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| ## CPE modeling results {#sec-sec_3_2_MOD_CPE} | |
| In this section, results to the \gls{cpevol} modeling explained in subsection [-@sec-subsec_2_4_1_CPE], shall be presented and assessed . | |
| First, a selection of equations, which defines the \gls{cpevol} are given for one model configuration. | |
| Next, representative plots of the \gls{cpevol} for different models are analyzed. | |
| Finally, the predicted centroid position is compared with the actual clustered centroid position.\newline | |
| Modeling the *CPE* returns, among other results, analytical equations. | |
| These equations describe the behavior of the centroid positions across the range $\vec{\beta}$ and can also be used for making predictions for $\vec{\beta}_{unseen}$. | |
| The model configuration for which they are be presented is *SLS*, i.e. Lorenz system @eq-eq_6_Lorenz, $K= 10,\, \vec{\beta }_{tr} = [\, \beta_0 = 28 ; \, \beta_{end} =33 \,], \, n_{\beta, tr} = 7$. | |
| The analytical \gls{cpevol} expressions are listed in @eq-eq_27 to @eq-eq_29 for the centroids $[\,1,\, 2,\,7\,]$, respectively. | |
| Recalling that the behavior across the 3 different axes (x, y, z) can vary greatly, each axis has its own regression model $(\tilde x,\, \tilde y,\, \tilde z)$. | |
| Thus, for each label, 3 different analytical expressions are provided. \newline | |
| <!-- Minpages here --> | |
| ::: {layout="[[1],[1]]"} | |
| $$ | |
| \begin{equation} | |
| \begin{aligned} | |
| \tilde x &= -0.1661 \, cos(3 \, \beta) \\ | |
| \tilde y &= -0.1375 \, cos(3 \, \beta) \\ | |
| \tilde z &= 0.8326 \, \beta | |
| \end{aligned} | |
| \end{equation}$$ {#eq-eq_27} | |
| {#fig-fig_45} | |
| ::: | |
| ::: {layout="[[1], [1]]"} | |
| $$ | |
| \begin{equation} | |
| \begin{aligned} | |
| \tilde x &= 0.1543 \, sin(3 \, \beta) + 0.2446 \, cos(3 \, \beta) \\ | |
| \tilde y &= 0.2638 \, sin(3 \, \beta) + 0.4225 \, cos(3 \, \beta) \\ | |
| \tilde z &= 0.4877 \, \beta | |
| \end{aligned} | |
| \label{eq_28} | |
| \end{equation}$$ {#eq-eq_28} | |
| {#fig-fig_46} | |
| ::: | |
| ::: {layout="[[1], [1]]"} | |
| $$ | |
| \begin{equation} | |
| \begin{aligned} | |
| \tilde x &= -0.1866 \, \beta + 0.133 \, sin(3 \, \beta) \\ | |
| & \quad + 0.1411 \, cos(3 \, \beta) \\ | |
| \tilde y &= -0.3 \, \beta \\ | |
| \tilde z &= -1.0483+ 0.6358 \,\beta | |
| \end{aligned} | |
| \label{eq_29} | |
| \end{equation} $$ {#eq-eq_29} | |
| {#fig-fig_47} | |
| ::: | |
| Right to the equations the corresponding plots are depicted in figures @fig-fig_45 to @fig-fig_47 . | |
| Here, the blue and green curves indicate true and modeled CPE, respectively. | |
| Each of the figures supports the choice of allowing each axis to be modeled separately. | |
| The z-axis appears to undergo less alteration or to be more linear than the x- and y-axis. | |
| If a model is supposed to be valid for all 3 axes, a more complex model, i.e., a higher of terms, is required. | |
| Although more flexible models fit training data increasingly better, they tend to overfit. | |
| In other words, complex models capture the trained data well but could exhibit oscillations for $\vec{\beta}_{unseen}$. | |
| The latter is even more severe when the model is employed for extrapolation. | |
| The difference between interpolation and extrapolation is that for extrapolation the prediction is made with $\beta_{unseen}$ which are not in the range of the trained $\vec{\beta}_{tr}$. | |
| Therefore, less complexity is preferred.\newline | |
| Next, the performance of predicting the centroid for $\vec{\beta}_{unseen}$ is elaborated. | |
| For this purpose, figures @fig-fig_48 to @fig-fig_52 shall be examined. | |
| All figures depict the original centroid positions, which are obtained through the clustering step in green and the predicted centroid positions in blue. | |
| On closer inspection, orange lines connecting the true and predicted centroid positions can be identified. | |
| Note, that they will only be visible if the deviation between the true and predicted state is high enough. | |
| Figures @fig-fig_48_0 an @fig-fig_48_1 show the outcome for *SLS* with $\beta_{unseen} = 28.5$ and $\beta_{unseen} = 32.5$, respectively. | |
| Visually, both predictions are very close to the true centroid positions. | |
| Because of this high performance in showed in figures @fig-fig_49_0 and @fig-fig_49_1 two examples for extrapolation are given for $\beta_{unseen} = 26.5$ and $\beta_{unseen} = 37$, respectively. | |
| For the first one, the outcome is very applicable. | |
| In contrast, $\beta_{unseen} = 37$ returns some deviations which are notably high. | |
| \newline | |
| <!--% ----------------- Interpolation ------------------------------------------------> | |
| :::{#fig-fig_48 layout="[[1,1]]"} | |
| {#fig-fig_48_0} | |
| {#fig-fig_48_1} | |
| *SLS*, original vs. modeled centroid position, $\beta_{unseen} = 28.5$ and $\beta_{unseen} = 32.5$ | |
| ::: | |
| <!--% ----------------- EXTRAPOLATION ------------------------------------------------> | |
| :::{#fig-fig_49 layout="[[1,1]]"} | |
| {#fig-fig_49_0} | |
| {#fig-fig_49_1} | |
| *SLS*, original vs. modeled centroid position, extrapolated $\beta_{unseen} = 26.5$ and $\beta_{unseen} = 37$ | |
| ::: | |
| <!--% --------- MODEL LOrenz K= 20--> | |
| :::{#fig-fig_50 layout="[[1,1]]"} | |
|  | |
|  | |
| *LS20*, original vs. modeled centroid position, $\beta_{unseen} = 31.75$ and $\beta_{unseen} = 51.75$ | |
| ::: | |
| <!--% --------- MODEL 25_Four_Wing_1_K_15 -----------> | |
| :::{#fig-fig_52 layout="[[1,1]]"} | |
|  | |
|  | |
| *FW15*, original vs. modeled centroid position, $\beta_{unseen} = 8.7$ and $\beta_{unseen} = 10.1$ | |
| ::: | |
| Quantitative measurements are performed by applying the Mean Square Error (MSE) following equation @eq-eq_30_MSE . | |
| The variables are denoted as the number of samples $n$, which in this case is equal to the number of centroids $n = K$, the known $f(x_k)$ and the predicted $y_k$ centroid position.\newline | |
| $$ | |
| \begin{equation} | |
| MSE = \frac{1}{n} \, \sum_{i=1}^n \left(f(x_k) - y_k\right)^2 | |
| \label{eq_30_MSE} | |
| \end{equation} | |
| $$ {#eq-eq_30_MSE} | |
| The measured MSE errors for all displayed results are summarized in table @tbl-tab_5_MSE . | |
| The MSE for results of $\beta_{unseen} = 28.5$ and $\beta_{unseen} = 32.5$ in figures @fig-fig_48 is $0.622$ and $0.677$, respectively. | |
| Consequently, the performance of \gls{cnmc} is also confirmed quantitatively. | |
| Figures in @fig-fig_50 illustrate the outcome for *LS20* for $\beta_{unseen} = 31.75$ and $\beta_{unseen} = 51.75$. | |
| In section [-@sec-sec_3_1_Tracking_Results] it is explained that for *LS20* cluster network deformations appear . | |
| Nevertheless, the outcome visually and quantitatively endorses the *CPE* modeling capabilities. | |
| Figures in @fig-fig_52 depict the outcome for *FW15* for $\beta_{unseen} = 8.7$ and $\beta_{unseen} = 10.1$. | |
| A few orange lines are visible, however overall the outcome is very satisfactory.\newline | |
| **Figure** | **Model** | $\boldsymbol{\beta_{unseen}}$ | **MSE** | |
| -------------|-----------|-------------------------------|--------- | |
| @fig-fig_48 | *SLS* | $28.5$ | $0.622$ | |
| @fig-fig_48 | *SLS* | $32.5$ | $0.677$ | |
| @fig-fig_49 | *SLS* | $26.5$ | $1.193$ | |
| @fig-fig_49 | *SLS* | $37$ | $5.452$ | |
| @fig-fig_50 | *LS20* | $31.75$ | $1.857$ | |
| @fig-fig_50 | *LS20* | $51.75$ | $2.536$ | |
| @fig-fig_52 | *FW15* | $8.7$ | $1.617$ | |
| @fig-fig_52 | *FW15* | $10.1$ | $1.5$ | |
| : MSE for different Model configurations and $\vec{\beta}_{unseen}$ {#tbl-tab_5_MSE} | |
| It can be concluded that the *CPE* modeling performance is satisfying. | |
| In the case of a few cluster network deformations, \gls{cnmc} is capable of providing acceptable results. | |
| However, as shown with *SLS*, if the model's training range $\vec{\beta}_{tr}$ and the number of $K$ was selected appropriately, the MSE can be minimized. | |