## 6 ARIMAX Model-Based Approach  
As in the previous case, the same dataset is used for the following: (1) to estimate the model parameters (training), (2) to determine the forecast boundaries and bias (validation), and (3) finally, to evaluate the forecasting performance using test data. Table 6 displays the ADF test results for stationarity assessment across the time series, while Figure 9 plots the ACF and PACF used to determine the AR and MA orders. Figure 8: Distribution of validation residuals incorporating bias correction. Prediction limits (95% Confidence interval). \begin{table}
\begin{tabular}{c c c c} \hline
**Data** & **Pearson** & **Spearman** & **Kendall Tau** \\ \hline Validation & \(-0.3160\) & \(-0.4022\) & \(-0.2732\) \\ Test & \(0.9243\) & \(0.5135\) & \(0.3695\) \\ \hline \end{tabular}
\end{table} 
Table 5: Correlation coefficients between the actual and estimated data. From Figure 9, the autocorrelation and partial autocorrelation functions indicate the series needs first-order differencing (d = 1), with preliminary evidence supporting at least AR(1) and MA(1) terms in the model specification. The final model corresponds to an ARIMA(1, 1, 2), with the values of statistically significant parameters shown in Figure 10. The values were derived by incrementally testing ARIMA models up to fifth-order terms and retaining only those coefficients that achieved statistical significance (\(p\leq 0.05\))  
The MAE for the training data is 3.4692, which is clearly higher than the values obtained by the neural network (see Table 3). This suggests greater imprecision in the predictions, indicating that the model is unable to capture the underlying patterns of the data and may be experiencing underfitting (an overly simplistic model). Based on the fitted model and using validation data, the residuals exhibit a bias of 0.6007. After bias removal, the percentiles show P\({}_{2.5}\) = \(-0.4466\) and P\({}_{97.5}\) = 0.1741. Figure 11 depicts the last 1000 test data points, including the forecast limits, control limits, and the prediction provided by the neural network and ARIMAX with bias correction (+0.916 MPa and +0.600 MPa). The evident underfitting of the ARIMAX prediction model manifests in excessively stable forecasts that prove unresponsive to either perturbations or structural changes in the input time series, consequently resulting in increased prediction \begin{table}
\begin{tabular}{c c c c} \hline
**Time Series** & **ADF Statistic** & \(p\)**-Value** & **Stationary** \\ \hline Outlet pressure & & & \\ Training & \(-4.34435\) & 0.000372 & Yes \\ Validation & \(-7.04869\) & 0.000000 & Yes \\ Test & \(-3.53525\) & 0.007127 & Yes \\ \hline Current consumption & & & \\ Training & \(-4.80803\) & 0.000052 & Yes \\ Validation & \(-6.36995\) & 0.00000 & Yes \\ Test & \(-3.82928\) & 0.002622 & Yes \\ \hline \end{tabular}
\end{table} 
Table 6: ADF test results. Figure 10: Coefficients and signification for ARIMAX model. Figure 9: ACF and PACF for endogenous variables. error. Moreover, a drop in the outlet pressure exceeding the predefined control limits is observed. The primary cause of this issue is a leak in one of the upstream gaskets of the compressor, located between the hydrogen storage tank and the compressor. Although the compressor outlet pressure is successfully regulated, the inlet pressure exhibits erratic behavior characterized by fluctuations and a downward trend. Several inferences can be drawn from the neural network prediction model. First, the prediction lags delay the actual series. Second, disturbances are smoothed, exhibiting lower amplitude compared to the actual ones. Third, the neural network displays greater variability than the real data, indicating that the network has captured noise during its training and is reflecting it. This highlights the need for further study and a deeper understanding of the process and the current control system. It can be observed that the control limits estimated by the predictive model are offset relative to the process control limits.