Noise-assisted estimation of attractor invariants

In this article, the noise-assisted correlation integral (NCI) is proposed. The purpose of the NCI is to estimate the invariants of a dynamical system, namely the correlation dimension $\left(D\right)$, the correlation entropy $\left(K_2\right)$, and the noise level $\left(\sigma \right)$. This correlation integral is induced by using random noise in a modified version of the correlation algorithm, i.e., the noise-assisted correlation algorithm. We demonstrate how the correlation integral by Grassberger et al. and the Gaussian kernel correlation integral (GCI) by Diks can be thought of as special cases of the NCI. A third particular case is the $U$-correlation integral proposed herein, from which we derived coarse-grained estimators of the correlation dimension $\left(D_m^U\right)$, the correlation entropy $\left(K_m^U\right)$, and the noise level $\left({\sigma }_m^U\right)$. Using time series from the Henon map and the Mackey-Glass system, we analyze the behavior of these estimators under different noise conditions and data lengths. The results show that the estimators $D_m^U$ and ${\sigma }_m^U$ behave in a similar manner to those based on the GCI. However, for the calculation of $K_2$, the estimator $K_m^U$ outperforms its GCI-based counterpart. On the basis of the behavior of these estimators, we have proposed an automatic algorithm to find $D,K_2$, and $\sigma$ from a given time series. The results show that by using this approach, we are able to achieve statistically reliable estimations of those invariants.

Figure 1

Correlation algorithm: The distances ${\tilde{z}}_{\omega }$ between all pair of phase-space vectors are compared with a threshold ${\theta }_{\omega }=h$ to produce the binary outputs $c_{\omega }$, which are averaged to calculate the correlation sum ${\widehat{C}}_m\left(h\right)$.

Figure 2

Noise-assisted correlation algorithm, consisting of $Q$ quantizers with threshold ${\theta }_{\omega }=0$. Each quantizer receives an input that is composed of the subtraction of an iid noise realization ${\mu }_{\omega }$, whose PDF depends on a parameter $h$, from the distance between a pair of phase-space vectors ${\tilde{z}}_{\omega }$. The binary outputs of all quantizers $s_{\omega }$ are averaged to calculate the noise-assisted correlation sum ${\widehat{S}}_m\left(h\right)$.

Figure 3

Noiseless Henon map. Log-log plot of GCI (dotted blue line) and ${\text{UCI}}^{\beta =2}$ (solid orange line) with $m=\left\{2,4,6,8,10\right\}$ $(m=2$ top and $m=10$ bottom).

Figure 4

Estimation of the correlation dimension for the noiseless Henon map using $m=\left\{2,4,6,8,10\right\}$. Log-derivatives of GCI (dotted blue line) and ${\text{UCI}}^{\beta =2}$ (solid orange line). The reported value of correlation dimension $D=1.22$ (dashed black line).

Figure 5

Noiseless Henon map. Log-log plot of the GCI (dotted blue line) and ${\text{UCI}}^{\beta =m}$ (solid orange line) with $m=\left\{2,4,6,8,10\right\}$ $(m=2$ top and $m=10$ bottom).

Figure 6

Estimation of the correlation dimension for the noiseless Henon map using $m=\left\{2,4,6,8,10\right\}$. Log-derivatives of GCI (dotted blue line) and ${\text{UCI}}^{\beta =m}$ (solid orange line). The reported value of correlation dimension $D=1.22$ (dashed black line).

Figure 7

Estimation of $K_2$ for the noiseless Henon map: Estimation based on GCI (dotted blue line), and estimation based on ${\text{UCI}}^{\beta =m}$ (solid orange line) for $m=\left\{2,4,6,8\right\}$. The reported value of correlation entropy $K=0.3$ (dashed black line).

Figure 8

Noisy Henon map $\left(\sigma =0.1\right)$ log-log plot for $m=\left\{2,4,6,8,10\right\}$: GCI (dotted blue line) and ${\text{UCI}}^{\beta =m}$ (solid orange line).

Figure 9

Correlation dimension coarse-grained estimators for the Henon map. $D_m^T$ (dotted blue line) and $D_m^U$ (solid orange line) for $m=\left[4,6,8\right]$ and $Q=\left\{500,5000,10000\right\}$. Top row: $Q=500$ (a) noiseless, (b) $\sigma =0.05$, and (c) $\sigma =0.2$. Middle row: $Q=5000$ (d) noiseless, (e) $\sigma =0.05$, and (f) $\sigma =0.2$. Bottom row: $Q=10000$ (g) noiseless, (h) $\sigma =0.05$, and (i) $\sigma =0.2$. The reported correlation dimension value $D=1.22$ is shown by a dashed black line.

Figure 10

Correlation entropy coarse-grained estimators for the Henon map. $K_m^T$ (dotted blue line) and $K_m^U$ (solid orange line) for $m=\left[2,4,6,8\right]$ and $Q=\left\{500,5000,10000\right\}$. Top row: $Q=500$ (a) noiseless, (b) $\sigma =0.05$, and (c) $\sigma =0.2$. Middle row: $Q=5000$ (d) noiseless, (e) $\sigma =0.05$, and (f) $\sigma =0.2$. Bottom row: $Q=10000$ (g) noiseless, (h) $\sigma =0.05$, and (i) $\sigma =0.2$. The reported correlation entropy value $K_2=0.3$ is shown by a dashed black line.

Figure 11

Noise level coarse-grained estimators for the Henon map. ${\sigma }_m^T$ (dotted blue line) and ${\sigma }_m^U$ (solid orange line) for $m=\left[2,4,6,8\right]$ and $Q=\left\{500,5000,10000\right\}$. Top row: $Q=500$ (a) noiseless, (b) $\sigma =0.05$, and (c) $\sigma =0.2$. Middle row: $Q=5000$ (d) noiseless, (e) $\sigma =0.05$, and (f) $\sigma =0.2$. Bottom row: $Q=10000$ (g) noiseless, (h) $\sigma =0.05$, and (i) $\sigma =0.2$. The true value is shown in each plot (dashed black line).

Figure 12

Henon map. Box plot of the estimation of $\sigma ,D$, and $K_2$ for different numbers of available phase-space vectors, $Q=\left\{500,1000,3000,5000,10000\right\}$. First row: estimation of $\sigma$. (a) Noiseless, (b) $\sigma =0.05$, and (c) $\sigma =0.2$. Second row: estimation of $D$. (d) Noiseless, (e) $\sigma =0.05$, and (f) $\sigma =0.2$. Third row: estimation of $K_2$. (g) Noiseless, (h) $\sigma =0.05$, and (i) $\sigma =0.2$. The reported values are shown by a dashed black line.

Figure 13

Mackey-Glass box plot of the estimation of $\sigma ,D$, and $K_2$ for different numbers of available phase-space vectors, $Q=\left\{1000,3000,5000,10000\right\}$. First row: estimation of $\sigma$. (a) Noiseless, (b) $\sigma =0.05$, and (c) $\sigma =0.2$. Second row: estimation of $D$. (d) Noiseless, (e) $\sigma =0.05$, and (f) $\sigma =0.2$. Third row: estimation of $K_2$. (g) Noiseless, (h) $\sigma =0.05$, and (i) $\sigma =0.2$. The reported values are shown by a dashed black line.