<p>Missing data is a concept used to describe the values that are, for some reason, not observed in datasets. Most standard analysis methods are not feasible for datasets with missing values. The methods handling missing data may result in biased and/or imprecise estimates if methods are not appropriate. It is therefore important to employ suitable methods when analyzing such data. Cardiac surgery is a procedure suitable for patients suffering from different types of heart diseases. It is a physical and psychical demanding surgical operation for the patients, although the mortality rate is low. Health-related quality of life (HRQOL) is a popular and widespread measurement tool to monitor the overall situation of patients undergoing cardiac surgery, especially in elderly patients with naturally limited life expectancies [Gjeilo, 2009]. There has been a growing attention to possible differences between men and women with respect to HRQOL after cardiac surgery. The literature is not consistent regarding this topic. Gjeilo et al. [2008] studied HRQOL in patients before and after cardiac surgery with emphasis on differences between men and women. In the period from September 2004 to September 2005, 534 patients undergoing cardiac surgery at St Olavs Hospital were included in the study. HRQOL were measured by the self-reported questionnaires Short-Form 36 (SF-36) and the Brief Pain Inventory (BPI) before surgery and at six and twelve months follow-up. The SF-36 reflects health-related quality of life measuring eight conceptual domains of health [Loge and Kaasa, 1998]. Some of the patients have not responded to all questions, and there are missing values in the records for about 41% of the patients. Women have more missing values than men at all time points. The statistical analyses performed in Gjeilo et al. [2008] employ the complete-case method, which is the most common method to handle missing data until recent years. The complete-case method discards all subjects with unobserved data prior to the analyses. It makes standard statistical analyses accessible and is the default method to handle missing data in several statistical software packages. The complete-case method gives correct estimates only if data are missing completely at random without any relation to other observed or unobserved measurements. This assumption is seldom met, and violations can result in incorrect estimates and decreased efficiency. The focus of this paper is on improved methods to handle missing values in longitudinal data, that is observations of the same subjects at multiple occasions. Multiple imputation and imputation by expectation maximization are general methods that can be applied with many standard analysis methods and several missing data situations. Regression models can also give correct estimates and are available for longitudinal data. In this paper we present the theory of these approaches and application to the dataset introduced above. The results are compared to the complete-case analyses published in Gjeilo et al. [2008], and the methods are discussed with respect to their properties of handling missing values in this setting. The data of patients undergoing cardiac surgery are analyzed in Gjeilo et al. [2008] with respect to gender differences at each of the measurement occasions; Presurgery, six months, and twelve months after the operation. This is done by a two-sample Student's t-test assuming unequal variances. All patients observed at the relevant occasion is included in the analyses. Repeated measures ANOVA are used to determine gender differences in the evolution of the HRQOL-variables. Only patients with fully observed measurements at all three occasions are included in the ANOVA. The methods of expectation maximization (EM) and multiple imputation (MI) are used to obtain plausible complete datasets including all patients. EM gives a single imputed dataset that can be analyzed similar to the complete-case analysis. MI gives multiple imputed datasets where all dataset must be analyzed sepearately and their estimates combined according to a technique called Rubin's rules. Results of both Student's t-tests and repeated measures ANOVA can be performed by these imputation methods. The repeated measures ANOVA can be expressed as a regression equation that describes the HRQOL-score improvement in time and the variation between subjects. The mixed regression models (MRM) are known to model longitudinal data with non-responses, and can further be extended from the repeated measures ANOVA to fit data more sufficiently. Several MRM are fitted to the data of cardiac surgery patients to display their properties and advantages over ANOVA. These models are alternatives to the imputation analyses when the aim is to determine gender differences in improvement of HRQOL after surgery. The imputation methods and mixed regression models are assumed to handle missing data in an adequate way, and gives similar analysis results for all methods. These results differ from the complete-case method results for some of the HRQOL-variables when examining the gender differences in improvement of HRQOL after surgery.</p>

<p>A parametrization of a multidimensional Markov chain model (MDMC) is studied with the goal of capturing texture in training images. The conditional distribution function of each row in the image, given the previous rows, is described as a one-dimensional Markov random field (MRF) that depends only on information in the immediately preceding rows. Each of these conditional distribution functions is then an element of a Markov chain that is used to describe the entire image. The parametrization is based on the cliques in the MRF, using different parameters for different clique types with different colors, and for how many rows backward we can trace the same clique type with the same color. One of the advantages with the MDMC model is that we are able to calculate the normalizing constant very efficiently thanks to the forward-backward algorithm. When the normalizing constant can be calculated we are able to use a numerical optimization routine from R to estimate model parameters through maximum likelihood, and we can use the backward iterations of the forward-backward algorithm to draw realizations from the model. The method is tested on three different training images, and the results show that the method is able to capture some of the texture in all images, but that there is room for improvements. It is reasonable to believe that we can get better results if we change the parametrization. We also see that the result changes if we use the columns, instead of the rows, as the one-dimensional MRF. The method was only tested on images with two colors, and we suspect that it will not work for images with more colors, unless there are no correlation between the colors, due to the choice of parametrization.</p>

<p>In this text we will look at two parameter estimation methods for Markov random fields on a lattice. They are maximum pseudo-likelihood estimation and maximum general pseudo-likelihood estimation, which we abbreviate MPLE and MGPLE. The idea behind them is that by maximizing an approximation of the likelihood function, we avoid computing cumbersome normalising constants. In MPLE we maximize the product of the conditional distributions for each variable given all the other variables. In MGPLE we use a compromise between pseudo-likelihood and the likelihood function as the approximation. We evaluate and compare the performance of MPLE and MGPLE on three different spatial models, which we have generated observations of. We are specially interested to see what happens with the quality of the estimates when the number of observations increases. The models we use are the Ising model, the extended Ising model and the Sisim model. All the random variables in the models have two possible states, black or white. For the Ising and extended Ising model we have one and three parameters respectively. For Sisim we have $13$ parameters. The quality of both methods get better when the number of observations grow, and MGPLE gives better results than MPLE. However certain parameter combinations of the extended Ising model give worse results.</p>

<p>We present the physics behind general optical interference filters and the design of dielectric anti-reflective filters. These can be anti-reflective at a single wavelength or in an interval. We solve the first case exactly for single and multiple layers and then present how the second case can be solved through the minimisation of an objective function. Next, we present several optimisation methods that are later used to solve the design problem. Finally, we test the different optimisation methods on a test problem and then compare the results with those obtained by the OpenFilters computer programme.</p>

<p>The design of security protocols is given an increasing level of academic interest, as an increasing number of important tasks are done over the Internet. Among the fields being researched is formal methods for modeling and verification of security protocols. One such method is developed by Cremers and Mauw. This is the method we have chosen to focus on in this paper. The model by Cremers and Mauw specifies a mathematical way to represent security protocols and their execution. It then defines conditions the protocols can fulfill, which is called security requirements. These typically states that in all possible executions, given a session in which all parties are honest, certain mathematical statements hold. Our aim is to extend the security requirements already defined in the model to allow some parties in the session to be under control of an attacker, and to add a new definition of identity protection. This we have done by slightly extending the model, and stating a new set of security requirements.</p>

Common cause failures (CCFs) have been an important issue in reliability analysis for several decades, especially when dealing with safety instrumented systems (SIS). Different approaches have been used in order to describe this CCFs, but the topic is still subject to much research and there does not exist a general consensus as to which method is most suitable for dealing with CCFs. The $beta$-factor model is the most popular method today, even though this model has some well-known limitations. Other, more complicated methods, are also developed to describe situations where the $beta$-factor model is inadequate. The purpose of this thesis is to develop a strategy to suggest in which situations the different CCF methods are applicable. This is done by making a survey which includes several of the existing methods, before applying these in concrete SIS-examples. Observing the specific system in operation is a valuable tool and may help in acquiring feedback data to describe the lifetime of specific components and the number of failed components conditioned on the fact that the total system is failed. Since such feedback data usually are scarce and in our case totally absent, assessing whether the obtained results are accurate is difficult. Thus, the numerical results obtained from the analysis are compared to each other with respect to the assumptions of the particular model. For instance, the PDS method, a method developed for the Norwegian offshore industry, contains some assumptions which are different from the assumptions of the $beta$-factor model, and the report provides a study with respect to how these different assumptions lead to different results. Although other models are introduced, most focus is given to the following four, the $beta$-factor model, the PDS method, Markov analysis and stochastic simulation. For ordinary $M$ out of $N$ architectures with identical components, the PDS method is assumed adequate, and for $N=2$, the $beta$-factor model works well. Markov analysis and stochastic simulation are also well suited for modelling ordinary $M$ out of $N$ SIS, but because of the higher level of complexity, these approaches are not deemed necessary for simple systems. The need for Markov analysis becomes evident when working with SIS of a more complex nature, for instance non-identical components. Both the $beta$-factor model and the PDS method are not able to describe the system in full when dealing with certain types of systems that have different failure rates. An even more complex SIS is also included to illustrate when stochastic simulation is needed. This SIS is modelled by designing a computer algorithm. This computer algorithm describes how the system behaves in the long run, which in turn provides the estimate of interest, namely the average probability of failure on demand (PFD). Finally, it is always important to remember that if there exist any feedback data or expert knowledge describing the distribution of the number of components that fail in a CCF, this is vital in deciding the most descriptive CCF model. By the term ``descriptive model'', we mean a model that both describes the architecture of the system as accurately as possible, and also makes as few assumptions as possible. If it is known, either by applying expert opinion or from feedback data, that if a CCF occurs, all components of the SIS will always be disabled, then the $beta$-factor model is an adequate way of modelling most systems. If such knowledge does not exist, or it is known that a CCF may sometimes disable only a part of the SIS, then the $beta$-factor model will not be the most descriptive model.

This article contains discussions of the algorithms used for the construction of the continuation software implemented in this thesis. The aim of the continuation was to be able to perform continuation of equilibria and periodic solutions originating from a Hopf bifurcation point. Algorithms for detection of simple branch points, folds, and Hopf bifurcation points have also been implemented. Some considerations are made with regard to optimization, and two schemes for mesh adaptation of periodic solutions based on moving mesh equations are suggested.

We investigate what the graphics processing units (GPUs) have to offer compared to the central processing units (CPUs) when solving a sparse linear system of equations. This is performed by using a GPU to simulate fluid-flow in a porous medium. Flow-problems are discretized mainly by the mimetic finite element discretization, but also by a two-point flux-approximation (TPFA) method. Both of these discretization schemes are explained in detail. Example-models of flow in porous media are simulated, as well as CO2 -injection into a realistic model of a sub-sea storage-cite. The linear algebra is solved by the conjugate gradient (CG) method without a preconditioner. The computationally most expensive calculation of this algorithm is the matrix-vector product. Several formats for storing sparse matrices are presented and implemented on both a CPU and a GPU. The fastest format on the CPU is different from the format performing best on the GPU. Implementations for the GPU is written for the compute unified driver architecture (CUDA), and C++ is used for the CPU-implementations. The program is created as a plug-in for Matlab and may be used to solve any symmetric positive definite (SPD) linear system. How a GPU differs from a CPU is explained, where focus is put on how a program should be written to fully utilize the potential of a GPU. The optimized implementation on the GPU outperforms the CPU, and offers a substantial improvement compared to Matlab’s conjugate gradient method, when no preconditioner is used.

Some theory on Lévy processes and stochastic differential equations driven by Lévy processes is reviewed. Inverse Fast Fourier Transform routines are applied to compute the density of the increments of Lévy processes. We look at exact and approximate path integration operators to compute the probability density function of the solution process of a given stochastic differential equation. The numerical path integration method is shown to converge under the transition kernel backward convergence assumption. The numerical path integration method is applied on several examples with non-Brownian driving noises and nonlinearities, and shows satisfactory results. In the case when the noise is of additive type, a general code written for Lévy driving noises specified by the Lévy-Khintchine formula is described. A preliminary result on path integration in Fourier space is given.

Intrauterine growth retardation means that the growth of a fetus is restricted as compared with its biological growth potential. This contributes to an increased risk for illnesses or death of the newborn. Therefore it is important to characterize, detect and to follow up clinically any suspected or confirmed growth restriction of the fetus. In this master thesis we aim to describe the course of growth during the pregnancy based on repeated ultrasound measurements and study how the growth depends on different background variables of the mother in analyzing the data from the SGA (small-for-getational age) - project. The SGA-project contains data from 5722 pregnancies that took place in Trondheim, Bergen and Uppsala from 1986-1988, named The Scandinavian SGA-studies. In this thesis we have confined ourselves to a random sample of 561 pregnancies. A problem with many studies of this kind is that the data set contain missing values. In the SGA data set under study there were missing values from one or more of the ultrasound measurements for approximately 40% of the women. Until recently, the most popular used missing-data method available has been complete case analysis, where only subjects with a complete set of data are being analysed. There exist a number of alternative ways of dealing with missing data. Bayesian multiple imputation (MI) has become a highly useful paradigm for handling missing values in many settings. In this paper we compare 2 general approaches that come highly recommended: Bayesian MI and maximum likelihood (ML), and point out some of its unique features. One aspect of MI is the separation of the imputation phase from the analysis phase. It can be advantageous in settings where the models underlying the two phases are different. We have used a multilevel analysis for the course of fetal growth. Multilevel analysis has a hierarchic structure with two levels of variation: variation between points in time for the same fetus (level 1) and variation between fetuses (level 2). Level 1 is modeled by regression analysis with gestational age as the independent variable and level 2 is modeled by regarding the regression coefficients as stochastic with a set of (non directly observed) values for individual fetuses and some background variables of the mother. The model we ended up with describes the devolopment in time of the abdominal diameter (MAD) of the fetus. It had several ``significant'' covariates (p-value &lt; 0.05), they were gestational age (Time-variable), the body-mass index (BMI), age of the mother, an index varible wich tells if a mother has given birth to a low-weight child in an earlier pregnancy and the gender of the fetus. The last covariate was not significant in a strictly mathematical way, but since it is well known that the gender of the fetus has an important effect we included gender in the model as well. When we used the MI-method on the random sample (561) with missing values, the estimated standard deviations of the parameters have been reduced compared to those obtained from the complete case analysis. There were not a significant change in the parameter estimates except for the coefficient for the age of the mother. We also have found a procedure to verify if the MI-method gives us reasonable imputed values for the missing values by following the MCAR-procedure defined in Section 6. Another interesting observation from a simulation study is that estimates of the coefficients for variables used to generate the MAR and MNAR missing mechanism are ``suffering'' because they tend to be more biased compared to the values from the complete case analysis on the random sample (320) than the other variables. According to the MAR assumption such a procedure should give unbiased parameter estimates. {Key Words: Longitudinal data, multilevel analysis, missing data, multiple imputation (MI), Gibbs sampling, linear mixed-effects model and maximum likelihood (ML)-procedure.