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I'm trying to numerically estimate the Jacobian from a time-series. Following the paper, https://journals.aps.org/pra/abstract/10.1103/PhysRevA.43.2787.

Consider that I have a scalar time series $x = (x(1), x(2), \dots, x(N - 1), x(N))$. In total there are $N$ data-points. So the steps that I'm following are as follows.

  1. Compute the time-delay (call it $\tau$).
  2. Compute the embedding dimension (call it $d_{emb}$).
  3. Compute the delay vectors (call it $\bar y_{i}(k)$). $\bar y_{i}(k) = [x(i), x(i + \tau), x(i + 2\tau), x(i + 3\tau), \dots, x(i + (d_{emb} - 1) \tau]$.
  4. Choose some random initial point in the time series and call the delay vector corresponding to that point, $\bar y(k)$.
  5. Look at the neighbours of $\bar y(k)$ and store them in ascending order of their Euclidean distance in $\bar y^{(r)}(k)$. Say we keep $N_{b}$ neighbours.
  6. Store the distance between the delay vector and its neighbours in $\bar z^{(r)}(k) = \bar y^{(r)}(k) - \bar y(k)$.

The basic idea is that the underlying dynamics is governed by a map $\bar F$ which takes $\bar y(k) \rightarrow \bar y(k + 1)$.

  1. Compute the next evolution of $\bar y(k)$. Therefore, $\bar y(k+1) = \bar F(\bar y(k))$. Similarly do it for the $r$th neighbour of $\bar y(k)$. One gets, $\bar y(r; k+1) = \bar F(\bar y^{(r)}(k))$.
  2. Compute the distance between these vectors, $\bar z(r; k+1) = \bar y(r; k+1) - \bar y(k+1)$.
  3. Now, $\bar z(r; k+1) = \bar y(r; k+1) - \bar y(k + 1) = \bar F(y^{(r)}(k)) - \bar F(y(k)) = \bar F(\bar y(k) + \bar z^{(r)}(k)) - \bar F(\bar y(k))= M_{\alpha \beta} z^{(r)}_{\beta}(k) + Q_{\alpha \beta \gamma} z^{(r)}_{\beta}(k) z^{(r)}_{\gamma}(k) + \cdots$.

We assume that $\bar z^{(r)}(k)$ are small so that one can do a Taylor expansion about it. $M$ is the Jacobian matrix of dimension, $((d_{emb} - 1)\tau, (d_{emb} - 1)\tau)$. $Q$ is the Hessian matrix.

  1. Idea now is to compute the unknown coefficients of $M$ using least-square fit keeping $H$ (for accuracy and dropping it finally).

The authors suggest that if $\tau$ be the step-size for the next iteration, so $\bar y(k) \rightarrow \bar y(k + 1)$, then $\bar z(r; k + 1) = M_{\alpha \beta} z^{(r)}(k)$. And then the only components unknown are the bottom row of $M$ Rest are $1$'s along the superdiag, and rest are all $0$.

What I don't understand it is that $\bar z^{(r)}(k) \rightarrow \bar F(\bar z^{(r)}(k)) = \bar F(\bar y^{(r)}(k) - \bar y(k)) \ne \bar F(\bar y^{(r)}(k)) - \bar F(\bar y (k)) = z(r; k+1)$. So how do they write -- $\bar z(r; k+1) = M_{\alpha \beta} z^{(r)}_{\beta}(k)$?

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