This thesis focuses on optimising the mortgage acceptance process at Achmea Bank, based on improved probability of default (PD) modelling. The objective is to improve the existing Advanced Internal Ratings-Based (A-IRB) model, originally designed for calculating capital requirements, and better tailor it to the credit acceptance mechanism. This research identifies areas where adjustments are necessary, including feature selection optimisation, the use of a more suitable target variable, and the exploration of multivariate isotonic regression as a non-parametric model for better estimating the nonlinear interactions among features and the target variable.

Earlier on, the thesis introduces an adapted dataset to mimic the setting of a mortgage acceptance process. This is where only the first observation of each facility in the dataset is used, unlike the original A-IRB model, which worked with more than one observation per facility.

The research employs logistic regression to model the probability of default, with a focus on the feature selection process that involves both univariate and multivariate analysis. Comparison among different models with target defaulting in 12 months and 24 months reveals that changing the target variable and improving the feature selection process results in better model performance.

The second focus of this thesis is isotonic multivariate regression, giving greater flexibility by fitting a nonlinear relationship between the risk drivers and target, but with the constraint of monotonicity. The minimum Redundancy Maximum Relevance (mRMR) algorithm is used for the selection of features to reduce computational time.

Comparison across models reveals that the isotonic regression model, with greater recall and more accurate default detection, has a higher false-positive rate. Logistic regression with a 24-month target, on the other hand, strikes a better trade-off between precision and recall, leading to fewer false alarms and a lower number of flagged cases.

Overall, the thesis demonstrates that logistic regression and isotonic regression models both provide valuable information to Achmea Bank's mortgage acceptance. While logistic regression with a 24-month goal is an appropriate balance between recall and precision, multivariate isotonic regression can provide recall-improvement potential at the cost of precision. Subsequent studies should focus on reducing false positives within the isotonic regression model, exploring rejection inference procedures to adjust for potential sample bias, and examining other forms of classification procedures that can more effectively handle class imbalance.

This research consists of two applications of image processing, namely, image compression and image denoising. Image compression aims to reduce the size of an image without losing too many features. This is often used to store a large number of images such as fingerprints. Denoising is a technique for removing noise from an image while preserving as many of the edges and other detailed features as possible. This research studies the use of different discrete wavelets and the Dual Tree Complex Wavelets in the image compression and denoising process. The wavelet transform decomposes the original image into approximation and detail coefficients, where the approximation coefficients are calculated by averaging and the detail coefficients by taking differences. The wavelet transform is also invertible so that the image can be reconstructed again using the approximation and detail coefficients. The compression and denoising method consists of three steps: decomposition, thresholding and reconstruction. The difference between compression and denoising lies in the threshold part. For image compression, a percentage of the detail coefficient is chosen as the threshold. The wavelets db6, sym5, coif3, bior4.4 and rbio1.5 are chosen to compress the images. The images are tested with compression rates ranging from 5:1 to 43:1. Based on the Structural Similarity Index Measurement (SSIM), the bior4.4 wavelet performs best. For denoising, the threshold is optimised to obtain a denoised image. The discrete wavelets used are db4, coif3, bior2.8. Of these, the bior2.8 wavelet performs the best on images used in this research. Therefore, the bior2.8 wavelet is compared with the Dual Tree Complex Wavelet (DTCW). Based on the Peak Signal-toNoise-Ratio (PSNR), the denoised image using the DTCW performs better than the bior2.8 wavelet. Overal, wavelets are a powerful tool in image processing. The different wavelets each have their own characteristics. The choice of the optimal wavelet depends on the application and cannot be generalised.