# Understanding method of time series delay embedding

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.