Estimation of large-scale dimension densities

We propose a technique to calculate large-scale dimension densities in both higher-dimensional spatio-temporal systems and low-dimensional systems from only a few data points, where known methods usually have an unsatisfactory scaling behavior. This is mainly due to boundary and finite-size effects. With our rather simple method, we normalize boundary effects and get a significant correction of the dimension estimate. This straightforward approach is based on rather general assumptions. So even weak coherent structures obtained from small spatial couplings can be detected with this method, which is impossible by using the Lyapunov-dimension density. We demonstrate the efficiency of our technique for coupled logistic maps, coupled tent maps, the Lorenz attractor, and the Roessler attractor.