Spaces:
Running
Running
\section{Tracking results} | |
\label{sec_3_1_Tracking_Results} | |
In this section, some outputs of tracking data and workflow, described in subsection \ref{subsec_2_3_1_Tracking_Workflow}, shall be presented. | |
After that, in short, the current \gls{cnmc} shall be compared to \emph{first CNMc} \newline | |
First, two illustrative solutions for the assignment problem from the final path, as explained in subsection \ref{subsec_2_3_1_Tracking_Workflow}, are provided in figures \ref{fig_27} and \ref{fig_28}. | |
The axes are denoted as $c_k$ and $c_p$ and represent the labels of the $\beta_j$ and $\beta_i$ centroids, respectively. | |
The color bar on the right side informs about the euclidean distance, which is equivalent to the cost. | |
Above the solution of the assignment problem in figures \ref{fig_27} and \ref{fig_28}, the corresponding $\beta_i$ and $\beta_j$ centroid labels are given in the respective two figures, i.e., \ref{fig_27_1}, \ref{fig_27_2} and \ref{fig_28_1}, \ref{fig_28_2}. | |
\begin{figure}[!h] | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{Ordered state, $\beta_i =32.167$ } | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/16_lb_32.167.pdf} | |
\label{fig_27_1} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{Ordered state, $\beta_j = 33$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/17_lb_33.000.pdf} | |
\label{fig_27_2} | |
\end{subfigure} | |
\smallskip | |
\centering | |
\begin{subfigure}{\textwidth} | |
\caption{Solution to the assignment problem} | |
\includegraphics[width =\textwidth]{2_Figures/3_Task/1_Tracking/1_LSA.pdf} | |
\label{fig_27} | |
\end{subfigure} | |
\vspace{-0.3cm} | |
\caption{Illustrative solution for the assignment problem, $\beta_i =32.167,\, \beta_j = 33 ,\, K =10$} | |
\label{fig_27_All} | |
\end{figure} | |
% | |
% | |
The centroid $c_{k=1} (\beta_j = 33)$ has its lowest cost to | |
$c_{p=3} (\beta_i = 32.167)$. In this case, this is also the solution for the assignment problem, outlined by the blue cross. | |
However, the solution to the linear sum assignment problem is not always to choose the minimal cost for one \emph{inter} $\beta$ match. | |
It could be that one centroid in $\beta_i$ is to found the closest centroid to multiple centroids in $\beta_j$. | |
Matching only based on the minimal distance does not include the restriction that exactly one centroid from $\beta_i$ must be matched with exactly one centroid from $\beta_j$. | |
The latter demand is incorporated in the solution of the linear sum assignment problem. \newline | |
\begin{figure}[!h] | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{Ordered state, $\beta_i =31.333$ } | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/18_lb_31.333.pdf} | |
\label{fig_28_1} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{Ordered state, $\beta_j = 32.167$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/16_lb_32.167.pdf} | |
\label{fig_28_2} | |
\end{subfigure} | |
\smallskip | |
\centering | |
\begin{subfigure}{\textwidth} | |
\caption{Solution to the assignment problem} | |
\includegraphics[width =\textwidth]{2_Figures/3_Task/1_Tracking/2_LSA.pdf} | |
\label{fig_28} | |
\end{subfigure} | |
\vspace{-0.3cm} | |
\caption{Illustrative solution for the assignment problem, $\beta_i =31.333,\, \beta_j = 32.167, \,K =10 $} | |
\label{fig_28_All} | |
\end{figure} | |
Comparing figure \ref{fig_27} with the second example in figure \ref{fig_28}, it can be observed that the chosen \emph{inter} $\beta$ centroid matches can have very different shapes. | |
This can be seen by looking at the blue crosses. | |
Furthermore, paying attention to the remaining possible \emph{inter} $\beta$ centroid matches, it can be stated that there is a clear trend, i.e., the next best \emph{inter} $\beta$ centroid match has a very high increase in its cost. | |
For example, considering the following \emph{inter} $\beta$ match. With $c_{k=1} (\beta_j = 32.167)$ and $c_{p=1} (\beta_i = 31.333)$, the minimal cost is around $cost_{min} \approx 0.84$. The next best option jumps to $cost_{second} = 13.823$. These jumps can be seen for each \emph{inter} $\beta$ match in figure in both depicted figures \ref{fig_27} and \ref{fig_28}. | |
The key essence behind this finding is that for the chosen number of centroids $K$ of this dynamical model (Lorenz system \eqref{eq_6_Lorenz}), no ambiguous regions, as explained at the beginning of this chapter, occur.\newline | |
Next, the tracking result of 3 different systems shall be viewed. | |
The tracked state for \emph{SLS} is depicted in figures \ref{fig_29}. | |
In each of the figures, one centroid is colored blue that denotes | |
the first centroid in the sequence of the underlying trajectory. | |
Within the depicted range $\vec{\beta}$, it can be observed, that each label across the $\vec{\beta}$ is labeled as expected. | |
No single ambiguity or mislabeling can be seen. | |
In other words, it highlights the high performance of the tracking algorithm. | |
% | |
% | |
% ============================================================================== | |
% ======================= SLS ================================================= | |
% ============================================================================== | |
\begin{figure}[!h] | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{$\beta =28$ } | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/3_lb_28.000.pdf} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{ $\beta = 28.833$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/4_lb_28.833.pdf} | |
\end{subfigure} | |
\smallskip | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{$\beta = 31.333$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/15_lb_31.333.pdf} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{ $\beta = 33$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/5_lb_33.000.pdf} | |
\end{subfigure} | |
\vspace{-0.3cm} | |
\caption{Tracked states for \emph{SLS}, $K = 10,\, \vec{\beta} = [\, 28, \, 28.333, \, 31.333, \, 31.14, \, 33 \, ]$} | |
\label{fig_29} | |
\end{figure} | |
% ============================================================================== | |
% ======================= SLS ================================================= | |
% ============================================================================== | |
% | |
The second model is the \emph{LS20}, i.e, $K= 20,\, \vec{\beta }_{tr} = [\, \beta_0 = 24.75 ; \, \beta_{end} = 53.75 \,], \, n_{\beta,tr} = 60$. | |
The outcome is depicted in figures \ref{fig_32}. | |
It can be noted that $\beta = 24.75$ and $\beta = 30.648$ exhibit very similar results to the \emph{SLS} model. | |
The same is true for intermediate $\beta$ values, i.e., $24.75 \leq \beta \lessapprox 30.648 $. | |
However, with $\beta \gtrapprox 30.64$ as depicted for $\beta = 31.14$, one centroid, i.e. the centroid with the label $20$ in the right ear appears unexpectedly. | |
With this, a drastic change to the centroid placing network is imposed. | |
Looking at the upcoming $\beta$ these erratic changes are found again.\newline | |
% ============================================================================== | |
% ======================= LS20 ================================================= | |
% ============================================================================== | |
\begin{figure}[!h] | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{$\beta =24.75$ } | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/6_lb_24.750.pdf} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{ $\beta = 28.682$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/7_lb_28.682.pdf} | |
\end{subfigure} | |
\smallskip | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{$\beta = 30.648$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/7_lb_30.648.pdf} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{ $\beta = 31.140$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/8_lb_31.140.pdf} | |
\end{subfigure} | |
\smallskip | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{$\beta = 42.936$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/9_lb_42.936.pdf} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{ $\beta = 53.750$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/10_lb_53.750.pdf} | |
\end{subfigure} | |
\vspace{-0.3cm} | |
\caption{Tracked states for \emph{LS20}, $K = 20,\, \vec{\beta} = [\, 24.75, \, 28.682, \, 30.648, \, 31.14, \, 31.14,$ $42.936, \, 53.75 \, ]$ } | |
\label{fig_32} | |
\end{figure} | |
% ============================================================================== | |
% ======================= LS20 ================================================= | |
% ============================================================================== | |
Generating a tracked state with these discontinuous cluster network deformations even manually can be considered hard to impossible because tracking demands some kind of similarity. | |
If two cluster networks differ too much from each other, then necessarily at least tracked label is going to be unsatisfying. | |
Hence, it would be wrong to conclude that the tracking algorithm is not performing well, but rather the clustering algorithm itself or the range of $\vec{\beta} $ must be adapted. If the range of $\vec{\beta} $ is shortened, multiple models can be trained and tracked.\newline | |
\FloatBarrier | |
The third model is referred to as \emph{FW15}. | |
Figures in \ref{fig_38} show the tracked state for 4 different $\beta$ values. It can be observed that for $\beta = 11$ the centroid placing has changed notably to the other $\beta$ values, thus tracking the centroids in the center for $\beta = 11$ becomes unfavorable. | |
Overall, however, the tracked state results advocate the performance of the tracking algorithm.\newline | |
% ============================================================================== | |
% ======================= FW15 ================================================= | |
% ============================================================================== | |
\begin{figure}[!h] | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{$\beta =8$ } | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/11_lb_8.000.pdf} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{ $\beta = 8.25$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/12_lb_8.250.pdf} | |
\end{subfigure} | |
\smallskip | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{$\beta = 10$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/13_lb_10.000.pdf} | |
\end{subfigure} | |
\hfill | |
\begin{subfigure}{0.5\textwidth} | |
\centering | |
\caption{ $\beta = 11$} | |
\includegraphics[width =\textwidth] | |
{2_Figures/3_Task/1_Tracking/14_lb_11.000.pdf} | |
\end{subfigure} | |
\vspace{-0.3cm} | |
\caption{Tracked states for \emph{FW15}, $K = 15,\, \vec{\beta} = [\, 8, \, 8.25, \, 10, \, 11 \, ]$} | |
\label{fig_38} | |
\end{figure} | |
% ============================================================================== | |
% ======================= FW15 ================================================= | |
% ============================================================================== | |
It can be concluded that the tracking algorithm performs remarkably well. However, when the cluster placing network is abruptly changed from one $\beta$ to the other $\beta$, the tracking outcome gets worse and generates sudden cluster network deformation. | |
As a possible solution, splitting up the $\vec{\beta}_{tr}$ range into smaller $\vec{\beta}_{tr,i}$ ranges, can be named. This is not only seen for the \emph{LS20}, but also for other dynamical systems as illustratively shown with the center area of the \emph{FW15} system for $\beta= 11$. | |
\FloatBarrier | |