Geodesic distance on Gaussian manifolds for the robust identification of chaotic systems

The identification of chaotic systems and their dynamical properties, from the embedding dimension to the Lyapunov exponents, is always based on the definition of a distance. The metric adopted for this distance can influence significantly the obtained results, particularly in the case of experimental data, which are always affected by noise and can include outliers. In this paper, we propose to adopt a metric explicitly developed for the analysis of experimental data typically corrupted by additive Gaussian noise. This metric is the geodesic distance on Gaussian manifolds (GDGM). A series of numerical tests prove that the GDGM provides qualitatively better results than the Euclidean distance, traditionally used in most applications. The GDGM can much better counteract the effects of noise providing systematically improved estimates of the dynamical quantities. An improvement of almost a factor of two in the estimates of the basic dynamical quantities can be achieved even for low levels of noise. The application to videos confirms the great potential of the proposed metric also for the analysis of higher-dimensional data.