時間序列在資料重建之下嵌入相空間

Abstract

When studying a physical phenomenon experimentally following the evolution of time, we measured and collected relevant one dimensional data and considered it correct even when the data appeared chaotic, we assumed the phenomenon is controlled by a strange attractor in an unknown phase space. This point of view induces the delay reconstruction method and embedding theorems due to Whitney, Takems, Sauer, Yorke, Casdagli. What follows then is to estimate the dimension of that strange attractor by Grassberger and Procaccia (D2 dimension) method in that embedded space with the dimension, or higher. Before doing so I tried the idea of making a description of the classical Cantor set which is defined only through logic and is an uncountable set while any time series is at most countable. Then I tried the same method to any relaxed Cantor set and “calculate” the dimension and demonstrate that time series description is applicable. Furthermore, from two sets of experimental data (1. Nuclear Magnetic Resonance (NMR) 2.Arrhythmias), they and we use the same algorithm to estimate the “fractal” dimension of the attractor of the dynamical system.