Denoising Noisy Chaotic Time Series Using Phase Space Local Projection Method

The phase space local projection method is an effective nonlinear signal processing technique, particularly suitable for denoising noisy chaotic time series or signal separation tasks. The core concept involves reconstructing the phase space of chaotic signals and utilizing local linear projection technology to remove noise or separate specific components from mixed signals.

First, phase space reconstruction of the chaotic time series is required. By selecting appropriate embedding dimensions and delay times, one-dimensional time series can be mapped to a high-dimensional phase space, restoring the topological structure of the chaotic attractor. This step is crucial because correct phase space reconstruction accurately reflects the characteristics of the original dynamical system. In code implementation, this typically involves using functions like delay_embed() or custom algorithms to calculate optimal parameters through mutual information or false nearest neighbors methods.

In the reconstructed phase space, the local projection method achieves denoising or signal separation by analyzing the neighborhood structure of phase points. Specifically, for each phase point, principal component analysis (PCA) is performed within its neighborhood, preserving the major principal component directions (corresponding to the main dynamic patterns of the chaotic signal) while projecting out minor directions (typically corresponding to noise or interference signals). For signal separation tasks (such as mixed chaotic and sinusoidal signals), differences in geometric characteristics of different signals in phase space can be utilized for separation. Algorithm implementation often involves calculating local covariance matrices and performing eigenvalue decomposition using libraries like numpy.linalg.eig().

The advantage of this method compared to traditional linear filtering techniques lies in its consideration of nonlinear characteristics of chaotic signals, better preserving the dynamic properties of the original signal. However, attention must be paid to the impact of parameter selection (such as neighborhood size, embedding parameters, etc.) on results, which typically requires optimization based on specific signal characteristics through techniques like cross-validation or parameter sweeps.