Estimating the correlation dimension of an attractor from noisy and small datasets based on re-embedding

Abstract

A modified version of the Grassberger-Procaccia algorithm is proposed to estimate the correlation dimension of an attractor. Firstly, a measured time series is embedded into an M dimensional phase space spanned by time delay coordinates. This, in turn, is linearly transformed into an equivalent space spanned by an orthogonal basis derived from singular value decomposition. Secondly, a subspace composed of the directions of the (first few) principal eigenvectors, is again embedded into a higher dimensional space, which is called re-embedding. Finally, the Grassberger-Procaccia algorithm is applied on a re-embedding space instead of the Takens' embedding and thereby the correlation dimension ( D 2) is calculated. This leads to a modified version of the Grassberger-Procaccia algorithm, which is aimed at dealing with the estimation of the D 2 from noisy and relatively small data sets. In order to make full use of the available data, the delay time for the first embedding is always set to the sampling time. In order to reduce the noise level, only the principal components which are clearly above the “noise level” are used for the re-embedding. This modified algorithm is tested using low dimensional dynamical models with random noise of different levels. Here we have used the Lorenz model with D 2 about 2.0 and the Mackey-Glass equation with D 2 about 5.0. The results show that the present procedure gives a clearer scaling region in the D 2( M, r)-ln( r) diagram and thus a better estimate of D 2, especially when the data set is noisy and relatively small. This modified algorithm is applied to meteorological data and some of the problems associated with estimating the dimension of the weather and climate attractors are discussed based on the results.