten worden aangepast. Aan de hand van deze aannames en aanpassingen worden de theoretische resultaten bewezen. De theoretische resultaten zijn belangrijk in de toepassing van niet-geneste model selectie, omdat er nu gecommuniceerd kan worden tussen experts vanuit een ander paradigma.

In dit onderzoek zijn twee manieren van A/B-testen met elkaar vergeleken. A/B-testen is het vergelijken van verschillende website versies om te achterhalen welke versie voor een hogere opbrengt zorgt. Consumenten krijgen afzonderlijk meerdere versies van een website te zien: versie A, versie B, versie C, et cetera. De websites verschillen op basis van één onderscheidend kenmerk van elkaar.
De twee manieren van A/B-testen zijn de klassieke methode en de meer recent ontwikkelde bandit-methode. In de klassieke methode van A/B-testen krijgen meerdere groepen van consumenten de verschillende versies van de website te zien en wordt achteraf bepaald welke website versie de betere versie is op basis van het aantal conversies (conversies zijn bijvoorbeeld aankopen, clicks, et cetera). Het nadeel van deze test is dat je pas achteraf weet welke website de hoogste opbrengst oplevert, terwijl deze uitkomst misschien al tijdens de steekproef duidelijk wordt. De bandit-methode heeft meerdere varianten, waarvan vijf varianten in dit onderzoek zijn onderzocht en vergeleken. De bandit-methode van A/B-testen blijft gedurende de steekproef de groep consumenten die naar de verschillende versies van de website gestuurd worden aanpassen zodat al tijdens het testen zo min mogelijk consumenten naar de slechter presterende website gestuurd worden. Dit betekent namelijk misgelopen opbrengsten, ook wel spijt in A/B-testen. In dit onderzoek wordt daarom de klassieke methode vergeleken met een vijftal bandit-methoden.
Aan de hand van een gesimuleerde steekproef en drie fictieve versies van een website (A,B en C) worden de uitkomsten van de zes methoden op basis van statistische analyses vergeleken. De analyses zijn in RStudio geprogrammeerd en uitgevoerd. Hieruit blijkt dat, alhoewel alle zes de methoden uiteindelijk de best presterende versie kunnen aanwijzen, er duidelijke verschillen zichtbaar zijn in de totale conversies en de spijt. De bandit-methode verbetert op die onderdelen de klassieke methode. Daarnaast zijn er aanwijzingen dat de bandit-methoden de verschillende conversieratio’s van de website versies eerder statistisch significant kunnen aantonen.

In today’s complex, modern world, where more data and knowledge is available then ever before, consensus over approaches in statistics is far away. To understand what this means for forensic statistics, both the frequentist and Bayesian approach are considered in quantifying evidence. Classical frequentist approaches are elaborated according to an example of knife data, where both univariate and multivariate approaches are applied. As an alternative to the previous methods, the likelihood ratio is introduced. To arrive at a unified framework for calculating likelihood ratios, the European Union funded a project to calculate likelihood ratios using a software package with a user friendly interface called SAILR. Calculating the likelihood ratio is mostly done by using feature based models. A feature based model used by SAILR will be elaborated according to a glass comparison example. When the feature based model is illustrated, an extension on the model will be given using a non-parametric function. The discussed models will be elaborated and applied on test data from and international drugs comparison project using SAILR. As an alternative to feature based models, a score based model is introduced and compared to the previous drugs comparison results of the feature based model.

What is the actual value of a house and which factors contribute the most to it? In this thesis we do a thorough research and try to come up with an answer. We will set up models that approximate the current market values for all houses in the Netherlands. To do this, we use a lot of data from different sources. Various characteristics of houses, the location and the trend on the housing market will be used.

This thesis is dedicated to the application of data science to sports data. The research for this thesis is part of a bigger project on injury prevention and sport performance called Citius Altius Sanius (CAS). Two data sets from two different projects within CAS are analysed, with two different goals; one focusses on sports injury prevention in soccer, the other on performance prediction in baseball. First we analyse a data set on acceleration during exercise from project P6, generated while testing a prototype of wearable sensor trousers during soccer drills. The aim of P6 is to design special leg wear with wearable sensors in order to gain more knowledge on hamstring injuries. Therefore, an algorithm needs to be developed to identify the intensity of certain movements using sensor data. Features were extracted from the acceleration data in order to classify the intensity. Four methods are then tested on the data, of which the decision tree seems to produce the best results. Analysis showed that this model seemed to be able to predict low intensity well (99.1% accuracy), although it struggles signicantly more with medium and high intensity exercise (75.5%). The second data set covered the growth in throwing speed of a group of young baseball athletes between the ages of 12 and 18. The aim of the research was to identify a common growth curve for throwing speed of pitchers during adolescence and provide personalised growth curve models. A mixed effects or multilevel design was chosen to model the growth in throwing speed, due to its ability to model the hierarchical nature of the longitudinal data. After analysing the data set and covariates, we found we could reduce the number of predictors, and thus the cost of collecting data. Furthermore, it is possible to predict throwing speed on a personal level using only age and one measurement on the predictors and throwing speed, although predictions are improved when more measurements are available. The results of this research can be implemented in the projects, although some complications and opportunities for improvement still exist. Recommendations for future research have therefore been discussed.

Banks are financial institutions that lend money from other parties and provide loans to individuals and organisation for a higher interest. Lending out money is associated with the risk that debtors are not able to fully or partially repay the loans. This is called credit risk. Banks have to make an estimate of the credit risk in their portfolios and have to keep reserves for potential losses. The way this risk is to be determined, is decided by the government where the bank is established. In Europe, the United States, Russia, China among others, the legislation on credit risk is derived from Basel III. Basel III is an international framework to homogenise banking regulation across the world. There are three important factors to determine credit risk In Basel III, namely Probability of Default, Loss Given Default and Exposure at Default. In this thesis I investigate Probability of Default (PD) modelling. The size of the portfolio, for which the Probability of Default has to be estimated, can vary greatly. When the amount of defaults in a portfolio is low and the amount of explanatory variables is high, there is a risk of overfitting. Variable selection methods can be used to counteract overfitting and give understanding of the important predictors. I apply variable selection methods on on a logistic regression. I look at three Frequentist variable selection methods, namely Forward Selection, Lasso and Relaxed Lasso. I compare these three methods with Predictive Projection combined with a Horseshoe prior, which is a Bayesian approach to variable selection. Forward Selection starts with only the intercept in the model and adds variables one by one to the model. The variables are added in such a way that each step increase the estimated performance the most. The Horseshoe prior and Lasso Regression are types of regularisation, where the estimates of the regression coefficients of the logistic regression get shrunk to zero. In Lasso regression, this is done by adding a L1 penalty of the regression coefficients to the logistic regression. This causes weak signals to be pulled to zero. Lasso shrinks all regression coefficients to zero to some degree, even those with a strong signal.
Lasso can also be used to and an order of importance for the regression coefficients by varying the strength of the L1 penalty. Regression coefficients are set to zero one-by-one as the penalty increases. Relaxed Lasso uses this rank and refits the variables without regularisation.
In Bayesian statics, regularisation is added via the prior. The Horseshoe prior can adjust to the average sparsity in the model and the Horseshoe prior either shrinks a signal aggressively to zero, or leaves the signal almost unchanged. The posterior of the model is never truly sparse. Predictive Projection can induce sparsity by setting the Monte Carlo samples of the posterior to zero for certain variables. This is done in such a way that the Kullback-Leibler divergence between the full posterior and the projected sparser posterior is minimised. I investigate the behaviour of the variable selection methods. The main focus is on the predictive performance, the sparsity, the computation time and the reliability of the estimated performance for the selected models. I apply the methods to various types of simulated data to compare the variable selection methods. The simulated data consist of data with independent predictors, collinear predictors and non-normal predictors, among others. The simulations studies show that Lasso and Predictive Projection lead to models with the highest performance overall and the predictive performance is more stable over different realisation of the data. For the same performance the Predictive Projection produces models with less variables. This makes Predictive Projection the most attractive method. I also employ the techniques to FreddieMac data, which is a data set on single-family mortgages. The results are similar to the simulated data and Predictive Projection with the Horseshoe prior is the most attractive variable selection method. Both the simulation studies and the FreddieMac application imply that the estimated performance of the Predictive Projection and Lasso are better than those of Forward Selection and Relaxed Lasso. However, the behaviour of the estimated performance remain unclear to a certain degree. More simulations per data type and more data types are needed for more insight into the estimated performance. Additional resources are needed to achieve this.

In this report, inhomogeneous Lévy processes are studied in a discrete observational model based on derivatives of the process. First, homogeneous Lévy models are defined and an already known nonparametric method, using Fourier techniques and call and put option prices, for estimating the parameters of the model is described based on Belomestny and Reiẞ (2006a). Previous research suggests that there is a need for an extension of this concept since option prices with different maturities produce significantly different results. After all, the assumption that the parameters of the model are the same for any time window is not realistic and better results could be achieved once this premise is rejected. That is why inhomogeneous Lévy processes are introduced and studied in this report. The estimation method for the homogeneous model from Belomestny and Reiẞ (2006a) is extended to fit into the inhomogeneous framework. Next, asymptotic normality of the estimators is proven for these processes in this setting and confidence intervals are constructed using the finite sample variance method. Asymptotic normality has already been shown and confidence intervals have been constructed in the homogeneous framework in the continuous observational model by Söhl (2014). Finally, data is simulated from an inhomogeneous Merton model to test the performance of the method and options from the S&P 500 index are used as a real-world application.