The first question is about the heuristic we learned about in the lecture
And it wants to know whether A, B, or C is the correct selection of tau based on Fraser and Swinneys heuristic, and thats A
You should select the first minimum of the mutual information according to Fraser and Swinney
Question 2 asks if the tau suggested by this heuristic is the only tau that will provide a faithful reconstruction according to the Takens theorem, and this is false
All the Takens theorem requires is that tau is greater than zero
All the other heuristics that appear in the literature are simply for choosing a good numerical approximation of this value
Question 3 asks which of the following is the logic behind choosing tau at the value marked A in the figure
As we learned in the lecture, this is the smallest tau that minimizes the shared information between coordinates
Question 4 is the same question, but now were wondering about B
The logic behind choosing B would be to maximize the shared information between coordinates, while obeying the theoretical constraint that tau is greater than zero
And finally, the logic behind choosing the tau at the value marked C in the figure is to minimize the shared information between coordinates while allowing enough lag between coordinates to unfold the dynamics more fully
Question 6 asks which of the following points A, B, and C are false neighbors
As you can see, in dimension 1, A, B, and C are all neighbors
However, when we embed it in two dimensions, A and C are still neighbors, so they are true neighbors
But now A and B, and B and C, are no longer neighbors, so these would be false neighbors
So this would be answer V: both A and B, and B and C, are false neighbors
And finally, if the ratio of false-nearest neighbors between dimension m and dimension m + 1 is less than 10%, that provides theoretical proof that m satisfies the conditions of the Takens theorem
And that the reconstructed attractor is diffeomorphic to the original attractor
This is most definitely false
You should keep in mind that all of these heuristics, including false nearest neighbor, are purely heuristics
They provide no theoretical proof whatsoever that you have satisfied the embedding theorems
That is, in practice, theres no way to theoretically prove from a time series, unless you already know the dimension of the system, that you have chosen an embedding dimension which satisfies the Takens theorem
This means that you can never prove in practice that the reconstructed attractor is actually diffeomorphic to the original attractor, unless more is known about the dynamics
This is something to be really careful about when youre using these heuristics in practice