In the theory of operators on Riesz spaces an important result states that Riesz homomorphisms on a C(Ω)-space are composition multiplication operators. Our aim is to extend this theorem to, not...Show moreIn the theory of operators on Riesz spaces an important result states that Riesz homomorphisms on a C(Ω)-space are composition multiplication operators. Our aim is to extend this theorem to, not necessarily Riesz, subspaces of such a C(Ω)-space. The main result entails the following, Riesz∗ homomorphisms on a pointwise order dense subspace X of C(Ω) are composition multiplication operators. Furthermore, we use this result to find additional results on Riesz∗ homomorphisms on these subspaces. We will exhibit, for example, that the inverse of a bijective Riesz∗ homomorphism on X is again a Riesz∗ homomorphism. As another corollary of the result we characterize which Riesz∗ homomorphisms on X are even complete Riesz homomorphisms. Results developed on pointwise order dense subspaces of C(Ω) can be applied in Sobolev space theory. As an analogy of the above we will develop a similar theory on subspaces of L p for a finite measure space. Most results carry over easily from the C(Ω) case. We will investigate difference in structure of Riesz∗ homomorphisms between these two type of space.Show less
High dimensional classification problems become increasingly frequent and these problems are notoriously difficult. Classifying Alzheimer patients using MRI data or fMRI data is one such challenge:...Show moreHigh dimensional classification problems become increasingly frequent and these problems are notoriously difficult. Classifying Alzheimer patients using MRI data or fMRI data is one such challenge: often no more than 50 subjects are measured while the number of variables or features observed per patient or object can be as high as 10000. Specialized statistical learners attempt to combat the challenges these high dimensional classification problems present. In this thesis we propose an extension of an ensemble learner called Stacked Generalization that combines the idea of Stacking multiple classification techniques and sub setting the feature space. We call it Stacked Domain Learning. We argue that Stacked Domain Learning may improve prediction performance in high dimensional classification problems. Performance increase is mainly expected in situations where the data presents different modalities. We investigate this claim in a simulation study. We apply state of the art (high dimensional) classification techniques as part of the ensemble learner and as comparison for the extension. Differential performance between the learners and the extension when applied to relatively simple data sets, without different modalities, shows that the extension could improve performance of both Stacked Generalization in general and choosing, through cross-validation, the single best performing statistical learner. Performance improvement is highly dependent on the characteristics of the data and most notable in conditions that are relatively noiseless. Performance increase is however not universal, even in the most favorable conditions, and application of Stacked Domain Learning is therefore best used not as a replacement of existing techniques but rather as an addition to the library of techniques the statistician might consider. The results warrant further study of Stacked Domain Learning to investigate performance improvement in a practical setting: in settings with or without explicit modalities. Results of the simulation study also implicate that further improvements could be made, for instance in the way the ensemble is combined. We also attempt to measure the quality of the prediction performance of the Stacking ensemble by attempting to measure the size of the classifier space (or hypothesis space), the enlargement of which is the main argument in favor of the extension. Results interpreted favorably indicate a negligible relation but the results are not conclusive.Show less
Het onderwerp differentiaalvergelijkingen is een zeer breed gebied van de wiskunde met toepassingen in alle andere wetenschappelijke disciplines, van de kleinste scheikundige reacties tot grote...Show moreHet onderwerp differentiaalvergelijkingen is een zeer breed gebied van de wiskunde met toepassingen in alle andere wetenschappelijke disciplines, van de kleinste scheikundige reacties tot grote astronomische vergelijkingen. Door het belang van deze toepassingen is het ook niet verbazend dat er al enkele eeuwen onderzoek gedaan wordt naar differentiaalvergelijkingen en er in die jaren ook al veel methodes gevonden zijn deze vergelijkingen op te lossen of de oplossing in ieder geval te benaderen. In deze scriptie ga ik kijken naar lineaire differentiaalvergelijkingen met polynomiale co¨effici¨enten, waar we het begrip singuliere punten kunnen introduceren. Deze singuliere punten blijken we vervolgens op te kunnen delen in 2 klassen: de reguliere en de irreguliere singuliere punten. Ik zal beginnen met enkele definities rond dit onderwerp en enkele voorbeelden die deze definities hopelijk duidelijker zullen maken. Vervolgens geef ik een korte uitleg over vergelijkingen met reguliere singuliere punten, waarna ik dit uitbreid naar een analyse van de irreguliere singuliere punten. Ook in deze laatste paragrafen zal ik voorbeelden geven om de theorie te verduidelijken, of om juist te laten zien wanneer we op problemen stuitenShow less
A growing body of literature studies the association between measures of hospital volume and patient outcomes after a surgical treatment to evaluate whether hospitals with large case volumes are...Show moreA growing body of literature studies the association between measures of hospital volume and patient outcomes after a surgical treatment to evaluate whether hospitals with large case volumes are associated with better outcomes. Applying the appropriate statistical methodology to these so-called volume-outcome studies erases several challenges such as the selection of a longitudinal estimation method and the specification of an appropriate measure for hospital volume. In daily practice, difficulties involved in volume-outcome studies are often not recognized. Regularly, hospital volume is analysed as a categorical variable, thereby neglecting its time-dependent nature. In addition, many volume-outcome studies ignore bias that may occur in the estimation process when certain assumptions are violated and traditional methods are used. In this thesis we use the recurrent marked point process to approach a longitudinal volumeoutcome analysis of clustered data. Statistical issues in the selection of both non-aggregate and yearly aggregate measures for hospital volume are considered. An additional aspect sometimes associated with clustered data concerns the presence of informative cluster size, where outcome depends on cluster size conditional on covariates. The concept of informative cluster size within a volume-outcome study presents a unique situation since hospital volume is both the covariate of primary interest under study and it is closely linked to cluster size. Within cluster resampling (WCR) is an appropriate method to analyse informative cluster size data. The novelty of this thesis is to apply WCR in the framework of a recurrent marked point process to study a longitudinal volume-outcome association. A simulation study has been performed to asses the performance of the proposed method and to evaluate whether the use of aggregate measures for hospital volume leads to bias in the estimation of the volumeoutcome association. Simulations show that when informative cluster size is present, the proposed method estimates the parameter for volume with small bias. In addition simulations suggest that bias might be introduced when an aggregate measure for present hospital volume is used.Show less
Multi-state models are powerful tools to understand and describe complex disease. Patients can move among a certain number of states defined by specific conditions of disease level often including...Show moreMulti-state models are powerful tools to understand and describe complex disease. Patients can move among a certain number of states defined by specific conditions of disease level often including death. Typically in these studies, the issues of interest include overall survival, effects of prognostic factors on disease progress and estimation of transition probabilities. Often multi-state models are employed under the Markov assumption, or it is assumed that the multi-state model can be described by a Markov renewal process. These assumptions are mainly made for mathematical convenience, since it is easier to estimate transition intensities and covariate effects. Moreover the transition probabilities can be computed. Obviously these assumptions could be rather unrealistic or too restrictive. The Markov assumption might not hold because there is association between transition times. In case a positive association is present, later transition will show higher rate if earlier transitions had taken place earlier. This implies a violation of the Markov assumption because the future depends not only on the present status but also on the past. In this thesis the Markov Renewal assumption is relaxed and two methods are proposed to deal with a violation of this assumption. The first method focus on the illness-death model, which is a 3-states model where only an intermediate event can occur before the main event of interest takes place. By relaxing the Markov renewal assumption, the proposed method models the correlation between transition times in the framework of Cox model. To obtain predictions for patients with a given history, formulas for prediction of transition probabilities are developed. For application purpose, some general functions for prediction are also developed in R and their use is illustrated through a set of data coming from a breast cancer trial. Relying on the frailty theory, the second approach proposed in this thesis models a forward-going sequential process in a framework of hidden Markov model. By extending the two-point mixture frailty model, frailties are modeled as hidden states which can have an impact on the transition rates and eventually be observed by the sojourn times and occurrence of events. Based on the likelihood construction an Expectation-Maximization algorithm was proposed.Show less
In this thesis an analysis of the resonance slice as found in MRFM experiments is developed and the derived model is used to gain insight in the role of experimental parameters. With this knowledge...Show moreIn this thesis an analysis of the resonance slice as found in MRFM experiments is developed and the derived model is used to gain insight in the role of experimental parameters. With this knowledge experiments are proposed to test the model. The influence of the cantilever amplitude and the bandwidth of the RF-pulse according to this analysis are also shown. Lastly a mathematical analysis of partial differential equations arising from spin diffusion in inhomogeneous magnetic fields is performed.Show less
In this paper we introduce a new kind of game, called a deck building game, of which Dominion is the most prominent example. We focus on the question to what extent traditional game analysis...Show moreIn this paper we introduce a new kind of game, called a deck building game, of which Dominion is the most prominent example. We focus on the question to what extent traditional game analysis techniques can be used to analyze deck building games? To do this, we look at several simple strategies, like Random and Greedy, and some traditional techniques, namely Monte Carlo Tree Search and Dynamic Programming. We compare the strategies for mid (31 turns) to long games (100 turns). We conclude that our implementation of DP seems to be suitable only for games of medium length or shorter because of its space complexity, whereas our implementation of MCTS seems to fall behind other strategies with similar performance in regards of time complexity.Show less
Menigeen heeft wel eens gehoord van het `lights out'-probleem. Over dit probleem is een hoop op internet te vinden, met name met verscheidene roosters, zogenaamde `grids'. Het idee hiervan is dat...Show moreMenigeen heeft wel eens gehoord van het `lights out'-probleem. Over dit probleem is een hoop op internet te vinden, met name met verscheidene roosters, zogenaamde `grids'. Het idee hiervan is dat je een heel rooster, bestaande uit lampjes die ofwel aan ofwel uit staan, vanuit een semi-willekeurige configuratie volledig uit kunt krijgen door goed gekozen lampjes van toestand te laten veranderen, waarbij de volgende eigenschap geldt: Als lampje X van toestand verandert, oftewel van uit naar aan gaat, of van aan naar uit, dan zullen alle aangrenzende lampjes van X tevens van toestand veranderen.Show less
This thesis concerns mathematical models and statistical analysis of management of default risk for markets, individual obligors, and portfolios. Firstly, we consider to use CPV model to estimate...Show moreThis thesis concerns mathematical models and statistical analysis of management of default risk for markets, individual obligors, and portfolios. Firstly, we consider to use CPV model to estimate default rate of both Chinese and Dutch credit market. It turns out that our CPV model gives good predictions. Secondly, we study the KMV model, and estimate default risk of both Chinese and Dutch companies based on it. At last, we use two mathematical models to predict the default risk of investors’ entire portfolio of loans. In particular we consider the influence of correlations. Our models show that correlation in a portfolio may lead to much higher risks of great losses.Show less
In this paper a new method of distinguishing shower-like from track-like neutrino interaction events using event reconstruction data is introduced. This method is then used to minimise the error...Show moreIn this paper a new method of distinguishing shower-like from track-like neutrino interaction events using event reconstruction data is introduced. This method is then used to minimise the error that a shower-like event is misidenti ed as a track-like event and vice versa, which is necessary for measuring the neutrino mass hierarchy in ORCA. In its current format, the method leads to errors of around 0.40 for noisy simulation data and approximately 0.15 for noiseless data. However, the method can still be extended to take into account more information of the reconstructions, thereby possibly improving the results.Show less
This report presents a first attempt to introduce noise into the protocol of reference-frame- independent quantum key distribution. It is found that a frequently accepted manner to introduce noise,...Show moreThis report presents a first attempt to introduce noise into the protocol of reference-frame- independent quantum key distribution. It is found that a frequently accepted manner to introduce noise, according to the model of Eckert et al. proposed in ref. [1] leads to non-physical state matrices and therefore another model is proposed: the $\beta_{\pi}$-noise model. In this model the basis states composing the state matrix are perturbed by a complex quantity. For pure states this approach is applied to all state matrix elements, whereas for mixed states it is applied only to the diagonal elements. The off-diagonal elements in the mixed state are perturbed by a complex quantity that is independent of the perturbations on the basis states that the matrix element consists of. Using a Monte Carlo simulation, statistics on the quantum bit error rate as well as the transverse correlation factor are obtained for this model. However, although the $\beta_{\pi}$-noise model solves the main issues that lead to the conclusion that the model of Eckert et al. might infer non-physical state matrices, it does not yet guarantee the state matrix is always physical: a mixed state may still violate positive semi-definiteness. Therefore the original model is improved by perturbing all basis states as before and using this approach for all state matrix elements. In this improved version of the $\beta_{\pi}$-noise model Eve is present as a (complex) scaling of the off-diagonal state matrix elements. Thus, positive semi-definiteness is guaranteed for this noise model. Also for this improved version of the model statistics on the quantum bit error rate and the transverse correlation factor are presented, thereby describing the implications on an experiment.Show less
