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
And this is true
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
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
Giving you a more precise location of where the point actually lies
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
Remember that chaotic trajectories can be thought of as having infinite periods
That means all frequencies are present in a chaotic trajectory
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
So doing low-pass filtering of a chaotic time series is generally a very bad idea
For Question 3, were interested in the following topology-based approaches for identifying noisy points in a trajectory from a dynamical system
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
If the forward images of two nearby points are not close, this would violate the continuity properties of the dynamical systems were looking at
For part b, if the attractor contains isolated points, they may be the result of noise, and this is also true
Attractors at least the attractors were looking at in this course are connected sets
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