ReDim: Quantifying dependence via dimension reduction

Detecting and estimating statistical association among random quantities is a problem that arises in numerous fields of application (insurance and finance, biology, geology, medicine, etc.) and at the same time it is one of the most important and thrilling research questions in the field of dependence modeling. Often, the statistical association among random quantities is considered to be direction-free, i.e. the influence of quantity X on quantity Y is just as strong as the influence of Y on X. In such a situation, both the Pearson correlation coefficient and the rank correlation coefficients according to Spearman and Kendall are popular and frequently used indices to derive information about the strength of dependence. In contrast, when causality is present, one variable may have a stronger influence on the other variable than vice versa. In such a situation, it is reasonable to use coefficients that take into account the direction of the relationship between the random quantities. Two extremes are in focus: (1) If the target variable Y can be completely described as a function of the variable X, this is called perfect dependence. (2) If, on the other hand, it is not possible to derive information about the target variable Y from the variable X, this is known as independence. This project deals with the latter type: directed dependence. The aim is to develop methods that allow quantifying the extent of dependence of a target variable on one or more explanatory variables. The more explanatory variables are involved, i.e. the higher the dimension, the more difficult the estimation usually becomes ("curse of dimensionality"). To avoid this problem, a dimension reduction is to be carried out, which leaves the relevant information about the directed dependence of the variables involved unaffected. The motivation behind this project is manifold: On the one hand, the underlying dimension reduction principle is a challenging and fascinating mathematical problem. On the other hand, the methods described can be used to effectively measure the predictability and explainability of a target variable by means of several potential explanatory variables. This allows the relevant variables to be filtered out in the case of high-dimensional data sets which frequently occur in practice, and to restrict the modeling to these relevant variables (variable selection, feature selection).