# Tree-Structured Regression Model Using a Projection Pursuit Approach
## 4 Results  
### Application: White and Red Wine Data  
To explain how to explore data and find interesting features with the projection pursuit regression tree, wine data from the UCI Machine Learning repository (\"[[https://archive.ics.uci.edu/ml/datasets.php](https://archive.ics.uci.edu/ml/datasets.php)]([https://archive.ics.uci.edu/ml/datasets.php](https://archive.ics.uci.edu/ml/datasets.php))\" [28]; accessed on 15 September 2021) is used. These data are for Portugal wines. Eleven sensory variables were collected from 4898 white wines and 1599 red wines. Additionally, the quality of wines was evaluated. The wine quality is between 0 (very bad) and 10 (excellent). In this paper, the projection pursuit regression tree is fitted to the white wine data and red wine data separately and compare the differences between white wine and red wine. For this exploration, the projection pursuit regression tree is fitted first. Figure 7 shows the final projection pursuit regression tree. Table 4 shows the best projection coefficients for all internal nodes. Node 1 divides the data into a high-quality group (quality \(\geq\) 7) and a low-quality group (quality \(\leq\) 6). In both wines, if the alcohol is high and the density is low, the wine is classified in the high-quality group. Additionally, there are differences in the significant sensory variables between the white and red wine quality. White wines with high residual sugar are classified in the high-quality group. In contrast, red wines with high fixed acidity and high sulphates are classified in the high-quality wine group. In node 2, the projection pursuit regression tree divides the low-quality group into two small groups, one with quality 6 and the other with the lowest quality (\(\leq\)5). In white wine,  
Figure 6: The projection pursuit regression tree with depth 2 for the simulated data. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{Lm} & \multirow{2}{*}{RandomForest} & \multirow{2}{*}{Rpart} & \multicolumn{3}{c}{**Projection Pursuit Regression Tree**} & \multirow{2}{*}{**Model 1**} & \multirow{2}{*}{**Model 2**} & \multirow{2}{*}{**Model 3**} & \multirow{2}{*}{**Model 4**} & \multirow{2}{*}{**Model 5**} \\ \hline MSE & 0.0353 & 0.0096 & 0.0083 & 0.0067 & 0.0073 & 0.0024 & 0.0024 & 0.0017 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MSEs of lm, randomForest, rpart, and the projection pursuit regression tree. Figure 7: The projection pursuit regression tree for the wine data: (**a**) white wine and (**b**) red wine. high volatile acidity, low residual sugar, high density, and low alcohol tend to produce lower quality, and red wines with high volatile acidity, high total sulfur dioxide, low sulphates, and low alcohol have the lowest quality. Node 3 divides the top-quality group (\(\geq\)8) and the group with quality 7. High residual sugar and low density are still important to classify a wine as a top-quality white wine. The red wine shows a bit of complicated sensory evaluation. Low fixed acidity, high volatile acidity, low chlorides, low pH, and high alcohol tend to produce top-quality wine. Node 4 divides the lowest-quality group (\(\leq\)4) and the group with quality 5. White wines with low residual sugar, high density, and high alcohol have the lowest quality. In red wine, high acidity, low density, and high pH tend to produce the lowest quality. Table 5 shows the importance measures of each variable in the white wine data. For the projection pursuit regression tree, the importance measure of each variable is a weighted sum of the absolute value of the projection coefficients of each node. For the weights, the number of observations in each node is used. In the white wine data, four variables--density, residual sugar, alcohol, and volatile acidity--have important roles in dividing five groups; the lowest-quality group, the groups of quality 5, 6, and 7, and the top-quality group. These results are compared with the importance measure in randomForest.