## 4 Case Study and Results Discussion
### Analysis 2: Covariate Selection Sensitivity  
In the second analysis, the objective was to evaluate the framework's covariate selection process and assess how these selections impacted the final decision policy. Consequently, all the vital signals listed in Table 1 were considered. Additionally, based on the results from the preceding analysis, the optimization engine IPOPT, the NS scaler, and the IPCRIDGE bound technique were employed in this case study. To initiate the analysis, a model including all the covariates was constructed to determine their respective weights. However, to gain a deeper understanding of how the weights varied with different covariate combinations, their weights were calculated for every possible combination, and their median values were computed (this approach helped to mitigate biases from extreme weight values compared to using the average). The results are presented in Table 10, offering a comprehensive analysis across various covariate combinations. For example, out of the total 15 covariates, when considering only 2 covariates, a total of \(\binom{15}{2}\) combinations were calculated. Then, the median values for each covariate were obtained from these cases. As evident from Table 10, the covariates Tint, Dift, C\({}_{2}\)H\({}_{4}\), TGC, R.Die, and % Hum consistently demonstrated substantial weight values across all scenarios. This suggested that a more parsimonious model could potentially include only these covariates without compromising the log-likelihood score. In contrast, covariates CH\({}_{4}\), GTF, and TGC-CO initially exhibited high weight values. However, their weights decreased significantly when more than nine covariates were considered. To evaluate the consistency of the covariates, a cross-validation analysis using K folds was applied. The underlying principle was to vary the dataset utilized for estimating the covariate weights in each fold. Thus, if a covariate demonstrated a consistent and relatively low coefficient of variation compared to others, it enhanced its suitability for inclusion in the final model. \begin{table}
\begin{tabular}{c}  
In this analysis, 5 and 10 folds were utilized, and the results, including the average, standard deviation, and coefficient of variation, are presented in Tables 11 and 12. To enhance readability, the tables exclude the covariates CH4, GTF, and TGC-CO, which exhibited extremely high CV values. The covariates C2H4, R.Die, and % Hum demonstrated relatively lower CV compared to the other covariates. This suggested that the weights assigned to C2H4 and R.Die were more consistent across all models. While % Hum had a higher CV than the aforementioned covariates, its CV remained significantly lower when compared to the rest. Consequently, based on this analysis, C2H4, R.Die, and % Hum should be prioritized when selecting the final model from the framework. Moreover, these covariates also received substantial weights in Table 10 when considering a set of 15 covariates. This may indicate that using the model with all covariates could allow us to assess the importance of each covariate, aiding decision-making in covariate selection. To extend the analysis, a focused examination was conducted on the impact of each covariate on the log-likelihood score, as presented in Table 13. Interestingly, C2H4 significantly influenced the log-likelihood score. Results for TGC and % Hum are not included as there is no solution when considering each separately.