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\subsection{Assessment of SLS} | |
\label{subsec_3_5_1_SLS} | |
In this subsection, the prediction capability for the \emph{SLS} will be analyzed in detail. All the presented output is generated with \gls{svd} as the decomposition method and \gls{rf} as the $\bm Q / \bm T$ regressor.\newline | |
The final objective of \gls{cnmc} is to capture the characteristics of the original trajectory. | |
However, it is important to outline that \gls{cnmc} is trained with the \gls{cnm} predicted trajectories. | |
Thus, the outcome of \gls{cnmc} highly depends on the ability of \gls{cnm} to represent the original data. | |
Consequently, \gls{cnmc} can only be as effective as \gls{cnm} is in the first place, with the approximation of the true data. | |
Figures \ref{fig_72} and \ref{fig_73} show the true, \gls{cnm} and \gls{cnmc} predicted trajectories and a focused view on the \gls{cnm} and \gls{cnmc} trajectories, respectively. | |
The output was generated for $\beta_{unseen} = 28.5$ and $L =1$. | |
First, it can be observed that \gls{cnm} 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 $K$, \gls{cnm} extracts the characteristics of the Lorenz system well. | |
Second, the other aim for \gls{cnmc} is to match the \gls{cnm} data as closely as possible. | |
Both figures \ref{fig_72} and \ref{fig_73} prove that \gls{cnmc} has fulfilled its task very well. \newline | |
\begin{figure}[!h] | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{True, \gls{cnm} and \gls{cnmc} predicted trajectories} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/4_SLS/0_lb_28.5_All.pdf} | |
\label{fig_72} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{\gls{cnm} and \gls{cnmc} predicted trajectories} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/4_SLS/1_lb_28.5.pdf} | |
\label{fig_73} | |
\end{subfigure} | |
\vspace{-0.3cm} | |
\caption{\emph{SLS}, $\beta_{unseen}=28.5,\, L=1$, true, \gls{cnm} and \gls{cnmc} predicted trajectories} | |
\end{figure} | |
A close-up of the movement of the different axes is shown in the picture \ref{fig_74}. | |
Here, as well, the same can be observed as described above. Namely, the predicted \gls{cnmc} 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 \cite{Fernex2021a}. | |
In \gls{cnmc} the predicted trajectories are obtained through the \gls{cnm} propagation, which itself is based on a probabilistic model, i.e. the $\bm Q$ 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 \gls{cpd} in figure \ref{fig_75} and \ref{fig_76}, respectively, shall be considered. | |
It can be inspected that the \gls{cnm} and \gls{cnmc} autocorrelations are matching the true autocorrelation in the shape favorably well. | |
Nonetheless, the degree of reflecting the magnitude fully decreases quite fast. | |
Considering the \gls{cpd}, it can be recorded that the true \gls{cpd} could overall be reproduced satisfactorily.\newline | |
\begin{figure}[!h] | |
\centering | |
\includegraphics[width =0.75\textwidth] | |
{2_Figures/3_Task/4_SLS/2_lb_28.5_3V_All.pdf} | |
\caption{\emph{SLS}, $\beta_{unseen}=28.5, \, L=1$, true, \gls{cnm} and \gls{cnmc} predicted trajectories as 2d graphs } | |
\label{fig_74} | |
\end{figure} | |
\begin{figure}[!h] | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{autocorrelation} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/4_SLS/3_lb_3_all_28.5.pdf} | |
\label{fig_75} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{\gls{cpd}} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/4_SLS/4_lb_28.5.pdf} | |
\label{fig_76} | |
\end{subfigure} | |
\vspace{-0.3cm} | |
\caption{\emph{SLS}, $\beta_{unseen}= 28.5, \, L =1$, autocorrelation and \gls{cpd} for true, \gls{cnm} and \gls{cnmc} predicted trajectories} | |
\end{figure} | |
To illustrate the influence of $L$, figure \ref{fig_77} shall be viewed. | |
It depicts the MAE error for the true and \gls{cnmc} 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 \ref{fig_78} 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 \ref{fig_75}. 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 $K =50$. | |
\begin{figure}[!h] | |
\begin{minipage}{0.47\textwidth} | |
\centering | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/4_SLS/5_lb_1_Orig_CNMc.pdf} | |
\caption{\emph{SLS}, MAE error for true and \gls{cnmc} predicted autocorrelations for $\beta_{unseen}= [\, 28.5,$ $32.5 \, ]$ and different values of $L$} | |
\label{fig_77} | |
\end{minipage} | |
\hfill | |
\begin{minipage}{0.47\textwidth} | |
\centering | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/4_SLS/6_lb_3_all_32.5.pdf} | |
\caption{\emph{SLS}, $\beta_{unseen}=32.5, \, L=7$, \gls{cnm} and \gls{cnmc} predicted autocorrelation } | |
\label{fig_78} | |
\end{minipage} | |
\end{figure} | |
\FloatBarrier | |