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| ## Transition property regression models {#sec-sec_3_4_SVD_Regression} | |
| In this section, the results of the 3 different regression methods, \glsfirst{rf}, AdaBoost and Gaussian Process (GP) are compared. | |
| All the 3 regressors are implemented in \gls{cnmc} and can be selected via *settings.py*. | |
| The utilized model configuration is *SLS* and the decomposition method is \gls{svd}.\newline | |
| First, it shall be noted that \gls{cnmc} also offers the possibility to apply *pySindy*. | |
| However, *pySindy* has struggled to represent the training data in the first place, thus it cannot be employed for predicting $\beta_{unseen}$. | |
| The latter does not mean that *pySindy* is not applicable for the construction of a surrogate model for the decomposed $\boldsymbol Q / \boldsymbol T$ modes, but rather that the selected candidate library was not powerful enough. | |
| Nevertheless, only results for the 3 initially mentioned regressors will be discussed.\newline | |
| In figures @fig-fig_66 to @fig-fig_71 the true (dashed) and the approximation (solid) of the first 4 $\boldsymbol Q / \boldsymbol T$ modes are shown for the methods RF, AdaBoost and GP, respectively. | |
| To begin with, it can be noted that the mode behavior over different model parameter values $mod(\beta)$ is discontinuous, i.e., it exhibits spikes or sudden changes. | |
| In figures @fig-fig_66 and @fig-fig_67 it can be observed that \gls{rf} reflects the actual behavior of $mod(\beta)$ quite well. | |
| However, it encounters difficulties in capturing some spikes. | |
| AdaBoost on the other hand proves in figures @fig-fig_68 and @fig-fig_69 to represent the spikes better. | |
| Overall, AdaBoost outperforms \gls{rf} in mirroring training data. \newline | |
| :::{layout="[[1,1]]"} | |
| {#fig-fig_66} | |
| {#fig-fig_67} | |
| *SLS*, \gls{svd}, $\boldsymbol Q / \boldsymbol T$ modes approximation with \gls{rf} for $L=1$ | |
| ::: | |
| :::{layout="[[1,1]]"} | |
| {#fig-fig_68} | |
| {#fig-fig_69} | |
| *SLS*, \gls{svd}, $\boldsymbol Q / \boldsymbol T$ mode approximation with AdaBoost for $L=1$ | |
| ::: | |
| :::{layout="[[1,1]]"} | |
| {#fig-fig_70} | |
| {#fig-fig_71} | |
| *SLS*, \gls{svd}, $\boldsymbol Q / \boldsymbol T$ mode approximation with GP for $L=1$ | |
| ::: | |
| Gaussian Process (GP) is a very powerful method for regression. | |
| Often this is also true when reproducing $mod(\beta)$. | |
| However, there are cases where the performance is insufficient, as shown in figures @fig-fig_70 and @fig-fig_71 . | |
| Applying GP results in absolutely incorrect predicted tensors | |
| $\boldsymbol{\tilde{Q}}(\beta_{unseen}),\, \boldsymbol{\tilde{T}}(\beta_{unseen})$, | |
| where too many tensors entries are wrongly forced to zero. | |
| Therefore, $\boldsymbol{\tilde{Q}}(\beta_{unseen}),\, \boldsymbol{\tilde{T}}(\beta_{unseen})$ will eventually lead to an unacceptably high deviation from the original trajectory. | |
| Consequently, the GP regression is not applicable for the decomposed $\boldsymbol Q / \boldsymbol T$ modes without further modification.\newline | |
| The two remaining regressors are \glsfirst{rf} and AdaBoost. | |
| Although AdaBoost is better at capturing the true modal behavior $mod(\beta)$, there is no guarantee that it will always be equally better at predicting the modal behavior for unseen model parameter values $mod(\beta_{unseen})$. | |
| In table @tbl-tab_8_RF_ABoost the MAE errors for different $L$ and $\beta_{unseen} = [\, 28 .5,\, 32.5\,]$ are provided. | |
| Since the table exhibits much information, the results can also be read qualitatively through the graphs @fig-fig_72_QT_28 and @fig-fig_72_QT_32 for $\beta_{unseen} = 28.5$ and $\beta_{unseen} = 32.5$, respectively. | |
| For the visual inspection, it is important to observe the order of the vertical axis scaling. | |
| It can be noted that the MAE errors themselves and the deviation between the \gls{rf} and AdaBoost MAE errors are very low. | |
| Thus, it can be stated that \gls{rf} as well ad AdaBoost are both well-suited regressors.\newline | |
| **$L$** | $\beta_{unseen}$ | $\boldsymbol{MAE}_{RF, \boldsymbol Q}$ | $\boldsymbol{MAE}_{AdaBoost, \boldsymbol Q}$ | $\boldsymbol{MAE}_{RF, \boldsymbol T}$ | $\boldsymbol{MAE}_{AdaBoost, \boldsymbol T}$ | |
| ---------|------------------|----------------------------------------|----------------------------------------------|----------------------------------------|---------------------------------------------- | |
| $1$ | $28.5$ | $0.002580628$ | $0.002351781$ | $0.002275379$ | $0.002814208$ | |
| $1$ | $32.5$ | $0.003544923$ | $0.004133114$ | $0.011152145$ | $0.013054876$ | |
| $2$ | $28.5$ | $0.001823848$ | $0.001871858$ | $0.000409955$ | $0.000503748$ | |
| $2$ | $32.5$| $0.006381635$ | $0.007952153$ | $0.002417142$ | $0.002660403$ | |
| $3$ | $28.5$ | $0.000369228$ | $0.000386292$ | $0.000067680$ | $0.000082808$ | |
| $3$ | $32.5$ | $0.001462458$ | $0.001613434$ | $0.000346298$ | $0.000360097$ | |
| $4$ | $28.5$ | $0.000055002$ | $0.000059688$ | $0.000009420$ | $0.000011500$ | |
| $4$ | $32.5$| $0.000215147$ | $0.000230404$ | $0.000044509$ | $0.000046467$ | |
| $5$ | $28.5$ | $0.000007276$ | $0.000007712$ | $0.000001312$ | $0.000001600$ | |
| $5$ | $32.5$ | $0.000028663$ | $0.000030371$ | $0.000005306$ | $0.000005623$ | |
| $6$ | $28.5$| $0.000000993$ | $0.000052682$| $0.000000171$ | $0.000000206$ | |
| $6$ | $32.5$ | $0.000003513$ | $0.000003740$ | $0.000000629$ | $0.000000668$ | |
| $7$ | $28.5$ | $0.000000136$ | $0.000000149$ | $0.000000023$ | $0.000000031$ | |
| $7$ | $32.5$| $0.000000422$ | $0.000000454$| $0.000000078$ | $0.000000082$ | |
| : *SLS*, Mean absolute error for comparing \gls{rf} and AdaBoost different $L$ and two $\beta_{unseen}$ {#tbl-tab_8_RF_ABoost} | |
| :::{#fig-fig_72_QT_28 layout="[[1,1]]"} | |
| {#fig-fig_72_Q_28} | |
| {#fig-fig_72_T_28} | |
| *SLS*, Mean absolute error for comparing \gls{rf} and AdaBoost different $L$ and $\beta_{unseen} = 28.5$ | |
| ::: | |
| :::{#fig-fig_72_QT_32 layout="[[1,1]]"} | |
| {#fig-fig_72_Q_32} | |
| {#fig-fig_72_T_32} | |
| *SLS*, Mean absolute error for comparing \gls{rf} and AdaBoost different $L$ and $\beta_{unseen} = 32.5$ | |
| ::: | |
| In summary, the following can be said, \gls{rf} and AdaBoost are both performing well in regression. Furthermore, no clear winner between the two regressors can be detected. | |
| The third option GP is dismissed as it sometimes has unacceptably low regression performance. | |
| Finally, there is the possibility to use *pySindy*, however, for that, an appropriate candidate library must be defined. | |
| \FloatBarrier |