# Tree Based Approaches for Predicting Concrete Carbonation Coefficient
## 3 Materials and Methods
### Methodology Adopted  
To predict the carbonation coefficient (_k_), the factors affecting the carbonation of concrete were considered. The data mentioned in the previous section were used, and three models were developed using MT, RF and MGGP--one for each tree-based approach. These global models eliminate the use of different models for different humidity levels. The data characteristics of MT, RF and MGGP models are as shown in Table 1.  
Model Tree models were developed in WEKA, software version 3.9, developed by University of Waikato, New Zealand [37], and the M5P algorithm developed by [PERSON] [24] was used. Random Forest was developed in Python through Jupyter notebook (a web-based interactive computing platform). For RF, the number of trees was selected between 100 and 500, and the bagging iterations and the tree size were selected, anticipating that these were those that yielded the best performance by a low Mean Square Error (MSE). Multi-Gene Genetic Programming was developed in Matlab 2017 using GPTIPS-2 (an open-source toolbox for MGGP). Readers are referred for features of GPTIPS to [31; 32]. The Root Mean Square Error (RMSE) function was adapted for error minimisation during runs [31; 32]. The parameters were selected as those that yielded the best performance of the models. A large population and generations were assessed to find models with minimum error. The programs were run until the specified number of generations was reached. The allowable number of genes and tree depth were, respectively, set to optimal values as trade-offs between the running time and the complexity of the evolved solutions [31; 32]. The best MGGP models were chosen based on providing the best fitness value on the training data as well as the simplicity of the models [32]. These settings in RF and MGGP were based on existing knowledge about the predictive modelling of other data sets of similar size, and so they may not be optimal. The MGGP parameters and their settings are as shown in Table 2.  
Variable or parameter importance is also a distinctive feature of the current work. The influential parameters in MT can be highlighted through the values of the standardised coefficients of the parameters, while in RF and MGGP, these can be shown through the variable/parameter permutation importance and the input frequency, respectively. The parameter influence analysis enhances the concept understanding of the algorithm and aids in its validation.  
In this investigation, 70% of the data were used for model calibration, while the remaining 30% were used for testing. The model calibration stops when the error in the validation dataset starts increasing--a phenomenon known as over fitting. Statistical metrics such as the correlation coefficient (\(r\)), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) were used to assess models' performance [38]. High prediction accuracy was expected for lower error statistics (RMSE, MAE), while the opposite was valid for the correlation coefficient. Further, the degree by which RMSE exceeded MAE was an indicator of the extent to which outliers (or variance in the differences between the modelled and observed values) existed in the data [38; 39; 40]. The results of the models (for testing data set) were also evaluated graphically in the form of scatter plots and hydrograph.