# Tree-Structured Regression Model Using a Projection Pursuit Approach
## 4 Results
### Comparison of the Performance of Tree-Based Regression Methods  
Even though the main purpose of the projection pursuit regression tree is to explore data, prediction is also an important feature of the tree-structured model. In this section, 18 data sets are used to examine the performance of the projection pursuit regression tree. These 18 data sets come from various sources, such as the UCI data repositories, Statlib, or R. For each data set, the data is divided into two-thirds training data and one-third test data, the projection pursuit regression model is fitted with the training data, and the mean squared error (MSE) and the mean absolute errors (MAE) of the training data and the test data are calculated, respectively. 1m, rpart, and randomForest are used to compare the performance. rpart is the most commonly used tree-based model and randomForest is also popular among ensemble tree methods. Usually a single regular tree-structured model cannot outperform ensemble tree methods. Tables 7 and 8 show the results of the MSE for the training and test data, respectively. The average of the ratios of the MSE of each method to the MSE of lm are calculated. If this average of ratios is less than 1, the performance of the corresponding method is better than that of the linear model. For the training data, the projection pursuit regression tree shows better performance than rpart in seven data sets and shows competitive performance in three data sets. Surprisingly, in reactor and yacht data sets, the projection pursuit regression tree shows much better performance than randomForest. The projection pursuit regression tree shows much better performance in the test data. For 15 data sets, the projection pursuit regression tree shows similar or better performance than rpart. For three data sets--mussels, reactor, and yacht, the performance is much better than that of randomForest. Tables 9 and 10 show the ratios of MAE in the training and test data, respectively. In the MAE results, the projection pursuit regression tree shows similar or slightly better \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{3}{c}{**Projection Pursuit Regression Tree**} & \multicolumn{3}{c}{RandomForest} \\
**ID** & **Variables** & **Importance** & **ID** & **Variables** & **Importance** \\ \hline X11 & alcohol & 545.82 & X11 & alcohol & 194.85 \\ X2 & volatile acidity & 364.27 & X10 & sulphates & 135.30 \\ X10 & sulphates & 336.86 & X2 & volatile acidity & 127.13 \\ X8 & fixed acidity & 331.98 & X8 & density & 86.55 \\ X1 & density & 313.67 & X7 & total sulfur dioxide & 79.42 \\ X7 & total sulfur dioxide & 288.75 & X3 & citric acid & 69.13 \\ X5 & chlorides & 182.27 & X5 & chlorides & 68.24 \\ X4 & residual sugar & 166.65 & X1 & fixed acidity & 61.71 \\ X3 & pH & 151.30 & X9 & pH & 57.77 \\ X9 & citric acid & 150.70 & X4 & residual sugar & 55.05 \\ X6 & free sulfur dioxide & 109.87 & X6 & free sulfur dioxide & 50.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Importance measures of the red wine data. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Projection Pursuit Regression Tree**} & \multicolumn{2}{c}{RandomForest} \\
**Variables** & **Importance** & **Variables** & **Importance** \\ \hline density & 646.23 & alcohol & 621.77 \\ residual sugar & 587.62 & density & 413.10 \\ alcohol & 436.91 & volatile acidity & 389.11 \\ volatile acidity & 252.90 & free sulfur dioxide & 385.95 \\ pH & 143.63 & chlorides & 306.12 \\ free sulfur dioxide & 117.04 & total sulfur dioxide & 293.33 \\ sulphates & 114.89 & residual sugar & 277.97 \\ fixed acidity & 91.90 & pH & 258.04 \\ total sulfur dioxide & 75.78 & citric acid & 255.65 \\ chlorides & 41.67 & fixed acidity & 235.73 \\ citric acid & 21.52 & sulphates & 222.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Importance measures of the white wine data. performance than rpart in most data sets, and it is outperformed in three data sets--mussels, tecator, and yachtdata.