New Methodological Developments for GAMs and VGAMs

Additive and Generalized Additive Models are frequently used in nonparametric regression analysis, since they avoid overfitting by the use of smoothing spline penalties. The underlying assumption is that the ground truth is overall smooth. However, in practice this assumption is often unrealistic, since there can be outliers in the response, spatially highly varying curvature, or jump signals. Violations of the model assumptions can lead to inaccurate results, and thus it is necessary to weaken the assumptions. In this project we cope with these issues by robustifying the definition of the smoothing splines in different ways. The modifications are made such that the computations are still feasible. The resulting nonparametric estimates will allow for quick local changes while still preserving an overall smoothing property. The newly developed methodology will therefore yield a flexible and powerful framework which ensures that data being multivariate in the response and covariates can be analyzed appropriately. The developed methods will be implemented in the prominent VGAM package of the software environment R.

Many phenomena that we observe in nature have an underlying nonlinear relationship. An example are concentrations of chemical elements measured in the soil. In case of a location which is promising for mineral exploration, we would expect an increase in some element concentration values around this target, and the increase might follow a nonlinear trend. There might also be different sources of error which cause noise in the signal, and this could result in problems for estimating the underlying signal. One of the objectives of this project was to develop new methods for nonlinear trend smoothing which are less sensitive to noise in the measurements, but which are still sensitive enough to identify relevant peaks. As the methods are less sensitive to spikes and other forms of artifacts, they are called robust against outliers. Robust smoothing methods have also been considered for other models, being more general than just smoothing methods. Once the signals are robustly smoothened, it can still happen that signals from some observations are atypical. For example, Covid infection data from different countries over time might be very heterogeneous due to varying policies in the countries. Even after smoothing, these differences in the signals are visible. If the focus is on several signals, e.g. Covid infections, hospitalizations, deaths, etc., it is no longer straightforward to visually investigate if some countries show atypical joint behavior. Moreover, the degree of heterogeneity should not be evaluated based on the absolute numbers, as they are mainly driven by the population sizes. In statistics, there is a way to look at relative information, which is called compositional data analysis. We developed a new approach to identify outlying observations (countries) of smoothed multivariate signals which are treated as functions over time, and where the functional observations are considered as compositions. Further work has been devoted to the identification of groups of variables in multivariate data that interact together. For example, for metabolomic data in bioinformatics it is known that some metabolites interact jointly as a group, and that several of these groups can explain an underlying phenomenon, such as the type of a disease. At the same time, many other measured metabolites are completely irrelevant for the disease identification. Several methods to cope with this problem have been proposed the literature, but they do not consider the data as compositions, where only relative information is relevant. An example are metabolomic data, where the measured values depend on external settings, and by modifying the settings, the measured values could increase or decrease by a certain factor. We linked the compositional data analysis methodology with the identification of a network or graph structure in variables, which allows for a significantly simplified interpretable model outcome.