State space reconstruction techniques and the accuracy of prediction

Abstract

If the data is dominated by deterministic dynamics, then a one-dimensional measurement of a single observable is sufficient to essentially reconstruct a potentially multidimensional state portrait of the entire governing dynamics. Our goal in this study was to find out which method of reconstruction is best to choose when the criterion is the quality of prediction of the studied observable. Several methods of reconstructing the state space portrait from a single time series were tested: uniform, non-uniform and weighted delay coordinates, an approach using principal components, and Xu’s differential reconstruction. In addition to predictability, we also evaluated the accuracy of estimating the complexity of a reconstructed attractor by a correlation dimension. We found that the numerical estimates of the correlation dimension were practically the same using different reconstructions. The prediction, on the other hand, was significantly influenced by the type of reconstruction. When trying to predict as accurately as possible, it is worth considering weighted delay coordinates, or approximations of derivatives instead of the standard uniform time delay embedding.