## 3 Results and Discussion
### Model Development
#### 3.1.2 Machine Learning Models  
To develop predictive models via machine learning, we should first define the independent variables and the dependent variable. With the experimental fact during the practical measurements, the temperature and pressure of R227 ea are easy to obtain. Here, we define the temperature (K) and pressure (bar) of the determinant as the independent variables, while the density (kg\(\cdot\) m\({}^{-3}\)) is set as the dependent variable. With the design that users can only input the values of the temperature and pressure to a developed model, we let the machine learning models in our study \"learn\" the existing data and make precise predictions. The experimental data of [PERSON] _et al._[1], [PERSON] _et al._[24], and [PERSON] _et al._[25] were used for model developments respectively. In each model, 80% of the data were set as the training set, while 20% of the data were set as the testing set. The SVMs were developed by Matlab software (Libsvm package [42]) and the ANNs were developed by NeuralTools(r) software (trial version, Palisade Corporation, NY, USA). General regression neural network (GRNN) [43; 44; 45] and multilayer feed-forward neural networks (MLFNs) [46; 47; 48] were chosen as the learning algorithms of ANNs. Numbers of nodes in the hidden layer of MLFNs were set from 2 to 35. In this case study, the number of hidden layer was set as one. Trials of all ANNs were set as 10,000. All these settings of ANNs were set directly in the NeuralTools(r) software. Linear regression models were also developed for comparisons. To measure the performance of the model and make suitable comparisons, RMSE (for testing), training time, and prediction accuracy (under the tolerance of 30%) were used as indicators that evaluate the models. Model results using experimental data from [PERSON] _et al._[1], [PERSON] _et al._[24], and [PERSON] _et al._[25] are shown in Tables 1-4 respectively. Error analysis results are shown in Figure 4.  
Figure 3: Theoretical calculated surface and experimental densities of R227 ea. The surface represents the theoretical calculated results by Equations (1)–(9); black points represent the experimental results from [PERSON] _et al._[1]; red crosses represent the experimental results from [PERSON] _et al._[24]; blue asterisks represent the experimental results from [PERSON] _et al._[25]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Type** & **RMSE (for Testing)** & **Training Time** & **Prediction Accuracy** \\ \hline Linear Regression & 10.90 & 0:00:01 & 85.0\% \\ SVM & 0.11 & 0:00:01 & 100\% \\ GRNN & 1.62 & 0:00:01 & 100\% \\ MLFN 2 Nodes & 1.13 & 0:03:46 & 100\% \\ MLFN 3 Nodes & 0.40 & 0:04:52 & 100\% \\ MLFN 4 Nodes & 0.25 & 0:06:33 & 100\% \\ MLFN 5 Nodes & 0.37 & 0:07:25 & 100\% \\ MLFN 6 Nodes & 0.59 & 0:10:38 & 100\% \\ MLFN 7 Nodes & 0.47 & 0:13:14 & 100\% \\ MLFN 8 Nodes & 0.32 & 0:14:10 & 100\% \\... &... &... &... \\ MLFN 29 Nodes & 0.13 & 2:00:00 & 100\% \\ MLFN 30 Nodes & 0.16 & 2:00:00 & 100\% \\ MLFN 31 Nodes & 0.10 & 2:00:00 & 100\% \\ MLFN 32 Nodes & 0.15 & 2:00:00 & 100\% \\ MLFN 33 Nodes & 0.13 & 2:00:00 & 100\% \\ MLFN 34 Nodes & 0.12 & 2:00:00 & 100\% \\ MLFN 35 Nodes & 0.13 & 2:00:00 & 100\% \\ \hline \hline \end{tabular} Root mean square error (RMSE); Support vector machine (SVM); General regression neural network (GRNN); Multilayer feed-forward neural network (MLFN).  
\end{table} 
Table 1: Prediction models using experimental data by [PERSON] _et al._[1]. Figure 4: Root mean square error (RMSE) _versus_ number of nodes of multilayer feed-forward neural networks (MLFNs). Bars represent the RMSEs; black dashed lines represent the RMSEs of general regression neural network (GRNN) and support vector machine (SVM).