16 CNMc predictions
In this section, some representative outputs for the CNMccontrol-oriented Cluster-based Network Modeling predicted trajectories shall be discussed. For that, first, the quality measurement abilities implemented in CNMccontrol-oriented Cluster-based Network Modeling are elaborated. Next, the model SLS is analyzed and explained in detail in the subsection 16.0.1. Finally, the outcome for other models shall be presented briefly in subsection 16.0.2.
There are several methods implemented in CNMccontrol-oriented Cluster-based Network Modeling to assess the quality of the predicted trajectories. The first one is the autocorrelation, which will be calculated for all \(\vec{\beta}_{unseen}\) and all provided \(\vec{L}\), for the true, CNMCluster-based Network Modeling and CNMccontrol-oriented Cluster-based Network Modeling predicted trajectories. As usual, the output is plotted and saved as HTML files for a feature-rich visual inspection. For qualitative assessment, the MAE errors are calculated for all \(\vec{\beta}_{unseen}\) and \(\vec{L}\) for two sets. The first set consists of the MAE errors between the true and the CNMCluster-based Network Modeling predicted trajectories. The second set contains the MAE errors between the true and the CNMccontrol-oriented Cluster-based Network Modeling predicted trajectories. Both sets are plotted as MAE errors over \(L\) and stored as HTML files. Furthermore, the one \(L\) value which exhibits the least MAE error is printed in the terminal and can be found in the log file as well.
The second technique is the CPDCluster Probability Distribution, which will also be computed for all the 3 trajectories, i.e., true, CNMCluster-based Network Modeling and CNMccontrol-oriented Cluster-based Network Modeling predicted trajectories. The CPDCluster Probability Distribution depicts the probability of being at one centroid \(c_i\). For each \(\vec{\beta}_{unseen}\) and all \(L\) the CPDCluster Probability Distribution is plotted and saved. The third method displays all the 3 trajectories in the state space. Moreover, the trajectories are plotted as 2-dimensional graphs, i.e., each axis as a subplot over the time \(t\). The final method calculates the MAE errors of the \(\boldsymbol Q / \boldsymbol T\) tensors for all \(L\).
The reason why more than one quality measurement method is integrated into CNMccontrol-oriented Cluster-based Network Modeling is that CNMccontrol-oriented Cluster-based Network Modeling should be able to be applied to, among other dynamical systems, chaotic systems. The motion of the Lorenz system 7.1 is not as complex as of the, e.g., the Four Wing 7.4 . Nevertheless, the Lorenz system already contains quasi-random elements, i.e., the switching from one ear to the other cannot be captured exactly with a surrogate mode. However, the characteristic of the Lorenz system and other chaotic dynamical systems as well can be replicated. In order to prove the latter, more than one method to measure the prediction quality is required.
16.0.1 Assessment of SLS
In this subsection, the prediction capability for the SLS will be analyzed in detail. All the presented output is generated with SVDSingular Value Decomposition as the decomposition method and RFRandom Forest as the \(\boldsymbol Q / \boldsymbol T\) regressor.
The final objective of CNMccontrol-oriented Cluster-based Network Modeling is to capture the characteristics of the original trajectory. However, it is important to outline that CNMccontrol-oriented Cluster-based Network Modeling is trained with the CNMCluster-based Network Modeling predicted trajectories. Thus, the outcome of CNMccontrol-oriented Cluster-based Network Modeling highly depends on the ability of CNMCluster-based Network Modeling to represent the original data. Consequently, CNMccontrol-oriented Cluster-based Network Modeling can only be as effective as CNMCluster-based Network Modeling is in the first place, with the approximation of the true data. Figures 16.1 and 16.2 show the true, CNMCluster-based Network Modeling and CNMccontrol-oriented Cluster-based Network Modeling predicted trajectories and a focused view on the CNMCluster-based Network Modeling and CNMccontrol-oriented Cluster-based Network Modeling trajectories, respectively. The output was generated for \(\beta_{unseen} = 28.5\) and \(L =1\). First, it can be observed that CNMCluster-based Network Modeling is not able to capture the full radius of the Lorenz attractor. This is caused by the low chosen number of centroids \(K=10\). Furthermore, as mentioned at the beginning of this chapter, the goal is not to replicate the true data one-to-one, but rather to catch the significant behavior of any dynamic system. With the low number of centroids \(K\), CNMCluster-based Network Modeling extracts the characteristics of the Lorenz system well. Second, the other aim for CNMccontrol-oriented Cluster-based Network Modeling is to match the CNMCluster-based Network Modeling data as closely as possible. Both figures 16.1 and 16.2 prove that CNMccontrol-oriented Cluster-based Network Modeling has fulfilled its task very well.
SLS, \(\beta_{unseen}=28.5,\, L=1\), true, CNMCluster-based Network Modeling and CNMccontrol-oriented Cluster-based Network Modeling predicted trajectories
A close-up of the movement of the different axes is shown in the picture 16.3 . Here, as well, the same can be observed as described above. Namely, the predicted CNMccontrol-oriented Cluster-based Network Modeling trajectory is not a one-to-one reproduction of the original trajectory. However, the characteristics, i.e., the magnitude of the motion in all 3 directions (x, y, z) and the shape of the oscillations, are very similar to the original trajectory. Note that even though the true and predicted trajectories are utilized to assess, whether the characteristical behavior could be extracted, a single evaluation based on the trajectories is not sufficient and often not advised or even possible. In complex systems, trajectories can change rapidly while dynamical features persist (Fernex, Semaan, and Noack 2021). In CNMccontrol-oriented Cluster-based Network Modeling the predicted trajectories are obtained through the CNMCluster-based Network Modeling propagation, which itself is based on a probabilistic model, i.e. the \(\boldsymbol Q\) tensor. Thus, matching full trajectories becomes even more unrealistic. The latter two statements highlight yet again that more than one method of measuring quality is needed. To further support the generated outcome the autocorrelation and CPDCluster Probability Distribution in figure 16.4 and 16.5, respectively, shall be considered. It can be inspected that the CNMCluster-based Network Modeling and CNMccontrol-oriented Cluster-based Network Modeling autocorrelations are matching the true autocorrelation in the shape favorably well. Nonetheless, the degree of reflecting the magnitude fully decreases quite fast. Considering the CPDCluster Probability Distribution, it can be recorded that the true CPDCluster Probability Distribution could overall be reproduced satisfactorily.
SLS, \(\beta_{unseen}= 28.5, \, L =1\), autocorrelation and CPDCluster Probability Distribution for true, CNMCluster-based Network Modeling and CNMccontrol-oriented Cluster-based Network Modeling predicted trajectories
To illustrate the influence of \(L\), figure 16.6 shall be viewed. It depicts the MAE error for the true and CNMccontrol-oriented Cluster-based Network Modeling predicted trajectories for \(\beta_{unseen}= [\, 28.5,\, 32.5 \, ]\) with \(L\) up to 7. It can be observed that the choice of \(L\) has an impact on the prediction quality measured by autocorrelation. For \(\beta_{unseen}=28.5\) and \(\beta_{unseen}=32.5\), the optimal \(L\) values are \(L = 2\) and \(L = 7\), respectively. To emphasize it even more that with the choice of \(L\) the prediction quality can be regulated, figure 16.7 shall be considered. It displays the 3 autocorrelations for \(L = 7\). Matching the shape of the true autocorrelation was already established with \(L =1\) as shown in figure 16.4 . In addition to that, \(L=7\) improves by matching the true magnitude. Finally, it shall be mentioned that similar results have been accomplished with other \(K\) tested values, where the highest value was \(K =50\).
16.0.2 Results of further dynamical systems
In this subsection, the prediction results for other models will be displayed. The chosen dynamical systems with their configurations are the following.All the presented outputs were generated with SVDSingular Value Decomposition as the decomposition method and RFRandom Forest as the \(\boldsymbol Q / \boldsymbol T\) regressor. Furthermore, the B-spline interpolation in the propagation step of CNMCluster-based Network Modeling was replaced with linear interpolation. The B-spline interpolation was originally utilized for smoothing the motion between two centroids. However, it was discovered for a high number of \(K\), the B-spline interpolation is not able to reproduce the motion between two centroids accurately, but rather would impose unacceptable high deviations or oscillations into the predictions. This finding is also mentioned in (Pierzyna 2021) and addressed as one of first CNMc’s limitations.
Two illustrative examples of the unacceptable high deviations caused by the B-spline interpolation are given in figures 16.8 and 16.9 . The results are obtained for LS20 for \(\beta = 31.75\) and \(\beta = 51.75\) with \(L=3\). In figures 16.8 (a) and 16.8 (b) it can be inspected that the B-spline interpolation has a highly undesired impact on the predicted trajectories. In Contrast to that, in figures, 16.8 (c) and 16.8 (d), where linear interpolation is utilized, no outliers are added to the predictions. The impact of the embedded outliers, caused by the B-spline interpolation, on the autocorrelation is depicted in figures 16.9 (a) and 16.9 (b) . The order of the deviation between the true and the CNMccontrol-oriented Cluster-based Network Modeling predicted autocorrelation can be grasped by inspecting the vertical axis scale. Comparing it with the linear interpolated autocorrelations, shown in figures 16.9 (c) and 16.9 (d), it can be recorded that the deviation between the true and predicted autocorrelations is significantly lower than in the B-spline interpolation case.
Nevertheless, it is important to highlight that the B-spline interpolation is only a tool for smoothing the motion between two centroids. The quality of the modeled \(\boldsymbol Q / \boldsymbol T\) cannot be assessed directly by comparing the trajectories and the autocorrelations. To stress that the CPDCluster Probability Distribution in figure 16.10 and 16.11 shall be considered. It can be observed that CPDCluster Probability Distribution does not represent the findings of the autocorrelations, i.e., the true and predicted behavior agree acceptably overall. This is because the type of interpolation has no influence on the modeling of the probability tensor \(\boldsymbol Q\). Thus, the outcome with the B-spline interpolation should not be regarded as an instrument that enables making assumptions about the entire prediction quality of CNMccontrol-oriented Cluster-based Network Modeling. The presented points underline again the fact that more than one method should be considered to evaluate the prediction quality of CNMccontrol-oriented Cluster-based Network Modeling.
Illustrative the B-spline interpolation and its impact on the CPDsCluster Probability Distributions, LS20, \(\beta = 31.75\) and \(\beta =51.75\), \(L=3\)
The results generated with the above mentioned linear interpolation for FW50, Rössler15 and TS15 are depicted in figures 16.12 to 16.14, respectively. Each of them consists of an illustrative trajectory, 3D and 2D trajectories, the autocorrelations, the CPDCluster Probability Distribution and the MAE error between the true and CNMccontrol-oriented Cluster-based Network Modeling predicted trajectories for a range of \(\vec{L}\) and some \(\vec{\beta}_{unseen}\). The illustrative trajectory includes arrows, which provide additional information. First, the direction of the motion, then the size of the arrows represents the velocity of the system. Furthermore, the change in the size of the arrows is equivalent to a change in the velocity, i.e., the acceleration. Systems like the TS15 exhibit a fast change in the size of the arrows, i.e., the acceleration is nonlinear. The more complex the behavior of the acceleration is, the more complex the overall system becomes. The latter statement serves to emphasize that CNMccontrol-oriented Cluster-based Network Modeling can be applied not only to rather simple systems such as the Lorenz attractor (Lorenz 1963), but also to more complex systems such as the FW50 and TS15.
All in all, the provided results for the 3 systems are very similar to those already explained in the previous subsection 16.0.1. Thus, the results presented are for demonstration purposes and will not be discussed further. However, the 3 systems also have been calculated with different values for \(K\). For FW50, the range of \(\vec{K}= [\, 15, \, 30, \, 50 \, ]\) was explored with the finding that the influence of \(K\) remained quite small. For Rössler15 and TS15, the ranges were chosen as \(\vec{K}= [\, 15, \, 30, \, 100\,]\) and \(\vec{K}= [\, 15, \, 75 \,]\), respectively. The influence of \(K\) was found to be insignificant also for the latter two systems.
