A pr 1 99 9 Determination of the proper embedding parameters for noisy time series

We suggest an algorithm for determining the proper delay time and the minimum embedding dimension for Takens’ delay-time embedding procedure. This method resorts to the rate of change of the spatial distribution of points on a reconstructed attractor with respect to the delay time, and can be successfully applied to a noisy time series which is too noisy to be discriminated from a structureless noisy time series by means of the correlation integral, and also indicates that the proper delay time depends on the embedding dimension. PACS numbers : 05.45.+b Electronic mail: [email protected] Electronic mail: [email protected] Since the discovery of the method to reconstruct state spaces from a time series [1,2], various attempts have been implemented to choose the embedding parameters, the proper delay time and the minimum embedding dimension [3–10]. By the theorem of Takens [2], the reconstructions are generically diffeomorphic to the original dynamics if d > 2dA where dA is the dimension of the compact invariant smooth manifold A corresponding to the continuous time dynamical system and d is the embedding dimension. From the theorem, one can not, however, determine the delay time and the minimum embedding dimension. Nevertheless because the quality of the reconstructed attractor affects strongly the success of noise reduction processes [11] and forecasting methods as well as accurate calculations of various invariant quantities such as Lyapunov exponents, it is essential to choose the proper embedding parameters before carrying out the above mentioned jobs. In computational analysis with a noisy time series, because what the proper delay time is depends on the purpose for reconstructing attractors, it is suggested that the delay time is much less critical to success of the reconstruction process than the embedding dimension. In general, with the chosen delay time too small, the reconstructed points coincide approximately within the range of errors so that the reconstructed attractor appears “stretched out and randomly crowded” along the identity line in the time-delayed coordinates. If the delay time is too large, operations such as Grassberger-Procaccia algorithm [12] for the correlation dimension can be lead to wrong results [13] due to the nonlinearity of the embedding process not of the original dynamics. On the other hand, working in any larger dimension than the minimum leads to a more computational burden in obtaining the invariant quantities such as Lyapunov exponents, prediction, etc. And it also enhances the problem of contamination by round-off or measurement errors since the noises will populate and dominate the additional dimensions of the embedding space. The most important reason for looking for the minimum embedding dimension is to obtain an intuition for understanding and modeling the underlying dynamics of a noisy time series. Also in many schemes for noise reduction, the first job to do is to have at least a vague idea about the minimum embedding dimension [11]. In this paper, we propose a method to determine simultaneously the minimum embedding dimension dM and the proper delay time T d d for a noisy time series. The suggested method can be applied successfully to a noisy time series contaminated by a measurement noise of which the size is so much large that one can not disentangle the deterministic part from the random part by means of the correlation integral [14]. Also on the basis of this method, we find that the proper window length W d ≡ (d − 1)T d d remains approximately constant if d ≥ dM . Let a continuous signal w(t) be measured at each “sampling interval” Ts to yield a time series {v(k)}: {v(k)} ≡ {v(k) | v(k) = w(kTs), k = 1, 2, · · · , NT}, (1) where NT is the total number of time series. We then assign to ~x(t) the delay vector: ~y(p|d, Td) = (v(1 + (p− 1)j), v(1 + (p− 1)j + Td), · · · , v(1 + (p− 1)j + (d− 1)Td)) (2) p = 1, 2, · · · , N In ~y(p|d, Td), d and Td represent the embedding dimension and the delay time in the unit of the sampling time Ts, respectively. j is the interval in the unit of sampling time Ts