In this report, the conversion from spin-echo signals, obtained with a low-field hand-held MRI scanner that was designed and built at the Leiden University, to images of the proton density within the sample is considered. This scanner does not make use of switchable gradient coils, but instead relies solely on the natural inhomogeneity of the field and on translations of the sample over this field for its spatial encoding. Specifically, an attempt is made to answer the question of how we can reconstruct a phantom using this kind of scanner. This is done by deriving a signal model, discretising it and writing it as a linear least squares problem. Then, we can make use of the techniques of Cojugate Gradient for Least Squares (CGLS) wih `2-regularization and Generalized Conjugate Gradient Minimal Error (GCGME) with `1-regularization for the difference between neighbouring pixels in order to solve this inverse problem. Firstly, we theoretically consider combinations for magnetic field geometry and measurement strategy for their usability for image reconstruction. After this, the obtained strategies are tested in three experiments, with two magnets and two samples. We start by doing this numerically, using a simulated phantom in combination with a measured magnetic field and the translation strategy. By doing this, we can determine if reconstruction is possible using that combination of field and strategy. Finally, the strategy is tested on real samples. Using numerical phantoms in combination with the magnetic field and translation strategy used in the measurements, we were able to correctly reconstruct the phantoms. However, the reconstruction broke down when data from real samples was considered. A variety of possible improvements is discussed. The improvement that would have the most impact would be to design a magnet that has a less uniform gradient in the z-direction, and instead has some locations in the xy-plane where the field falls slowly as function of z, but quickly in other locations.

This thesis concerns the modelling, risk analysis and deposit rate optimization for Non-Maturing Deposits (NMDs), applied on aggregate data for the Dutch banking sector. The final NMD model consists of three parts, where there is a clear separation between the model for the term structure, the deposit rate and the deposit volume. The final model is calibrated on public data from 2004 to 2023. For the term structure, we calibrate a shifted CIR model using a genetic algorithm on historical bond rates that were obtained via a Svensson model. The deposit volume is modelled via the change in the logarithm of the volume, which is regressed using Ordinary Least Squares on market and deposit rate parameters. The model is made autoregressive to remove serial correlation in the residuals. Additionally, we describe and conduct a deposit rate model optimization for several classes of deposit rate models. We define two properties, the profit and the shortage, and maximize over a weighted sum of the two. For the optimization, we use a Robbins-Monro type model for its ability to deal with expectations, without the requirement of approximating the expectation at every step. With the calibrated model and optimized deposit rate models, we introduce a way of risk analysis, by looking at the quantiles of the deposit volume, conditioned on the minimum of the market rate being in the lowest quantiles. For our model, this risk analysis indicates that a low interest rate environment is not problematic for the final, maximum and minimum volumes over a period of at least 10 years. On the contrary, the quantiles of the volume are higher when conditioned on rates being very low, indicating that the worst scenario for the deposit volume is not a scenario with low interest rates. For simple models, we find that the procedure converges to an optimum value within an acceptable timeframe; for more complicated models, we find that the convergence is either slow, or that the these parameters do not influence the criterion that is optimized, compared to, for instance, the random component in the volume evolution. When the volume is considered as an input for the deposit rate model, an optimal model is obtained that outperforms models only considering the market rate significantly on the two considered properties. This suggests that taking the volume into account is beneficial for maintaining a strong NMD portfolio. Our methods are applied to public, aggregate data. As a result, we are unable to observe the effects of consumers switching banks within the Dutch system. We suspect that our approach works better when single bank data is available to the experimenter. All the methods that were described in this thesis are applicable to such data, yielding a possibly interesting way of conducting optimization, as well as risk analysis within a bank. This thesis adds to the literature by introducing a way of risk analysis that is incorporated in the model itself. For this analysis quantiles of the distribution of the dependent variable are calculated for both unconditioned and conditioned scenarios. Concretely for this thesis, quantiles of the deposit volume, conditioned on the market rate minimum over the whole period being in the lowest quantile, are calculated. Secondly, the thesis adds to the literature by introducing a method for optimizing the model for setting the deposit rate. Optimal models obtained via this approach can be used to challenge current ways that banks set their deposit rate.