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

Abstract

By using the techniques of Manifold Embedding and Laplacian Eigenmaps, a novel strategy has been proposed in this paper to detect the chaos of Dow Jones Industrial Average. Firstly, the chaotic attractor of financial time series is assumed to lie on a low-dimensional manifold that is embedded into a high-dimensional Euclidean space. Then, an improved phase space reconstruction method and a nonlinear dimensionality reduction method are introduced to help reveal the structure of the chaotic attractor. Next, the empirical study on the financial time series of Dow Jones Industrial Average shows that there exists an attractor which lies on a manifold constructed by the time sequence of Moving average convergence divergence; finally, Determinism Test, Poincaré section, and translation analysis are used as test approaches to prove both whether it is a chaos and how it works.