Denoising Time Series by Way of a Flexible Model for Phase Space Reconstruction

We present a denoising technique in the domain of time series data that presumes a model for the uncorrupted underlying signal rather than a model for noise. Specifically, we show how the nonlinear reconstruction of the underlying dynamical system by way of time delay embedding yields a new solution for denoising where the underlying dynamics is assumed to be highly non-linear yet low-dimensional. The model for the underlying data is recovered using a non-parametric Bayesian approach and is therefore very flexible. The proposed technique first clusters the reconstructed phase space through a Dirichlet Process Mixture of Exponential density, an infinite mixture model. Phase Space Reconstruction is accomplished by time delay embedding in the framework of Taken's Embedding Theorem with the underlying dimension being determined by the False Neighborhood method. Next, an Infinite Mixtures of Linear Regression via Dirichlet Process is used to nonlinearly map the phase space data points to their respective temporally subsequent points in the phase space. Finally, a convex optimization based approach is used to restructure the dynamics by perturbing the phase space points to create the new denoised time series. We find that this method yields significantly better performance in noise reduction, power spectrum analysis and prediction accuracy of the phase space.