Delay embedding

The delay-embedding of a signal $S : ZZ -> RR^n$ is a signal $R : ZZ -> RR^(kn)$ such that

$R(t) = ( (S(t - (k - 1) Delta t)), (...), (S(t - Delta t)), (S(t)))$

where

$Delta t in ZZ$ is the embedding delay.
$k in ZZ$ is the embedding factor.
$kn$ is the embedding dimension.

A specific delay-embedding can be identified with the pair $(Delta t, k)$ .

The future of the signal $S$ under a given delay-embedding $(Delta t, k)$ is given by the signal $Q : ZZ -> RR^n$ such that

$Q(t) = S(t + Delta t)$

Practice

In the delay embedding, we do not allow for an embedding shift, i.e. translation of the signal before delay-embedding. This retains the time instant zero associated with the first element of the embedded signal. A possible time shifting is done in the estimators via the lag parameters.

In practice the signals are only defined on a bounded support. The signals are extended to the whole domain by padding them with NaNs. This results in NaNs in the beginning of each delay-embedded signal. For example, in TIM Matlab:

>> A = delay_embed([1, 2, 3, 4], 3, 1);

Yields:

A = NaN NaN 1 2 
    NaN   1 2 3 
      1   2 3 4

The samples containing NaNs at the first element are clipped away when the control transfers from TIM Matlab to TIM Core and replaced with additional lags. For example, in the previous example A is equivalent to the signal [1, 2;2, 3;3, 4] with a lag of 2.

Files

Delay embedding

Practice

Files

Delay embedding

Delay-embedding

Future of a delay-embedded signal