3.3 State-Space Reconstruction
3.3.3 Application Examples and Discussion
The presented approach has been tested on several higher dimensional chaotic time series and gave convincing results. In each example, the states of the original system were reconstructed from 15000 samples of the first state variable,y =x1.
According to our experience, the presented approach is quite robust with respect to the choice of the clustering algorithm parameters. In all examples, the number of clusters isc= 20, the termination tolerance²= 10−4and the weighting exponentm= 2. The applied threshold is equal to 0.93 according to [160] (see (3.56)).
Example 3.4 (State-space reconstruction of the R¨ossler attractor). The following three dif-ferential equations define this system:
˙
x1 = −(x2+x3) (3.57)
˙
x2 = x1+ax2 (3.58)
˙
x3 = b+x3(x1−c) (3.59)
The initial conditions are x1(0) = 0,x2(0) = 0 andx3(0) = 0 and the parameters used are a= 0.38, b = 0.3 and c = 4.5, according to [89]. Under these conditions, the trajectories are as depicted in Figure 3.7 (a). The variabley=x1is shown in the time domain in Figure 3.7 (b). The lag time was 20τ in this case whereτ is the sampling timeτ= 0.05. The lag time was chosen as the first minimum of the average mutual information function [61, 89].
Figure 3.8 shows that the proposed index based on the one-step-ahead prediction error correctly reflects the embedding dimension of the original three-dimensional system, for both the MIMO and MISO method. After the embedding dimension has been chosen, in this case it is de = 3, the local dimension,dl can be estimated based on the screeplot (see Section 3.3.2). In Figure 3.8 the subfigures (c) and (d) show that the local dimension is equal to 2 because the first two eigenvalues weighted by the a prioriprobability of the clusters contain 99.8% of the total variance. The Correlation dimension of the analyzed 15000 data is dC = 1.9838 based on the result of the TSTOOL Toolbox, which is a software package for signal processing with emphasis on nonlinear time-series analysis [125]. Since the correlation dimension is a global measure of the local dimension (the calculation of the correlation dimension does not use the local approximation of the reconstruction space), the agreement of the results of the two approaches with the visual inspection of phase space (see Figure 3.9) indicate that the presented method gives good results in terms of the estimation of the dimensionality of the system.
Furthermore, the identified MIMO and MISO models give excellent prediction performance (see Figure 3.9), the original and the predicted data take the same subspace. The predicted data were generated by a free run simulation of 5000 samples based on the identified MIMO model. It can be
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5 10
15
−20
−10 0 10
0 10 20 30 40 50 60
x1 x2
x3
(a) Trajectories in the original state space in case of the R¨ossler system.
0 50 100 150 200 250 300 350
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−5 0 5 10 15
t x1
(b) The variabley=x1 in time in case of the R¨ossler system.
Figure 3.7: Original data in case of the the R¨ossler system.
seen that although the system is chaotic, the trajectories are similar, so the approximation is very good.
¤
1 2 3 4 5 6
(a) Estimation of the embedding dimensionde
based on MIMO approach.
(b) Estimation of the embedding dimensionde
based on MISO approach.
(c) Estimation of the local dimension dl based on MIMO approach forde= 3.
(d) Estimation of the local dimensiondlbased on MISO approach forde= 3.
Figure 3.8: Estimation ofde anddl of the reconstruction space for the R¨ossler system.
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Figure 3.9: Prediction performance for the R¨ossler system. Cluster centers are denoted by diamonds.
Example 3.5 (State-space reconstruction of the Lorenz system). The Lorenz system, de-scribing the thermal driving of convection in the lower atmosphere, is defined by the following three differential equations:
˙
x1 = σ(x2−x1) (3.60)
˙
x2 = x1(r−x3)−x2 (3.61)
˙
x3 = x1x2−bx3 (3.62)
where the variablesx1,x2andx3are proportional to the intensity of the convection rolls, the horizontal temperature variation, and the vertical temperature variation, respectively, and the parameters σ, b and r are constants representing the properties of the system. The initial conditions are x1(0) = 0, x2(0) = 1andx3(0) = 1 and the parameter values used arer= 28,b=83 andσ= 10, following [172].
As shown in Figure 3.10, the results as good as in the case of the R¨ossler system, and the same statements can be given. The Correlation dimension of the data set isdC= 2.0001.
¤
1 2 3 4 5 6
(a) Estimation of the embedding dimensionde
based on MIMO approach.
(b) Estimation of the embedding dimensionde
based on MISO approach.
(c) Estimation of the local dimension dl based on MIMO approach forde= 3.
(d) Estimation of the local dimensiondlbased on MISO approach forde= 3.
Figure 3.10: Estimation ofdeanddl dimensions of the reconstruction space for the Lorenz system
Example 3.6 (State-space reconstruction of a four dimensional system). In this example let us consider the four dimensional system published by Yao [179]. The system equations are
˙
x1 = x3 (3.63)
˙
x2 = x4 (3.64)
˙
x3 = −(α+x22)x1+x2 (3.65)
˙
x4 = −(β+x21)x2+x1. (3.66)
With α= 0.1, β = 0.101 and the initial conditionx1(0) = 0.1, x2(0) =−0.1, x3(0) = 0.1, x4(0) =
−0.1, this system is highly chaotic [179]. The sampling rate was 0.05, and the lag time is τ = 50 estimated by the average mutual information function [89].
In Figure 3.11 one can see that the estimated embedding dimension is four. The correct embedding dimension can be found by comparing the ratio of the neighboring M SE values. While M SE(de = 3)/M SE(de = 4) = 2.031, M SE(de = 4)/M SE(de = 5) = 1.2329 in subfigure (a) of Figure 3.11 (MIMO approach). The values in case of MISO method are similar. The screeplots forde= 4 can be seen in Figure 3.11. The local dimension is equal to 2 according to the applied threshold value, 0.93.
The Correlation dimension of the data set is dC= 2.0445.
In Figure 3.12 (a) the first three dimensions, in Figure 3.12 (b) the second three dimensions of the original (solid line) and the predicted (dashed line) four dimensional data can be seen given by a free run simulation of 5000 data. The prediction performance is as good as in the previous examples.
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1 2 3 4 5 6
(a) Estimation of the embedding dimensionde
based on MIMO approach
(b) Estimation of the embedding dimensionde
based on MISO approach
(c) Estimation of the local dimension dl based on MIMO approach forde= 4.
(d) Estimation of the local dimensiondlbased on MISO approach forde= 4.
Figure 3.11: Estimation of thedeanddldimensions of the reconstruction space for the four dimensional system.
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Figure 3.12: Prediction performance for the four dimensional chaotic system.