As discussed e.g. in the paper by Muskulus and Verduyn-Lunel http://www.math.leidenuniv.nl/reports/files/2009-12.pdf the delay vector reconstruction of time series for a dynamical model allows to represent its trajectories in a Euclidean space. Below I copy few sentences from the paper.

A time series $$x = (x_1, . . . , x_N)$$ of N measurements of a single observable X, a dynamical system is reconstructed by mapping each consecutive block $$x_{[i]} = (x_i, x_{i+q}, . . . , x_{i+(k−1)q})$$ of $k$ values, sampled at discrete time intervals $q$, into a single point $x_{[i]}$ in a Euclidean reconstruction space $\Omega= R^k$. The intuitive idea is that the information contained in the block $x_{[i]}$ fully describes the state of the (deterministic) system at time $i$, albeit in an implicit fashion.

I have following naive questions - as you see I am quite lost:

  1. How is each point of the original $x$ represented exactly?
  2. If the original time series has $N$ elements, is $k$ larger or smaller than $N$?
  3. What is 'recipe' for the embedding, how exactly is it done?
  4. What is the 'block $x_{[i]}$'? Is $i$ from $1...N$?

Any comments would be very appreciated.


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