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:
- How is each point of the original $x$ represented exactly?
- If the original time series has $N$ elements, is $k$ larger or smaller than $N$?
- What is 'recipe' for the embedding, how exactly is it done?
- What is the 'block $x_{[i]}$'? Is $i$ from $1...N$?
Any comments would be very appreciated.
