Dropout is one of the most popular regularization methods used in deep learning. The general form of dropout is to add random noise to the training process, limiting the complexity of the models and preventing overfitting. Evidence has shown that dropout can effectively reduce overfitting. This thesis project will show some results where dropout regularizes the deep neural networks only under certain circumstances. Potential explanations would be discussed. Our major contributions are 1. summarizing the regularization behaviors of dropout, including how different hyper-parameters could affect dropout's performance; 2. proposing possible explanations to the dropout's regularization behaviors.

Burn injuries occur daily and can have severe physical and mental effects both in short and long term, such as disabilities due to severe skin contraction. Even though the mortality rate has decreased over the years, the need for a higher quality of life after severe burns remains. Decreasing the probability of a severe contraction is essential for increasing the quality of life. Mathematical models have been developed to predict skin contraction over time. However, the computations are time-expensive and not suitable for applications that require many simulations, for example, when considering input uncertainty for patient-based predictions. To that end, the application of neural network surrogates is studied to accelerate the computations of a morphoelastic numerical model for the prediction of skin contraction. Two datasets are generated from a one- and two-dimensional model respectively and are used to train the neural networks. It is shown that a feedforward neural network can accurately learn the nonlinear mapping between the input parameters and the outputs of the considered morphoelastic models. The trained neural network provides fast and accurate predictions on skin contraction and strain energy. Furthermore, a first step is taken towards a hybrid model where a neural network is applied as a surrogate for computationally expensive time-stepping in the numerical model. The added value of fast neural network surrogates is demonstrated in two clinical case studies. It is shown that the surrogate can be used to perform input parameter studies by comparing an age study with the morphoelastic model with the age study using the neural network. The neural network can reproduce the age study with high accuracy in just a fraction of the time. Secondly, a concept application is designed to demonstrate patient-based predictions using Monte-Carlo simulations to cope with input parameter uncertainty. The application provides predictions for skin contraction and the strain energy, based on the age of the patient and the wound size.

This thesis evaluates the pass-through of the 2001 and 2012 Dutch Value Added Tax (VAT) increases to customer prices using a difference-in-differences model. To this end, the first difference and feasible generalised least squares estimators are introduced. Contrary to the conventional pooled OLS estimator, these estimators always show significant causal effects for both VAT hikes. These also dramatically improve the accuracy of the estimates compared to earlier research on the incidence of VAT. For the 2012 tax increase, the null hypothesis of full pass through is even rejected. This result is a novelty in the econometric literature. Even in more general settings, the estimators used in this thesis prove far superior over conventional causal estimation techniques of difference-in-differences models.