Title: Sturm-Liouville Framework for Dynamical Reconstruction by Delay Embedding
Abstract: Delay embedding is well-known for non-linear time-series analysis, and it is used in several research fields such as physics, informatics, neuroscience and so on. The celebrated theorem of Takens ensures validity of the delay embedding analysis: embedded data preserves topological properties, which the original dynamics possesses, if one embeds into some phase space with sufficiently high dimension. This means that, for example, an attractor can be reconstructed by the delay coordinate system topologically. However, configuration of an embedded dataset may easily vary with the delay width and the delay dimension, namely, ``the way of embedding". In a practical sense, this sensitivity may cause degradation of reliability of the method, therefore it is natural to require robustness of the result obtained by the embedding method in certain sense. In this study, we investigate the mathematical structure of the framework of delay-embedding analysis to provide Ansatz to choose the appropriate way of embedding, in order to utilize for time-series prediction. In short, mathematical theories of the Hilbert-Schmidt integral operator and the corresponding Sturm-Liouville eigenvalue problem underlie the framework. Using these mathematical theories, one can derive error estimates of mode decomposition obtained by the present method and can obtain the phase-space reconstruction by using the leading modes of the decomposition. In this talk, we will show some results for some numerical and experimental datasets to validate the present method.