The Chaotic Attractor Analysis of DJIA Based on Manifold Embedding and Laplacian Eigenmaps

Figure 1

Demonstration of the entire process of phase space reconstruction and manifold dimensionality reduction. Note: a one-dimensional time series (a) is firstly mapped to a D-dimensional Euclidean space according to embedding theorem, where we use phase space reconstruction () so that a compact but redundant dynamic space will be built (b). Then, NLDR such as Laplacian Eigenmaps is used to reduce the phase space to a d-dimensional manifold (), which helps to eliminate the redundancy and illustrate the chaotic attractor of the original dynamical system (c).