Delong, Ł., Kozak, A., 2021, The use of autoencoders for training neural networks with mixed categorical and numerical features
We focus on modelling categorical features and improving predictive power of neural networks with mixed categorical and numerical features. First, we study regular and denoising autoencoders for categorical features in unsupervised learning problems. Second, we discuss possible architectures of neural networks in supervised learning problems which differ in the way categorical features are concatenated with numerical features. Third, we investigate a learning algorithm where we initialize parameters of a neural network in subsequent layers with representations of inputs learned with autoencoders for categorical and numerical data. We illustrate our techniques on a real data set with claim numbers. We conclude that our new architecture of a neural network initialized with parameters derived from autoencoders and a joint embedding for all categorical features performs better, in terms of predictive power, than the classical architecture with random initialization of parameters and separate entity embeddings for each categorical feature.
Delong, Ł., Szatkowski, M., 2021, One-year and ultimate reserve risk in Mack Chain Ladder model
We investigate the relation between one-year reserve risk and ultimate reserverisk in Mack Chain Ladder model in a simulation study. The first goal is to validate the so-called linear emergence pattern formula, which maps the ultimate loss to the one-year loss, incase when we measure the risks with Value-at-Risk. The second goal is to estimate the true emergence pattern of the ultimate loss, i.e. the conditional distribution of the one-year loss given the ultimate loss, from which we can properly derive a risk measure for the one-year horizon from the simulations of ultimate losses. Finally, our third goal is to test if classical actuarial distributions can be used for modelling of the outstanding loss from the ultimate and the one-year perspective. In our simulation study we investigate several synthetic loss triangles with various duration of the claims development process, volatility, skewness and distributional assumptions of the individual development factors. We quantify the reserverisks without and with the estimation error of the claims development factors.
Delong, Ł., Lindholm, M., Wüthrich, M.V., 2021, Gamma Mixture Density Networks and their application to modelling insurance claim amounts
We discuss how mixtures of Gamma distributions with mixing probabilities, shape and rate parameters depending on features can be fitted with neural networks. We develop two versions of the EM algorithm for fitting so-called Gamma Mixture Density Networks, which we call the EM network boosting algorithm and the EM forward network algorithm, and we test their implementation together with the choices of hyperparameters. A simulation study shows that our algorithms perform very well on synthetic data sets. We further illustrate the application of the Gamma Mixture Density Network on a real data set of motor insurance claim amounts and conclude that Gamma Mixture Density Networks can improve the fit of the regression model and the predictions of the claim severities used for rate-making compared to classical actuarial techniques.