Part 2 - - Terminology — phase space embedding

I am facing difficulties in understanding the definition of Takens' delay embedding from a non-physicist point of view....too much technical jargons. Can somebody please provide a simpler way to explain? Thank you.

The embedding theorem treats 1-dimensional sequences of numbers...

A delay embedding theorem uses an observation function to construct the embedding function. An observation function α must be twice-differentiable and associate a real number to any point of the attractor A

So you have a sequence of real numbers (dimension 1 regardless of the system's original dimension $d$). The delay reconstruction has dimension L, which is defined by the user. It is a construct, (hence the name re-construction).

The "points" used in the reconstruction are originally 1-dimensional which means that the reconstruction is not straightforwardly related to the original dimension of the system the timeseries was recorded from (of course for your reconstruction to actually be useful it has some bounds depending on d, see the wiki page).

(adding the following for future reference, through edit) There are three different dimensions here, which impose bounds on each other but are not "straight-forwardly" related:

1. The dimension of the original phase-space of the system, out of which a 1D timeseries was recorded. This dimension is equal to the number of independent variables of the system.
2. The dimension of the reconstruction. This integer number is defined by the user.
3. The dimension of the attractor (IF IT EXISTS!) that the original system ends up on. For example, it is known that the Lorenz system is 3 dimensional, and for some specific parameter values, states are converging to an (almost -)2-Dimensional attractor. By carefully doing a reconstruction of some dimension R, you can estimate the attractor dimension (something different from the reconstruction dimension).

You question:

QUESTION: In general, the volume of an object is given by $\pi r^d$. What would be phase space or state space volume? Would the power over the radius, r beL or d?

is ill defined. Where is that "object" from? From what space? If it is from the reconstructed space its volume has power L. If it is from the original space it has power $d$. You can't say something as general as "I have an object, what is its dimensional volume scaling" without saying where does this object come from.

It actually gets even worse. What is the "object"? Does it have the same dimensionality as the space it lives in? A disk has scaling 2 no matter if it is embedded in a 2 dimensional space or a 15 dimensional one. The state-space or phase-space volume has no difference from a "volume" of any other space. The dimensional scaling of a volume of the object depends on the object itself, not on the space it is embedded in (it is upper bound by the space's dimension obviously).

Please consider asking precise and short questions that about a single and specific thing. These are easy to answer with clarity and accuracy. The original question has many questions and many "confusions". I'd considered completely removing the second page of your question and turning it into a new question instead.

• Thank you for your patience and taking out the time to go through the Question. I need some clarifications and i will also clarify certain misleading terms. By object I meant that the attractor formed by the time series occupies a finite "volume" in the phase space. Based On Takens Embedding theorem, the properties of the chaotic system is preserved in the embedded space. (1) So if points are separated by a certain distance in dimension d then in the embedded space, would the pairwise distances be preserved? (2) Would the shape of the attractor remain same in embedded space?.... – Srishti M Oct 4 '17 at 0:32
• For instance, if Logistic map is reconstructed into 3 D embedded space, it still has that parabola structure, but in general can we say that the delay embedding preserves the shape of the attractor or can be guess about the chaotic system by looking at the attractor after phase space reconstruction? A similar concept of embedding a lower dimension into higher dimension is used in dimensionality reduction in machine learning. Methods such as manifold learning which focus on finding a low-dimensional embedding. of high-dimensional data. – Srishti M Oct 4 '17 at 0:34
• In my question, the observations are observed in higher dimension space L. I don't want to estimate the lower dimension but I want to know what is the phase space volume from the distances r calculated between points in higher dimension L. (3)If I am calculating the distances in L dimension, then the r should be raised to the power of d or L where d<L. – Srishti M Oct 4 '17 at 0:36
• The "shape" of the attractor will not remain the same, no matter what kind of embedding you do. The dimension of the attractor can remain the same, provided you have correctly chosen parameters for D and τ. The dimension of the attractor, if you have chosen correct parameters will remain the same, irrespectively of how many dimensions you have the attractor embedded in. Use this software package to create reconstructions of different dimensions on the fly and see for yourself. – George Datseris Oct 4 '17 at 16:50
• As I said in my answer, the fact that you ask so many questions in one posting, and then continue into lengthy exchanges in the comments, means that you have not taken the time to truly understand what you do not understand. Consider thinking about this for some time, and ask a single, clear question about a specific thing you do not understand. This will be better not only for the one answering, but more importantly for you as well. – George Datseris Oct 4 '17 at 16:53