On reconstruction of strange attractors using their noise related directional properties

Abstract

We study directional properties of strange attractor trajectory vectors projected on selected subset of principal component eigenvectors in m-dimensional embedding spaces. We introduce a simple measure of anisotropicity for chosen principal component subspace, based on transformed projected trajectory matrix, which changes smoothly as a function of window width of the original trajectory matrix. The value of such defined measure is dependent on amount of noise in the data. For isotropically distributed noise (or close to isotropic), that fact allows us to set up the criterion for minimal window width for satisfactory reconstruction, as a function of amount of noise in the data. For data with low noise content we introduce a simple quantity that is closely related to “potential energy” of attractor's trajectory. It turns out that the minimum of such defined quantity determines the adequate window width for reconstruction of the attractor. The method is assessed through correlation dimension calculations.