The first questions asks which paper originally proposed delay-coordinate embedding
And it was the Packard et al. paper entitled Geometry from a time series
Technically, this paper didnt formally prove anything about delay-coordinate embedding
It actually just mentioned delay-coordinate embedding as an aside; that this might be something people might want to look into, and that it might be useful
But they proposed a slightly different method
However, this was the first paper to bring up that the delay-coordinate embedding process could be useful in reconstructing an attractor
The Sauer et al. 1991 paper entitled Embedology really extends and improves on Takens theorem
This is the paper that relaxed the embedding restriction from m greater than two dimensions to m greater than two times the capacity dimension
It also provided several other theoretical proofs and discussions that were necessary
The Kennel et al. paper entitled Determining embedding dimension using a geometrical construction is the original false-nearest-neighbor paper
And Kantz and Schreibers 1997 book entitled Nonlinear time-series analysis is a very general textbook on nonlinear time-series analysis
For Question 2, The delay-coordinate embedding machinery only works, in both theory and practice, if you have an infinite amount of noise-free data, and fortunately, this is false
If this were true, as we can never have an infinite amount of data, let alone noise-free data, we could never use delay-coordinate embedding in practice
Question 3 states that the delay-coordinate embedding machinery only works, in theory, if you have an infinite amount of noise-free data, but can still be useful in practice even if those conditions are not met
And this is thankfully true
If this were not true, we could not use delay-coordinate embedding in practice
Luckily, even though this infinite amount of noise-free data is never met in practice, delay-coordinate embedding is still a very useful piece of machinery in the field of nonlinear time-series analysis
For problem 4, were assuming that our time series only has a thousand data points
Question a asks, Its fine to embed this time series in as many dimensions as the false near neighbor method suggests are appropriate, and this is false
The rule of thumb I generally use is you should have ten to the m data points
So in this case, you really shouldnt choose an m any higher than two or three
If you chose to embed in four or five dimensions, for example, you would need ten- or a hundred-thousand data points, respectively
This doesnt mean, however, that the false near neighbor method will never work for this time series
For example, the time series may be just fine to embed in two or three dimensions, and in this case, this method would work just fine
So b is false
And part c asks, You really should not embed it in more than two or maybe three dimensions, and this is true
Since 1,000 is ten cubed, I would feel just fine embedding this in two or three dimensions
For Question 5, repeating the same nonlinear time-series analysis procedures separately on chunks of your data set for example, the first half and then the second half can tell you one of two effects
It can tell you that you have inadequate data, or it can tell you that you have a nonstationary system
But it cant really tell you which one of these is occurring
For example, if you did an analysis on a full time series, and then you did an analysis on the first and second half, and you got the same answer, then you can probably assume that you had a stationary time series and you had an adequate amount of data
However, if you got a different answer on the first and second half as compared to the full time series, you couldnt tell if the time series was nonstationary, and a transition had occurred between the first and second half
Or if you have inadequate data going to half the time series
Repeating the same nonlinear time-series analysis procedures on half the time series or small chunks of the time series is a really great idea
But if you get different answers, you need to be very careful about how you interpret these
It could just be inadequate data, but it could also be nonstationaries in the system
In either case, more exploration needs to occur