Partial Tail-Correlation Coefficient Applied to Extremal-Network Learning

We propose a novel extremal dependence measure called the partial tail-correlation coefficient (PTCC), which is an analogy of the partial correlation coefficient in the non-extreme setting of multivariate analysis. The construction of the new coefficient is based on the framework of multivariate regular variation and transformed-linear algebra operations. We show how this coefficient allows identifying pairs of variables that have partially uncorrelated tails given the other variables in a random vector. Unlike other recently introduced asymptotic independence frameworks for extremes, our approach requires only minimal modeling assumptions and can thus be used generally in exploratory analyses to learn the structure of extremal graphical models. Thanks to representations similar to traditional graphical models where edges correspond to the non-zero entries of a precision matrix, we can exploit classical inference methods for high-dimensional data, such as the graphical LASSO with Laplacian spectral constraints, to efficiently learn the extremal network structure via the PTCC. The application of our new tools to study extreme risk networks for two datasets extracts meaningful extremal structures and allows for relevant interpretations. Specifically, our analysis of extreme river discharges observed at a set of monitoring stations in the upper Danube basin shows that our proposed method is able to recover the true river flow network quite accurately, and our analysis of historical global currency exchange rate data reveals interesting insights into the dynamical interactions between major economies during critical periods of stress.