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Abstract
Many nonlinear or chaotic time series exhibit an innate broad spectrum, which makes noise reduction difficult. Local projective noise reduction is one of the most effective tools. It is based on proper orthogonal decomposition (POD) and works for both map-like and continuously sampled time series. However, POD only looks at geometrical or topological properties of data and does not take into account the temporal characteristics of time series. Here, we present a new smooth projective noise reduction method. It uses smooth orthogonal decomposition (SOD) of bundles of reconstructed short-time trajectory strands to identify smooth local subspaces. Restricting trajectories to these subspaces imposes temporal smoothness on the filtered time series. It is shown that SOD-based noise reduction significantly outperforms the POD-based method for continuously sampled noisy time series.
Highlights
Many natural and engineered systems generate nonlinear deterministic time series that are contaminated by random measurement/dynamical noise
We describe a new method for nonlinear noise reduction that is based on a smooth local subspace identification[7,8] in the reconstructed phase space[9,10]
In contrast to identifying the tangent subspace of an attractor using proper orthogonal decomposition (POD) of a collection of nearest neighbor points, we identify a smooth subspace that locally embeds the attractor using smooth orthogonal decomposition (SOD) of bundle of nearest neighbor trajectory strands
Summary
Many natural and engineered systems generate nonlinear deterministic time series that are contaminated by random measurement/dynamical noise. In contrast to identifying the tangent subspace of an attractor using proper orthogonal decomposition (POD) of a collection of nearest neighbor points (i.e., as in local projective noise reduction scheme), we identify a smooth subspace that locally embeds the attractor using smooth orthogonal decomposition (SOD) of bundle of nearest neighbor trajectory strands. This new method accounts for geometrical information in the data, but its temporal characteristics too. Models generating test time series are introduced along with the methods used in the algorithm evaluation, which is followed by the description of results and discussion
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