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| ## Tracking results {#sec-sec_3_1_Tracking_Results} | |
| In this section, some outputs of tracking data and workflow, described in subsection [-@sec-subsec_2_3_1_Tracking_Workflow], shall be presented . | |
| After that, in short, the current \gls{cnmc} shall be compared to *first CNMc* \newline | |
| First, two illustrative solutions for the assignment problem from the final path, as explained in subsection [-@sec-subsec_2_3_1_Tracking_Workflow], are provided in figures @fig-fig_27 and @fig-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 @fig-fig_27 and @fig-fig_28, the corresponding $\beta_i$ and $\beta_j$ centroid labels are given in the respective two figures, i.e., @fig-fig_27_1, @fig-fig_27_2 and @fig-fig_28_1, @fig-fig_28_2 . | |
| ::: {#fig-fig_27_All layout="[ [1,1], [1] ]"} | |
| {#fig-fig_27_1} | |
| {#fig-fig_27_2} | |
| {#fig-fig_27} | |
| Illustrative solution for the assignment problem, $\beta_i =32.167,\, \beta_j = 33 ,\, K =10$ | |
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| 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 *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 | |
| ::: {#fig-fig_28_All layout="[ [1,1], [1] ]"} | |
| {#fig-fig_28_1} | |
| {#fig-fig_28_2} | |
| {#fig-fig_28} | |
| Illustrative solution for the assignment problem, $\beta_i =31.333,\, \beta_j = 32.167, \,K =10$ | |
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| Comparing figure @fig-fig_27 with the second example in figure @fig-fig_28, it can be observed that the chosen *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 *inter* $\beta$ centroid matches, it can be stated that there is a clear trend, i.e., the next best *inter* $\beta$ centroid match has a very high increase in its cost. | |
| For example, considering the following *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 *inter* $\beta$ match in figure in both depicted figures @fig-fig_27 and @fig-fig_28 . | |
| The key essence behind this finding is that for the chosen number of centroids $K$ of this dynamical model (Lorenz system @eq-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 *SLS* is depicted in figures @fig-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. | |
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| ::: {#fig-fig_29 layout="[ [1,1], [1,1] ]"} | |
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| Tracked states for *SLS*, $K = 10,\, \vec{\beta} = [\, 28, \, 28.333, \, 31.333, \, 31.14, \, 33 \, ]$ | |
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| The second model is the *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 @fig-fig_32 . | |
| It can be noted that $\beta = 24.75$ and $\beta = 30.648$ exhibit very similar results to the *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 | |
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| Tracked states for *LS20*, $K = 20,\, \vec{\beta} = [\, 24.75, \, 28.682, \, 30.648, \, 31.14, \, 31.14,$ $42.936, \, 53.75 \, ]$ | |
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| 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 *FW15*. | |
| Figures in @fig-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 | |
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| Tracked states for *FW15*, $K = 15,\, \vec{\beta} = [\, 8, \, 8.25, \, 10, \, 11 \, ]$ | |
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| 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 *LS20*, but also for other dynamical systems as illustratively shown with the center area of the *FW15* system for $\beta= 11$. | |
| \FloatBarrier | |