Optimal state-space reconstruction using derivatives on projected manifold

A paradigm for optimal state-space reconstruction with nonuniform time delays is proposed. A comparison based on a diffeomorphic measure and a smoothness cost function shows that the proposed methodology achieves a better reconstruction compared to a reconstruction based on time delays that are multiples of the first minimum of mutual information. It is also shown how the proposed methodology is a more reliable approach to determining the embedding dimension.

Figure 1

${\varphi }_d^{\prime }$ for projection on $x_1$ and $x_2$ with $x_2$ viewed as a function of $x_1$ for two different delays in the Rossler system.

Figure 2

$L_k$ for full delay coordinate vector plotted against ${\tau }_w$ for the Rossler system. The value of ${\tau }_w$ for which $L_{k_{\mathrm{full}}}$ is minimum is used to set the upper bound in the search for optimal delays.

Figure 3

Reconstruction cycles for the Mackey-Glass equation. The ${\beta }_d\left(\tau \right)$ profile for each cycle has been shown on log scale. The optimal value of delay in each cycle is represented by a round dot.

Figure 4

Reconstruction cycles for the Mackey-Glass equation using conditional mutual information. The optimal value of delay in each cycle is represented by a round dot.

Figure 5

Reconstruction cycles for down-sampled Mackey-Glass equation. The ${\beta }_d\left(\tau \right)$ profile for each cycle has been shown on log scale. The optimal value of delay in each cycle is represented by a round dot.

Figure 6

Gluing of disparate parts in WINDMI system when multiples of first minimum of average mutual information, which is 44 in this case, are used as time delays for reconstruction. The left attractor is original state space and the right attractor is reconstructed. The reconstructed attractor has been translated by 50 points along the positive $x$ direction for better visualization. Points 14 419 and 19 077 in the original state space are glued together and artificially become nearest neighbor in the reconstructed state space. The $F_{\mathrm{RSTA}}$ was 22.44.

Figure 7

WINDMI attractor with MDOP reconstruction showing coordinates $x\left(t\right)$, $x(t-44)$, and $x(t-25)$. The reconstructed attractor has a low $F_{\mathrm{RSTA}}$ of 2.47.

Figure 8

Comparison $L_k$ distribution for Rossler system.