For Question 1, In the delay-coordinate embedding process, the time-delay parameter tau dictates how many teeth are in the data comb
This is false
The time-delay parameter dictates how far apart the teeth are in the data comb
The embedding dimension is how many teeth are in the data comb
For Question 2, In the delay-coordinate embedding process, the embedding dimension parameter m dictates how many coordinates are present in each delay vector
And this is true
Remember, the embedding dimension m dictates how many teeth are in the data comb, and theres one element in the delay vector or one coordinate in the delay vector for every tooth on the data comb
So this question is true
Question 3 states that all projections of a dynamical system can be undone perfectly with delay-coordinate embedding, and this is false
While delay-coordinate embedding does undo some types of projections
In particular projections caused by taking a scalar measurement of a dynamical system using a smooth generic observation function at regular time intervals
This is one particular projection, not all projections
Even just getting at this one class of projections, we can break this statement
For example, consider an observation function which is only continuous
Or, even worse, consider an observation function which is discontinuous
This would be an effective projection of this dynamical system for which delay-coordinate embedding should not be able to undo the projection perfectly
It is very important to keep in mind that, while delay-coordinate embedding is very powerful, and it can perfectly undo a specific kind of projection
There are very rigorous mathematical requirements on what kind of projection was performed
And it most definitely does not undo all projections of a dynamical system
Question 4 states that the conditions of the Takens delay-coordinate embedding theorem require that the observation function h that produces the time-series data be a continuous function on the state space
And this is false
While it is certainly true that the function must be continuous, for delay-coordinate embedding to work the observation function h must also be both differentiable and generic
That is, the observation function h must be a smooth, generic function on the state space, not just continuous
Question 5 states that, since tau > 0 satisfies the theoretical requirements of the Takens theorem, any choice of tau will result in an embedded attractor that looks the same as the original dynamics
And this is also false
Even if you choose a tau greater than 0, and you satisfy all other requirements of the Takens theorem, it is not guaranteed that the embedded attractor will look anything like the original dynamics
Were only guaranteed that theyll be topologically equivalent
However, they may look geometrically very different
Because they can look geometrically very different, theyll look very different to us visually
For example, consider these two Lorenz attractors
They look very different geometrically
However, topologically, they should be equivalent
Question 6 states that, in practice, any tau greater than 0 is as good as the next
That is, an tau greater than 0 will provide a correct, equally complicated, and equivalently useful embedding
And this is false
In theory, this statement is true: any tau greater than 0 is as good as the next
However, in practice, due to noise and finite data lengths, the choice of tau becomes critical in not only the usefulness of the embedding, but whether or not the embedding is over-complicated
For example, whether or not its folded in on itself
And if tau is chosen too large with finite data length, you can actually get embeddings which are incorrect
So, while in theory any tau greater than 0 is as good as the next, in practice this is a very different story, and tau must be chosen very carefully