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For Question 1, the Farmer/Sidorowich filtering strategy leverages the stable and unstable manifold structure of the attractor to remove noise from a chaotic trajectory
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And this is true
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Remember, it moves a cluster of points forward in time to stretch it along the unstable manifold, and backward in time to stretch it along the stable manifold
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And looks at the intersection of what was left of this noise ball, forward and backward in time, stretched along the stable and unstable manifolds
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Giving you a more precise location of where the point actually lies
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For Question 2, low-pass filtering of chaotic systems is a bad idea because it can remove signal, not just noise, and this is true
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Remember that chaotic trajectories can be thought of as having infinite periods
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That means all frequencies are present in a chaotic trajectory
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This means, if you did a low-pass filter, you would effectively be removing a good portion of the actual signal, not just the noise
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So doing low-pass filtering of a chaotic time series is generally a very bad idea
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For Question 3, were interested in the following topology-based approaches for identifying noisy points in a trajectory from a dynamical system
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Part a asks, if the forward images of two nearby points are not close, one of those two points may have been perturbed by noise, and this is true
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If the forward images of two nearby points are not close, this would violate the continuity properties of the dynamical systems were looking at
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For part b, if the attractor contains isolated points, they may be the result of noise, and this is also true
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Attractors at least the attractors were looking at in this course are connected sets
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If you have points that are isolated that is, theyre not connected to the rest of the set this may mean that that point is simply perturbed by noise, or thats a noisy point