This thesis investigated the influence of matrix structure, arising from various boundary conditions, on the performance of both direct and iterative solvers for linear systems resulting from discretized partial differential equations (PDEs). Using the negative one-dimensional Poisson equation as a model problem, system matrices were constructed for Dirichlet, Neumann, and mixed boundary conditions, including symmetrized variants.

The study classified the resulting matrices in terms of singularity and positive definiteness. Direct solvers such as LU decomposition and Cholesky factorization were tested for accuracy and applicability. Iterative solvers (Jacobi, Gauss-Seidel, and Successive Over-Relaxation or SOR) were analysed using the spectral radius of their iteration matrices to predict convergence.

Key findings include:
- Matrices from Dirichlet and mixed boundary conditions were non-singular; Neumann matrices were singular but still allowed convergence in iterative solvers under specific conditions.
- Cholesky factorization only applied to positive definite systems, while LU decomposition was more generally applicable to non-singular systems.
- Gauss-Seidel converged faster than Jacobi, with twice the convergence speed in tridiagonal cases.
- SOR achieved the fastest convergence when using the optimal relaxation parameter, though this optimal ω is generally not known a priori.
- Even when the spectral radius ρ(B) approached 1, as in Neumann systems, convergence was observed due to the structure of the eigenvalue spectrum.
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The thesis concludes that the structural properties of the matrix, such as singularity and diagonal dominance, are critical in determining the suitability and efficiency of numerical solvers. It also highlights the fragility of these methods when applied to nearly singular systems, motivating future research into more robust solution techniques and convergence criteria.

The Bayesian approach is a very important approach for tackling problems in statistics. It involves choosing a distribution that reflects the prior knowledge and  thus takes all knowledge into account in contrast to the frequentist approach. It also assumes that the parameters (the regression coefficients) follow a distribution called the posterior distribution instead of fixed constants. When a specific choice of this prior is made, this needs to be justified as the prior directly influences the posterior distribution of the regression coefficients. It is also possible to consider priors that do not carry a lot of information and such priors will be compared in this project. 

In this thesis, the Bayesian approach will be used to apply a multinomial logistic regression model to data concerning students’ study habits and beliefs. The data is provided by a research group called PRIME and they focus on mathematics education at the TU Delft. Multinomial logistic regression is used to find predictions of the choices expressed in probabilities. Bayesian statistics is not only useful in a sense that it offers the possibility to specify the prior knowledge, but also because the Bayesian way of thinking can be incorporated in evaluating results. This can be done by constructing credible intervals for the predicted probabilities. Overlap between intervals can then give insight on prediction quality.

In this project, the models are coded in R and here two packages are used: the UPG and the BRMS package. The priors that are compared are the Gaussian and Cauchy distributions. Other than that there are also default priors used in the packages, which can be compared to the Gaussian and Cauchy priors. In the end, a conclusion can be drawn about the performance of each model based on the prediction accuracy. It can be concluded that the BRMS package outperforms the UPG package in terms of accuracy both using default priors and overall using default priors gives more accurate results than specifying the prior. However, the difference in the accuracy of the model using the BRMS package is not significantly higher than the accuracy obtained from the UPG model and the running time is a lot higher for the BRMS package. From the models with a specified prior, the model with the Cauchy distribution as prior performed better.

The ability to accurately forecast sales volumes holds substantial significance for businesses. Current classical models struggle in capturing the impact of different variables upon the sales volume. These machine learning models are also not applicable to more than one specific product. The Temporal Fusion Transformer (TFT) is implemented to address these issues. The TFT is a powerful tool designed for time series forecasting. TFT leverages deep neural networks and self-attention to capture variable dependencies across all time steps,providing temporal context for improved accuracy. The developed TFT model showcases its efficacy in accurate
sales forecasting. By considering both past and future variables, TFT generates predictions with errors of 30%. Moreover, the interpretability of the model highlights the importance of variables such as Covid lockdown periods and product distribution. The scalability of the TFT model allows it to generate forecasts for every product-retailer combination, making it a versatile tool for businesses. As a multi-horizon forecaster, TFT incorporates both past and future variables to generate predictions. This characteristic makes it an excellent candidate for evaluating the impact of changes in future inputs controlled by the business, such as pricing and distribution strategies.

Frequentist statistics and Bayesian statistics are the two main approaches to statistical inference. The frequentist approach is commonly integrated into academic curricula, while the Bayesian approach is less frequently employed. However a comparison of the approaches, further investigating their shortcomings and advantages, might give a better comprehension of statistics and more insight in statistical inference. Therefore the current study applied both the frequentist and Bayesian approach to multinomial logistic regression.

The multinomial logistic regression model can be described as a generalized linear model and as a random utility model, and the current study has shown that these models generate an equivalent probability function. Moreover, the method of estimating coefficients in the frequentist and in the Bayesian approach were described. The multinomial logistic regression model was subsequently applied to data from educational research, conducted by PRIME. Three different R packages were used to perform the multinomial logistic regression: the VGAM package (frequentist, generalized linear model), the mlogit package (frequentist, random utility model) and the UPG package (Bayesian, random utility model). The results of the analysis of one dependent variable were subsequently compared.

The results indicated that the frequentist and Bayesian approach differ in their estimation time and model fit: the Bayesian approach required more computational time, but resulted in a better model fit. The frequentist 95% confidence intervals and Bayesian 95% credible intervals are comparable, but the interpretation of these is considerably different due to the philosophical underpinnings of both approaches. Moreover in the Bayesian approach, existing knowledge and information can be incorporated by choosing the prior distribution. Furthermore the Bayesian approach gives a posterior distribution, which is more informative than only a point estimate. In comparing the three different R packages it is noted that all three have a slightly different theoretical background. Since the packages all have their own shortcomings and advantages, combining them when conducting multinomial logistic regression could be desirable.