# Offline Identification of a Laboratory Incubator
## 3 Experiments
### Experimental Results
#### 3.2.4 Results for the Box-Jenkins Model  
The order of the BJ model was determined as \(n_{b}=2\), \(n_{c}=2\), \(n_{d}=2\) and \(n_{f}=2\) for the first scenario. The parameter vector \(\theta\) of BJ [2 2 2 2] was estimated on the identification dataset-1. The polynomials of the BJ model for this scenario were obtained as in Equations (47)-(50). \[B(q)=-5.36891\times 10^{-5}q^{-1}-5.66662\times 10^{-5}q^{-2}, \tag{47}\]  
\[C(q)=1-0.663758q^{-1}-0.33229q^{-2}, \tag{48}\]  
\[D(q)=1-0.0039244q^{-1}-0.995552q^{-2}, \tag{49}\]  
\[F(q)=1-1.99376q^{-1}+0.993759q^{-2}, \tag{50}\]  
For the second scenario, the order for the model was determined as \(n_{b}=2\), \(n_{c}=2\), \(n_{d}=2\) and \(n_{f}=2\). The parameter vector \(\theta\) of BJ [2 2 2 2] was estimated on the identification dataset-2. The polynomials for the BJ model were obtained as in Equations (51)-(54). \[B(q)=0.0001333q^{-1}-0.0001312q^{-2}, \tag{51}\]  
\[C(q)=1+0.1308q^{-1}-0.865q^{-2}, \tag{52}\]  
\[D(q)=1-1.296q^{-1}+0.2967q^{-2}, \tag{53}\]  
\[F(q)=1-1.996q^{-1}+0.9957q^{-2}, \tag{54}\]  
The 2-step-, 10-step- and 20-step-ahead prediction and simulation responses of the BJ [2 2 2 2] model for both scenarios were computed by using the verification dataset. The responses were compared with the actual output of the incubator and visualized in Figure 19, and in Figure 20 for the first and the second scenarios, respectively. MSE, MAE and G measures were computed for k-step predictions (k = 2, 10 and 20) and simulation. The results for the model are represented in Table 4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Validation Criteria**} & \multicolumn{2}{c}{**G**} & \multicolumn{2}{c}{**MSE**} & \multicolumn{2}{c}{**MAE**} \\ \cline{2-7}  & **Step** & **PRBS** & **Step** & **PRBS** & **Step** & **PRBS** \\ \hline Prediction k: 2 & 96.2799 & 99.4335 & 4.01182 \(\times 10^{-5}\) & 9.1998 \(\times 10^{-7}\) & 0.0041233 & 0.0007 \\ Prediction k: 10 & 91.4721 & 98.1200 & 0.000210816 & 1.0131 \(\times 10^{-5}\) & 0.0108614 & 0.0021 \\ Prediction k: 20 & 90.4577 & 96.4682 & 0.000263956 & 3.5756 \(\times 10^{-5}\) & 0.0124288 & 0.0040 \\ Simulation & 90.3288 & 95.0771 & 0.000271132 & 7.0355 \(\times 10^{-5}\) & 0.0123917 & 0.0074 \\ \hline \end{tabular}
\end{table} 
Table 4: Box–Jenkins [2 2 2 2] results for the first and second scenarios. Figure 19: Comparison of the responses for BJ [2 2 2] in the first scenario. Figure 23: G vs. prediction horizon for the first scenario. From Figure 23, for the first scenario, the following can be concluded: ARX has a better G value than ARMAX for k = 2, while ARMAX has better G values for k = 10 and 20 step-ahead-predictions and simulation. Although the G value of OE for simulation is greater than ARX, it is not a suitable model for the incubators used in practical applications. This is because OE has high deviations between the simulation and actual output, caused by changes in the input signal. BJ has the best G values for all the k-step-ahead predictions, including simulation. From Figure 24, for the second scenario, the following can be concluded: ARX, ARMAX and BJ have similar G values for k = 2, while ARMAX has better G values than ARX for k = 10 and 20 step-ahead-predictions and simulation. BJ has the best G values for all responses, and the worst G value for simulation belongs to OE. Residue analysis for both scenarios represents that all the models except OE have a good confidence level. The cross-correlation plots of the ARX, ARMAX and BJ models show that the cross-correlations between the residuals and input are within the confidence region, which is set to 99% for the study.