# Tree-Structured Regression Model Using a Projection Pursuit Approach
## 5 Conclusions and Discussion  
This paper proposed a new exploratory regression method: the projection pursuit regression tree. This tree is an extension of the projection pursuit classification tree to \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & & & & & \\
**Dataset** & **n** & **p** & **Model 1** & **Model 2** & **Model 3** & **Model 4** & **Model 5** & **Rpart** & **Forest** \\ \hline abalone & 4177 & 8 & 1.074 & 1.062 & 1.073 & 1.066 & 1.066 & 1.079 & 0.944 \\ airfoil & 1503 & 5 & 1.106 & 1.094 & 1.084 & 1.085 & 1.094 & 0.934 & 0.791 \\ ais & 202 & 10 & 1.272 & 1.278 & 1.226 & 1.164 & 1.163 & 1.204 & 1.078 \\ alcohol & 2467 & 17 & 1.067 & 1.006 & 1.103 & 1.102 & 1.103 & 1.002 & 1.043 \\ autompg & 392 & 7 & 1.032 & 1.026 & 0.901 & 0.871 & 0.947 & 1.039 & 0.765 \\ baseball & 263 & 18 & 1.063 & 1.038 & 1.087 & 1.024 & 1.019 & 0.938 & 0.706 \\ boston & 506 & 13 & 1.090 & 1.075 & 0.936 & 0.846 & 0.876 & 0.967 & 0.660 \\ budget & 1729 & 9 & 0.670 & 0.628 & 0.539 & 0.493 & 0.531 & 0.534 & 0.122 \\ cane & 3775 & 23 & 1.152 & 1.143 & 1.082 & 1.064 & 1.071 & 1.143 & 0.957 \\ mussels & 82 & 4 & 1.854 & 1.840 & 1.047 & 1.073 & 1.066 & 1.474 & 1.111 \\ ozone & 330 & 9 & 1.144 & 1.135 & 0.956 & 0.933 & 0.953 & 1.005 & 0.836 \\ parkinsons & 5875 & 20 & 1.116 & 1.108 & 1.064 & 1.048 & 1.048 & 1.082 & 0.804 \\ price & 159 & 15 & 1.140 & 1.106 & 1.101 & 1.122 & 0.945 & 0.979 & 0.681 \\ strikes & 628 & 4 & 1.013 & 0.975 & 1.049 & 1.030 & 1.035 & 1.011 & 0.869 \\ teacator & 240 & 22 & 2.458 & 2.425 & 1.762 & 0.907 & 1.494 & 2.726 & 1.999 \\ wine-red & 1599 & 11 & 0.907 & 0.907 & 0.969 & 0.979 & 0.975 & 1.048 & 0.864 \\ wine-white & 4898 & 11 & 1.026 & 0.983 & 1.116 & 1.099 & 1.101 & 1.032 & 0.774 \\ yachtdata & 308 & 6 & 0.298 & 0.303 & 0.114 & 0.084 & 0.104 & 0.240 & 0.360 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison of lm, rpart, randomForest, and the projection pursuit regression tree with various data sets; the mean MAE/MAE(lm)of 200 iterations for the test dataset. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & & & & & \\
**Dataset** & **n** & **p** & **Model 1** & **Model 2** & **Model 3** & **Model 4** & **Model 5** & **Rpart** & **Forest** \\ \hline abalone & 4177 & 8 & 1.079 & 1.062 & 1.072 & 1.068 & 1.068 & 1.044 & 0.446 \\ airfoil & 1503 & 5 & 1.095 & 1.083 & 1.073 & 1.070 & 1.084 & 0.854 & 0.669 \\ ais & 202 & 10 & 1.191 & 1.174 & 1.119 & 0.957 & 1.029 & 1.028 & 0.516 \\ alcohol & 2467 & 17 & 1.058 & 0.997 & 1.092 & 1.084 & 1.090 & 1.006 & 0.707 \\ autompg & 392 & 7 & 1.018 & 1.010 & 0.888 & 0.821 & 0.910 & 0.895 & 0.403 \\ baseball & 263 & 18 & 1.042 & 1.010 & 1.023 & 0.891 & 0.966 & 0.794 & 0.341 \\ boston & 506 & 13 & 1.052 & 1.040 & 0.880 & 0.741 & 0.823 & 0.843 & 0.309 \\ budget & 1729 & 9 & 0.660 & 0.620 & 0.528 & 0.462 & 0.518 & 0.514 & 0.054 \\ cane & 3775 & 23 & 1.131 & 1.122 & 1.060 & 1.034 & 1.044 & 1.104 & 0.545 \\ mussels & 82 & 4 & 1.996 & 1.954 & 1.078 & 0.999 & 1.041 & 1.161 & 0.656 \\ ozone & 330 & 9 & 1.135 & 1.118 & 0.952 & 0.876 & 0.935 & 0.813 & 0.383 \\ park the regression problem. In each node, all observations in the node are divided into two groups--a larger dependent-variable group and a smaller dependent-variable group. Then, the projection pursuit method is applied using indices for classification. One of the projection pursuit indices--the LDA, Lr, or PDA index--is maximized to find a projection that separates the two groups. With the projection pursuit regression tree, the partitioned data as well as the data as a whole can be explored. The main difference between the general tree-based regression method and the projection pursuit regression tree is the way the data is partitioned. The general tree-based regression methods focus on the space of independent variables and each final node tends to have much wider-ranging \(Y\) values. However, the projection pursuit regression tree starts from the dependent variable.