Dimensionless embedding for nonlinear time series analysis

Recently, infinite-dimensional delay coordinates (InDDeCs) have been proposed for predicting high-dimensional dynamics instead of conventional delay coordinates. Although InDDeCs can realize faster computation and more accurate short-term prediction, it is still not well-known whether InDDeCs can be used in other applications of nonlinear time series analysis in which reconstruction is needed for the underlying dynamics from a scalar time series generated from a dynamical system. Here, we give theoretical support for justifying the use of InDDeCs and provide numerical examples to show that InDDeCs can be used for various applications for obtaining the recurrence plots, correlation dimensions, and maximal Lyapunov exponents, as well as testing directional couplings and extracting slow-driving forces. We demonstrate performance of the InDDeCs using the weather data. Thus, InDDeCs can eventually realize “dimensionless embedding” while we enjoy faster and more reliable computations.

Figure 1

Recurrence plots of the Rössler model obtained by the infinite-dimensional delay coordinates (InDDeCs) and the conventional three-dimensional delay coordinates. Panels (a), (d), and (g) show the recurrence plots obtained by InDDeCs with $\lambda =0.1,0.5,$ and 0.9, respectively. Panels (b), (e), and (h) show the recurrence plots obtained by the conventional delay coordinates using all the distances with the embedding dimensions of 1, 3, and 10, respectively. Panels (c), (f), and (i) show the recurrence plots obtained by the k-d tree [10] with the embedding dimenisons of 1, 3, 10, respectively. In panels (a), (b), (d), (e), (g), and (h), exactly 1% of places have the plotted points, while in panels (c), (f), and (i), nearly 1% point of distances was used to plot the recurrence plots.

Figure 2

The average accuracy of the obtained recurrence plots among 10 trials against to those for the conventional delay coordinates with the time points between 1001 and 2000. In panel (a), we compared recurrence plots (the red solid line) obtained by the InDDeCs using $\lambda =0.1$ with recurrence plots obtained by the decomposition of Eq. (4) of Ref. [5] using the $L_1$ metric (the blue dashed line) and recurrence plots obtained by the k-d tree [10] (the green dash dotted line) using the embedding dimension of 1. In panels (b) and (c), the similar figures are shown with $\lambda =0.5$ and 0.9 and the embedding dimensions of 3 and 10, respectively.

Figure 3

The average times required among 10 trials for the InDDeCs (the red solid line) and the conventional delay coordinates with a normal implementation using the $L_2$ metric (the black dotted line), as well as the decomposition of Eq. (4) of Ref. [5] using the $L_1$ metric (the blue dashed line) to obtain the corresponding distance matrices given a length of time series. In addition, the calculation times required by the k-d tree were shown in the green dash-dotted line. In panel (a), we used $\lambda =0.1$ and the embedding dimension of 1. In panel (b), we used $\lambda =0.5$ and the embedding dimension of 3. In panel (c), we used $\lambda =0.9$ and the embedding dimension of 10. For these calculations, we used a computer with 2.7 GHz 12-Core Intel Xeon E5 with 64 GB memory.

Figure 4

Correlation dimensions and maximal Lyapunov exponents estimated using InDDeCs. Panels (a) and (b) are for correlation dimensions and Panels (c)–(f) are for maximal Lyapunov exponents. Panels (a) and (c) are for the Hénon map. Panels (b) and (d) are for the Lorenz model with sets of parameters ($R=28,\sigma =10,b=8/3$) and ($R=40,\sigma =16,b=4$), respectively. Panel (e) is for the Ikeda map and panel (f) is for the Rössler model. For each panel, the estimation using InDDeCs is shown in the blue solid line and the estimation in the literature is shown in the red dashed line.

Figure 5

Surrogate tests using correlation dimensions. We used the random shuffle surrogates [20] (the black dotted lines), phase randomized surrogates [20] (the blue dash-dotted lines), and iterative amplitude adjusted Fourier transform (IAAFT) surrogates [21] (the green dashed lines), respectively. Each of the two lines show 95% confidence intervals obtained by 40 surrogates each. The red solid thick line shows the lines obtained from the original data. In panel (a), we show the results for the autoregressive linear model, and in panel (b), we show the results for the Hénon map. The length of time series generated was 2000 each. We applied the end-to-end matching [35] as a preprocessing to avoid spurious high frequency components during the Fourier transforms.

Figure 6

Surrogate tests using the maximum Lyapunov exponents. See the caption of Fig. 5 to interpret the results.

Figure 7

Identifying directional couplings depending on parameter $\lambda$ of InDDeCs and coupling strengths ${\eta }_{yx}$ and ${\eta }_{xy}$ in the examples of coupled logistic maps. In the first and second rows, we used mutually coupled logistic maps $x$ and $y$. In the third and fourth rows, we used logistic maps $x$ and $y$ driven by another logistic map $z$. In the fifth and sixth rows, we used mutually coupled logistic maps $x$ and $y$ driven by another logistic map $z$. In each row, the left, center, and right columns correspond to the results of $\lambda =0.1,0.5$, and 0.9, respectively. In each panel, the logarithms for the $p$ values were shown. In the white regions, the $p$ values were smaller than 0.01, representing significance. The darker regions show greater $p$ values, which are not significant.

Figure 8

Driving forces and their reconstructions. (a) Correlation coefficients between driving forces and their reconstructions using InDDeCs depending on the parameter $\lambda$. The blue solid line corresponds to the driving force constructed by the Lorenz model, while the red dash-dotted line corresponds to the driving force constructed by the Rössler model. (b) The original driving force of the Lorenz model (the black solid line) and its reconstruction (the blue dashed line) when $\lambda =0.62$. (c) The original driving force of the Rössler model (the black solid line) and its reconstruction (the red dashed line) when $\lambda =0.62$. In Panels (b) and (c), we plotted the reconstructed driving forces so that the means, standard deviations, and direction are matched with the original driving forces.

Figure 9

The validation for the reconstructed driving forces using time series prediction in the example of Fig. 8. The prediction errors are shown by box plots. When we take into account the reconstructed slow-driving forces up to the second one, the prediction errors have improved significantly. We used $\lambda =0.62$ as an example.

Figure 10

The validations for the reconstructed driving forces for the weather data at Akita, Japan. Panels (a), (b), (c), and (d) correspond to the temperature, the solar irradiance, the precipitation, and the wind speed, respectively. For the temperature, the solar irradiance, and the precipitation, the first driving force significantly reduced the prediction errors. For the wind speed, we found the third driving force reduced the prediction errors. Thus, later we selected these driving forces to conduct the further analysis.

Figure 11

Reconstructed driving forces and directional couplings for the weather at Akita, Japan. Panels (a), (b), (c), and (d) are the validated principal components for the driving forces reconstructed by using InDDeCs for the temperature, solar irradiance, precipitation, and wind speed, respectively. Panels (e) and (f) represent the directional couplings identified using the method of joint distribution for distances using (e) the conventional delay coordinates and (f) InDDeCs. The gray scales show the $p$ values in the logarithm using base 10. Namely, the white regions correspond to significant pairs of the time and coupling direction with the significance level of 0.01. The darker regions have higher $p$ values, which are not significant.

Figure 12

Parts of time series for the weather at Akita, Japan. Panel (a) corresponds to the temperature, panel (b) corresponds to the precipitation, panel (c) corresponds to the solar irradiance, and panel (d) corresponds to the wind speed. Because the time series data shown correspond to the beginning of January, which means the winter season, we could not see the clear daily cycle for the temperature in panel (a) for the first four days, possibly due to the accumulated snow.

Figure 13

Estimated network structures. By classifying the estimated network structures using the $k$-mean algorithm [30], we identified two typical structures, which are denoted by cluster 0 [panel (a)] and cluster 1 [panel (b)].

Figure 14

The frequency of estimated network structures depending on the month within a year.

Figure 15

Validation for the estimated network structures. Based on the results for the cluster 0 of Fig. 13, the precipitation is driven by the temperature and/or the wind speed. Thus, we tested whether the measurements for the temperature and the wind speed help to improve the prediction for the precipitation. We found that by taking into account the temperature and the wind speed, the 1-h-ahead time series prediction for the precipitation has been improved significantly. Thus, the finding in Fig. 13 seems appropriate.