CoSy Seminar: Measuring attractors via extreme value theory: strengths, drawbacks and applications to the climate system.

Everyone is welcome to join. The first 40 people to register will be offered a free lunch sandwich. If you do not want lunch, you can still attend without registrating.

Register for a lunch sandwich (deadline 10th of December).

Abstract: The attractor Hausdorff dimension is an important quantity bridging information theory and dynamical systems, as it is related to the number of effective degrees of freedom of the underlying dynamical system. By using the link between extreme value theory (EVT) and Poincaré recurrences, it is possible to estimate this quantity from time series of high-dimensional systems without embedding the data. In general, d ≤ n, where n is the dimension of the full phase-space, as the dynamics constrains trajectories on a compact object - the attractor - thus freezing some of the available degrees of freedom. Although information theory predicts that the equality d = n holds for random systems, we systematically find estimated values d < n. In this seminar, we will see that this effect is due to the curse of dimensionality stemming from the 'concentration of the norm' phenomenon in EVT estimators, by deriving an explicit expression for d(n) for n-dimensional Gaussian random vectors. We will also compare the performance of several EVT-based local fractal dimension estimators, analyzing and comparing their performance across a range of systems. Results suggest that, for most real-world applications, an approximation based on the assumption of an underlying Axiom-A systems leads to a very simple estimator and is sufficient to draw meaningful conclusions. Finally, we will see a few real-world applications, most of which effectively employing the aforementioned simplified estimator.

This is a lecture in the seminar series given by CIM (Centre for Interdisciplinary Mathematics).