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