The shadowing problem is that of finding a deterministic orbit as close as possible to a given noisy orbit. We present an optimal solution to this problem in the sense of least-mean-squares, which also provides an effective and convenient numerical method for noise reduction for data generated by a dynamical system. Given a noisy orbit y and a dynamical system f, we derive a set of nonlinear equations whose solution x is the deterministic orbit with the smallest possible Euclidean distance to y. We present a numerical method for solving these equations. The quality of the solution depends on the initial noise level. When f is known exactly, the noise can be reduced to machine precision over long trajectory segments; with higher noise levels there are regions where the algorithm has difficulty, but significant overall noise reductions are still achieved. If f must be learned from the data the noise reduction is limited by the accuracy of the learning algorithm and the number of available data points, but large reductions are still possible in some cases.


Farmer, J.D. & Sidorowich, J.J. (1991). 'Optimal Shadowing and Noise Reduction'. Physica D, 47, pp.373-392.
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