Possibility of Reconstructing Nonlinear Dynamical System from Marked Point Process

Abstract

We can observe marked point processes in a wide variety of natural phenomena. Thus, it is important to establish theories and methods for analyzing such observed marked point processes. In a nonlinear time series analysis, the theories are the embedding theorems and the methods are the state space reconstruction from the observed time series. Although it has been revealed that we can reconstruct a state space from a point process without marked values, it has not been clarified yet whether we can reconstruct the state space from a marked point process. Therefore, in this study, we numerically investigated whether a state space could be reconstructed from a marked point process using a delayed coordinate system. To assess the issue, we evaluated the similarity between the interpoint distance distributions on an attractor of the original dynamical system that generated the marked point process and an attractor reconstructed from the marked point process. Results of numerical experiments show that the interpoint distance distributions in the original and reconstructed state spaces are similar, implying that it is possible to reconstruct nonlinear dynamical systems from the observed marked point processes.