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The first question is about the heuristic we learned about in the lecture
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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
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You should select the first minimum of the mutual information according to Fraser and Swinney
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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
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All the Takens theorem requires is that tau is greater than zero
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All the other heuristics that appear in the literature are simply for choosing a good numerical approximation of this value
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Question 3 asks which of the following is the logic behind choosing tau at the value marked A in the figure
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As we learned in the lecture, this is the smallest tau that minimizes the shared information between coordinates
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Question 4 is the same question, but now were wondering about B
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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
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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
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Question 6 asks which of the following points A, B, and C are false neighbors
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As you can see, in dimension 1, A, B, and C are all neighbors
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However, when we embed it in two dimensions, A and C are still neighbors, so they are true neighbors
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But now A and B, and B and C, are no longer neighbors, so these would be false neighbors
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So this would be answer V: both A and B, and B and C, are false neighbors
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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
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And that the reconstructed attractor is diffeomorphic to the original attractor
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This is most definitely false
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You should keep in mind that all of these heuristics, including false nearest neighbor, are purely heuristics
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They provide no theoretical proof whatsoever that you have satisfied the embedding theorems
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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
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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
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This is something to be really careful about when youre using these heuristics in practice