This thesis explores how forecasts of Dutch government bond yields can be improved by extending the current Dynamic Nelson-Siegel (DNS) model, used by the Dutch State Treasury Agency (DSTA), with stochastic volatility modeling and a Bayesian approach to parameter estimation and forecasting. The primary goal was to determine if the model extensions together with the Bayesian approach could improve the accuracy of yield forecasts given the highly volatile interest rate environment. In particular, we aimed to improve the "worst-case" forecasts, which we have defined as the upper bound of the 95% credible region with respect to the observed bond yields. To this end, we began with a baseline state-space model, resembling the current model in a state-space framework. Subsequently, we applied the findings from both in-sample and forecasting results as well as the findings from a literature review on volatility modeling to develop different models including two volatility models.

The volatility of the DNS model extensions is modeled as a GARCH process through the observation noise based on findings in the literature. This allowed for computationally efficient state estimation using a modified Kalman filter. Then, employing the Random Walk Metropolis algorithm for parameter estimation allowed us to use Bayesian multiple-step ahead forecasting. In particular, a comparative analysis of various models showed that while the current model performed better than expected, it was significantly outperformed in-sample by the DNS model with AR(1) observation noise (DNS-ARRW) and the DNS model with GARCH(1,1) observation noise volatility (DNS-OV). The Bayesian forecasting method particularly improved capturing the uncertainty of increasing yields in twelve-months ahead forecasts. Moreover, the two volatility models showed promising in-sample performance, but only one (DNS-OV) showed relatively good forecasting performance as well. Furthermore, the DNS-ARRW model consistently showed the best performance both in-sample and in forecasting.

In conclusion, the Bayesian approach to parameter estimation and forecasting proved effective in accounting for more variability in increasing forecast yields and simulating the direction of forecasts slightly better than the current MLE-based method. Moreover, the DNS-ARRW model showed significantly better worst-case forecasting performance, whereas the volatility models had a mixed performance.

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.
1

Since their introduction in 1993, particle filters are amongst the most popular algorithms for performing Bayesian inference on state space models that do not admit an analytical solution. In this thesis, we will present several particle filtering algorithms adapted to a class of models known as Piecewise Deterministic Markov Processes (PDMP), i.e. processes governed by one or more parameters that admit random jumps in their value at random times. Our work will focus on object tracking, the estimation of a target’s kinematic state over time from a sequence of noisy or incomplete measurements. Moreover, we will combine these techniques with Markov Chain Monte Carlo methods in order to infer the model parameters. We will perform sequential inference on both parameters and states by introducing an adaptation of the SMC2 to PDMPs. Finally, all algorithms will be tested both on simulated and real-world data (Piraeus AIS Dataset).

We live in an online world: we date online, we do business online and we communicate online. To make sure this happens securely, almost all transferred data is encrypted by cryptosystems. This paper focuses on one of these: the knapsack cryptosystem of Merkle and Hellman, which relies on the hardness of solving the knapsack problem (1978). In general it works as follows.
Bob, who wants to communicate with Alice, encrypts his message with a public key and sends this to her. If a third party now intercepts it, he must solve an instance of the knapsack problem, which is NP-hard in general. However, this becomes computationally infeasible if the number of items is large, and therefore Bob’s message is safe. Alice has access to a private key, which she uses to transform the hard knapsack problem into an easier one: one where the vector of weights is superincreasing. That is, each component of the vector is larger than the sum of all previous components. In this case she can solve the problem efficiently and read what Bob sent her.
This method seems to be reliable at first sight. However, a few years after it was published, cryptographer Shamir proposed an algorithm that breaks the system (1984). The algorithm finds a pair of numbers by solving two systems of inequalities. Then a third party can use this pair to transform the hard knapsack problem into one he can solve, which may be different from the one Alice finds. However, he will find the same solution, and therefore he can also read Bob’s message.
This algorithm can be implemented as a computer program and in this paper we used Python. The first system of inequalities is written as a optimization problem and since there is fixed number of unknowns, we can solve it in polynomial time using Lenstra’s integer programming algorithm (1983). In this paper we used the Gurobi optimizer to solve it, as Lenstra’s algorithm is hard to use in practice. The second system of inequalities is solved by comparing lower and upper bound, which can also be carried out in polynomial time. Nevertheless, the total algorithm finishes in polynomial time only with a certain high probability since we made some probabilistic assumptions. This implies that there is a small probability of failure. However, if it finds a solution, then the algorithm is fast, and therefore it is still valuable to use in real life.

In this thesis we analyse the class of maximum coverage problems. For all discussed problems, linear programs are formulated. Using the notion of submodularity, we prove that for the weighted version of the basic Maximum Coverage problem, where the weights differ per set, a polynomial-time greedy algorithm guarantees a (1 - 1/e)-approximation of the optimal solution. This improves the already known bound of (1 - 1/e - ε), for all ε > 0. We then show that the same result holds true, if we allow elements to be covered by multiple sets. Furthermore, a completely novel extension is introduced, where weights differ per combination of sets. It is proved that, under the assumption that the weights are submodular and increasing, a greedy algorithm still provides a (1 – 1/e)-approximation. 
The latter algorithm is tested in the framework of optimal sensor selection. To this end, we consider all official weather stations in the Netherlands as our sensor group. We test the performance of the approximation algorithm, if some of the assumptions do not hold and no theoretical bounds exists. Corresponding weights are calculated, using two approaches: inverse distance weighting and multiple linear regression. For both approaches the in practice performance of the greedy algorithm is shown to be even higher than (1 – 1/e), even though not all assumptions hold. Finally, the corresponding selection of weather stations is shown.

The clustering of events can have a large impact on society. The extremal index $\theta$ tells how much extreme events cluster. We will compare different types of estimators in this project. First, we review the extremes of different sequences which have different values of $\theta$. We have found significant differences between the extremes. Then, 2 different types of estimators are introduced which both use different ways to divide the data, using disjoint blocks and using sliding blocks. The optimal block lengths are simulated for all those estimators. Using those block lengths, $\theta$ is simulated with all the estimators. From the simulations we conclude that the estimators using sliding blocks perform better. The best-performed estimator that we found from the simulations is used to estimate $\theta$ on data from the KNMI, comparing wind gusts and precipitation at weather stations De Bilt and Vlissingen.

Many firms are occupied with determining the optimal replacement time of machinery. Machine replacement is a complex investment decision that requires the estimation of future cash flows and other parameters. The non-deterministic character of future cash flows has given rise to stochastic models, that take into account this uncertainty. This study has applied a theoretical stochastic asset replacement model in practice. It was found that the stochastic replacement model can be used on real data by performing a weighted least squares (WLS) regression. Decision-makers should however be aware of the model assumptions and limitations of the model. The replacement decision-making process can be automated using a Python script that is provided in this study. However, the CMMS that was used in the case study needs to be upgraded to have additional features. When one wants to perform an analysis of assets on a system level, the expected replacement year value can be used. Until now, the probability distribution of the expected replacement year had to be computed by means of Monte Carlo simulation. In this report, a closed form solution is used for the expected replacement year distribution when operating cost follows a geometric Brownian motion (GBM). With this contribution, decision-makers in engineering asset management now have the opportunity to rapidly analyse and perform probabilistic computations on the expected economic life of deteriorating machinery on system levels, such as geographic systems or weather systems. As only few studies on stochastic asset replacement are empirical, a second important contribution of this study is the application of the model in a case study. The case study concerns HVAC systems of petrol stations in the Netherlands. The paper describes how to perform a weighted least squares (WLS) regression so that model parameters can be easily estimated for real cases. Finally, several new insights and barriers on implementing theoretical models in practice are introduced.

In this thesis we are going to study outlier detection methods and propose a new method. Classical outlier detection is typically based on the assumption that the data is from a Gaussian/normal distribution. When the underlying distribution of a random sample is heavy tailed, so not normal , it is likely to have some extreme observations which would be identified as outlier using the classical procedure. This paper aims to address this issue by proposing a procedure
to identify real ‘outliers’ for heavy tailed data set. We first dive in the some existing methods and see how they work, try to understand them, simulate them and see their shortcomings in the case of a heavy tailed distribution. Then we study Extreme Value Theory (EVT) which we shall use to set up our proposed method of detecting outliers. Once we have constructed the proposed method, we are going to simulate and compare it with the existing methods. The goal in the case of normality is that the new method is not worse than the existing ones, at least not extremely, and in the case of a heavy tailed function to work better.

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.

In well-bore engineering, oil-well boreholes are made using specialized drilling rigs. The position of the drill bit needs to be indirectly determined through accelero- and magnetometer measurements. To this end, the measurement data is first converted into a survey of direction vectors by applying a series of coordinate transformations. Then, a method called Minimum Curvature Method (MCM) is applied, which outputs a close idealized approximation of the actual drill bit trajectory. However, systematic and random errors in the magnetometer measurements result in error in position vectors.

A novel solution called Multi-Station Analysis (MSA) determines the systematic (Scale and Bias) errors in the magnetometer measurement data. Using reference measurements, a non-linear least squares error function is minimized, which is equivalent to solving a non-linear system of equations. This is done numerically by the Newton-Raphson algorithm. Subsequently, the measurement data is corrected. Reapplying survey conversion and MCM results in an improved estimate for the actual position vectors.

The main objective of this thesis is to derive a method that describes the uncertainty of the MSA solution. This is primarily done through the method of Monte Carlo simulation. As part of validation, the effect of MSA on final drill bit position is studied and compared with results from an uncertainty model used by the well-bore industry. Secondary, a pessimistic quantification of the uncertainty of MSA solution is given through condition numbers of Jacobian matrices from the Newton-Raphson algorithm applied to MSA, which measure the sensitivity of the non-linear least-squares error. The question whether these condition numbers are a representative measure of MSA solution quality is answered. Finally, further potential research areas are described.