13 CPE modeling results
In this section, results to the CPECentroid Position Evolution modeling explained in subsection 10.0.1, shall be presented and assessed . First, a selection of equations, which defines the CPECentroid Position Evolution are given for one model configuration. Next, representative plots of the CPECentroid Position Evolution for different models are analyzed. Finally, the predicted centroid position is compared with the actual clustered centroid position.
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 7.1, \(K= 10,\, \vec{\beta }_{tr} = [\, \beta_0 = 28 ; \, \beta_{end} =33 \,], \, n_{\beta, tr} = 7\). The analytical CPECentroid Position Evolution expressions are listed in 13.1 to 13.3 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.
\[ \begin{equation} \begin{aligned} \tilde x &= -0.1661 \, cos(3 \, \beta) \\ \tilde y &= -0.1375 \, cos(3 \, \beta) \\ \tilde z &= 0.8326 \, \beta \end{aligned} \end{equation} \tag{13.1}\]
\[ \begin{equation} \begin{aligned} \tilde x &= 0.1543 \, sin(3 \, \beta) + 0.2446 \, cos(3 \, \beta) \\ \tilde y &= 0.2638 \, sin(3 \, \beta) + 0.4225 \, cos(3 \, \beta) \\ \tilde z &= 0.4877 \, \beta \end{aligned} \label{eq_28} \end{equation} \tag{13.2}\]
\[ \begin{equation} \begin{aligned} \tilde x &= -0.1866 \, \beta + 0.133 \, sin(3 \, \beta) \\ & \quad + 0.1411 \, cos(3 \, \beta) \\ \tilde y &= -0.3 \, \beta \\ \tilde z &= -1.0483+ 0.6358 \,\beta \end{aligned} \label{eq_29} \end{equation} \tag{13.3}\]
Right to the equations the corresponding plots are depicted in figures 13.1 to 13.3 . 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.
Next, the performance of predicting the centroid for \(\vec{\beta}_{unseen}\) is elaborated. For this purpose, figures 13.4 to 13.7 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 13.4 (a) an 13.4 (b) 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 13.5 (a) and 13.5 (b) 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.
Quantitative measurements are performed by applying the Mean Square Error (MSE) following equation 13.4 . 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. \[ \begin{equation} MSE = \frac{1}{n} \, \sum_{i=1}^n \left(f(x_k) - y_k\right)^2 \label{eq_30_MSE} \end{equation} \tag{13.4}\]
The measured MSE errors for all displayed results are summarized in table 13.1 . The MSE for results of \(\beta_{unseen} = 28.5\) and \(\beta_{unseen} = 32.5\) in figures 13.4 is \(0.622\) and \(0.677\), respectively. Consequently, the performance of CNMccontrol-oriented Cluster-based Network Modeling is also confirmed quantitatively. Figures in 13.6 illustrate the outcome for LS20 for \(\beta_{unseen} = 31.75\) and \(\beta_{unseen} = 51.75\). In section 12 it is explained that for LS20 cluster network deformations appear . Nevertheless, the outcome visually and quantitatively endorses the CPE modeling capabilities. Figures in 13.7 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.
| Figure | Model | \(\boldsymbol{\beta_{unseen}}\) | MSE |
|---|---|---|---|
| 13.4 | SLS | \(28.5\) | \(0.622\) |
| 13.4 | SLS | \(32.5\) | \(0.677\) |
| 13.5 | SLS | \(26.5\) | \(1.193\) |
| 13.5 | SLS | \(37\) | \(5.452\) |
| 13.6 | LS20 | \(31.75\) | \(1.857\) |
| 13.6 | LS20 | \(51.75\) | \(2.536\) |
| 13.7 | FW15 | \(8.7\) | \(1.617\) |
| 13.7 | FW15 | \(10.1\) | \(1.5\) |
It can be concluded that the CPE modeling performance is satisfying. In the case of a few cluster network deformations, CNMccontrol-oriented Cluster-based Network Modeling 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.