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| ### Results of further dynamical systems {#sec-subsec_3_5_2_Models} | |
| In this subsection, the \gls{cnmc} prediction results for other models will be displayed. | |
| The chosen dynamical systems with their configurations are the following. | |
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| \begin{enumerate} | |
| \item *FW50*, based on the *Four Wing* set of equations @eq-eq_10_4_Wing with $K=50, \, \vec{\beta }_{tr} = [\, \beta_0 = 8 ; \, \beta_{end} = 11 \,], \, n_{\beta, tr} = 13$. | |
| \item *Rössler15*, based on the *Rössler* set of equations @eq-eq_7_Ross with $K=15, \, \vec{\beta }_{tr} = [\, \beta_0 = 6 ; \, \beta_{end} = 13 \,], \, n_{\beta, tr} = 15$. | |
| \item *TS15*, based on the *Two Scroll* set of equations @eq-eq_9_2_Scroll with $K=15, \, \vec{\beta }_{tr} = [\, \beta_0 = 5 ; \, \beta_{end} = 12 \,], \, n_{\beta, tr} = 15$. | |
| \end{enumerate} | |
| All the presented outputs were generated with \gls{svd} as the decomposition method and \gls{rf} as the $\boldsymbol Q / \boldsymbol T$ regressor. | |
| Furthermore, the B-spline interpolation in the propagation step of \gls{cnm} 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 [@Max2021] 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 @fig-fig_82_Traject and @fig-fig_82_Autocorr . | |
| The results are obtained for *LS20* for $\beta = 31.75$ and $\beta = 51.75$ with $L=3$. | |
| In figures @fig-fig_82_Traj_B and @fig-fig_83_Traj_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, @fig-fig_82_Traj_L and @fig-fig_83_Traj_L, 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 @fig-fig_82_Auto_B and @fig-fig_83_Auto_B . | |
| The order of the deviation between the true and the \gls{cnmc} predicted autocorrelation can be grasped by inspecting the vertical axis scale. | |
| Comparing it with the linear interpolated autocorrelations, shown in figures @fig-fig_82_Auto_L and @fig-fig_83_Auto_L, it can be recorded that the deviation between the true and predicted autocorrelations is significantly lower than in the B-spline interpolation case. | |
| \newline | |
| 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 \gls{cpd} in figure @fig-fig_82_CPD_B and @fig-fig_83_CPD_B shall be considered. | |
| It can be observed that \gls{cpd} 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 \gls{cnmc}. The presented points underline again the fact that more than one method should be considered to evaluate the prediction quality of \gls{cnmc}. | |
| \newline | |
| :::{#fig-fig_82_Traject layout="[[1,1],[1,1]]"} | |
| {#fig-fig_82_Traj_B} | |
| {#fig-fig_83_Traj_B} | |
| {#fig-fig_82_Traj_L} | |
| {#fig-fig_83_Traj_L} | |
| Illustrative undesired oscillations cased by the B-spline interpolation and its impact on the predicted trajectory contrasted with linear interpolation, *LS20*, $\beta = 31.75$ and $\beta =51.75$, $L=3$ | |
| ::: | |
| <!--%----------------------------------- AUTOCOR -------------------------------------> | |
| :::{#fig-fig_82_Autocorr layout="[[1,1],[1,1]]"} | |
| {#fig-fig_82_Auto_B} | |
| {#fig-fig_83_Auto_B} | |
| {#fig-fig_82_Auto_L} | |
| {#fig-fig_83_Auto_L} | |
| Illustrative undesired oscillations cased by the B-spline interpolation and its impact on the predicted autocorrelations contrasted with linear interpolation, *LS20*, $\beta = 31.75$ and $\beta =51.75$, $L=3$ | |
| ::: | |
| <!-- % ------------- CPD ------------------------> | |
| :::{layout="[[1,1],[1,1]]"} | |
| {#fig-fig_82_CPD_B} | |
| {#fig-fig_83_CPD_B} | |
| Illustrative the B-spline interpolation and its impact on the \glspl{cpd}, *LS20*, $\beta = 31.75$ and $\beta =51.75$, $L=3$ | |
| ::: | |
| \FloatBarrier | |
| The results generated with the above mentioned linear interpolation for *FW50*, *Rössler15* and *TS15* are depicted in figures @fig-fig_79 to @fig-fig_81, respectively. | |
| Each of them consists of an illustrative trajectory, 3D and 2D trajectories, the autocorrelations, the \gls{cpd} and the MAE error between the true and \gls{cnmc} 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 \gls{cnmc} can be applied not only to rather simple systems such as the Lorenz attractor [@lorenz1963deterministic], but also to more complex systems such as the *FW50* and *TS15*.\newline | |
| All in all, the provided results for the 3 systems are very similar to those already explained in the previous subsection [-@sec-subsec_3_5_1_SLS]. | |
| 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. | |
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| :::{#fig-fig_79 layout="[[1,1],[1,1], [1,1]]"} | |
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| Results for *FW50*, $\beta_{unseen} = 8.1, \, L= 2$ | |
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| :::{#fig-fig_80 layout="[[1,1],[1,1], [1,1]]"} | |
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| Results for *Rössler15*, $\beta_{unseen} = 9.6,\, L =1$ | |
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| :::{#fig-fig_81 layout="[[1,1],[1,1], [1,1]]"} | |
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| Results for *TS15*, $\beta_{unseen} = 5.1,\, L =2$ | |
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