# How Trees are Built

### Building Decision Trees

The process used to split a node is the same whether the node is the root node with all of the rows or a child node many levels deep in the tree. The only difference is the set of rows in the node being split.

### Splitting Nodes

DTREG tries each predictor variable to see how well it can divide the node into two groups.

If the predictor is continuous, a trial split is made between each category of the variable. For example, if the predictor being evaluated is Age and there are 80 categories of Age ranging from 10 to 79, then DTREG makes a trial split putting the rows with a value of 10 for Age in the left node and the rows with values from 11 to 79 in the right node. The improvement gained from the potential split is remembered, and then the next trial split is done putting rows with Age values of 10 and 11 in the left group and values from 12 to 79 in the right group. The number of splits evaluated is equal to the number of categories of the predictor variable less one.

This process is repeated by moving the split point across all possible division points. The best improvement found from any split point is saved as the best possible split for that predictor variable in this node. The process is then repeated for each other predictor variable. The best split found for any predictor variable is used to perform the actual split on the node.

When examining the possible splits for a categorical predictor variable, the calculations are more complex and potentially much more time consuming.

If the predictor variable is categorical and the target variable is continuous, the categories of the predictor variable are sorted so that the mean value of the target variable for the rows having each category of the predictor are increasing. For example, if the target variable is Income and the predictor variable has three categories, single, married and divorced, the categories are ordered so that the mean value of Income for the people in each category is increasing. The splitting process then tries each split point between each category of the predictor. This is very similar to the process used for continuous predictor variables except the categories are arranged by values of the target variable rather than by values of the predictor variable. The number of splits evaluated is equal to the number of categories of the predictor less one.

If both the target variable and the predictor variable are categorical, the process gets more complex. In this case, to perform an exhaustive search DTREG must evaluate a potential split for every possible combination of categories of the predictor variable. The number of splits is equal to 2^(k-1)-1 where k is the number of categories of the predictor variable. For example, if there are 5 categories, 15 splits are tried; if there are 10 categories, 511 splits are tried; if there are 16 categories, 32,767 splits are tried; if there are 32 categories, 2,147,483,647 splits are tried. Because of this exponential growth, the computation time to do an exhaustive search becomes prohibitive when there are more than about 12 predictor categories. In this case, DTREG uses a clustering algorithm (see below).

There is one case where classification trees are efficient to build using exhaustive search even with categorical predictors having a large number of categories. That is the case where the target variable has only two possible categories. Fortunately, this situation occurs fairly often – the target categories might be live/die, bought-product/did-not-buy, malignant/benign, etc. For this situation, DTREG has to evaluate only many splits as the number of categories for the predictor variable less one.

In order to make it feasible to construct classification trees with target variables that have more than two categories and predictor variables that have a large number of categories, DTREG switches from using an exhaustive search to a cluster analysis method when the number of predictor categories exceeds a threshold that you can specify on the Model Design property page. This technique uses cluster analysis to group the categories of the target variable into two groups. DTREG is then able to try only (k-1) splits, where k is the number of predictor categories.

Once DTREG has evaluated each possible split for each possible predictor variable, a node is split using the best split found. The runner-up splits are remembered and displayed as “Competitor Splits” in the report.

Evaluating Splits

The ideal split would divide a group into two child groups in such a way so that all of the rows in the left child are the same (have the same value on the target variable) and all of the rows in the right group are the same – but different from the left group. If such a split can be found, then you can exactly and perfectly classify all of the rows by using just that split, and no further splits are necessary or useful. Such a perfect split is possible only if the rows in the node being split have only two possible values on the target variable.

Unfortunately, perfect splits do not occur often, so it is necessary to evaluate and compare the quality of imperfect splits. Various criteria have been proposed for evaluating splits, but they all have the same basic goal which is to favor homogeneity within each child node and heterogeneity between the child nodes. The heterogeneity – or dispersion – of target categories within a node is called the “node impurity”. The goal of splitting is to produce child nodes with minimum impurity.

The impurity of every node is calculated by examining the distribution of categories of the target variable for the rows in the group. A “pure” node, where all rows have the same value of the target variable, has an impurity value of 0 (zero). When a potential split is evaluated, the probability-weighted average of the impurities of the two child nodes is subtracted from the impurity of the parent node. This reduction in impurity is called the improvement of the split. The split with the greatest improvement is the one used. Improvement values for splits are shown in the node information that is part of the report generated by DTREG.

DTREG provides two methods for evaluating the quality of splits when building classification trees (categorical target) (1) Gini and (2) Entropy,. Only one method is provided when building regression trees (continuous target variable), and that is minimum variance within nodes. The minimum variance/least squares criteria is essential the same criteria used by traditional, numeric regression analysis (i.e., line and function fitting).

Experience has shown that the splitting criterion is not very important, and Gini and Entropy yield trees that are very similar. Gini is considered slightly better than Entropy, and it is the default criteria used for classification trees. See Breiman, Friedman, Olshen and Stone Classification And Regression Trees for a technical description of the Gini and Entropy criteria.

Assigning Categories to Nodes

When a decision tree is used to predict values of the target variable, rows are run through the tree down to the point where they reach a terminal node. The category assigned to the terminal node is the predicted value for the row being evaluated. So a natural question is how categories are assigned to nodes.

For regression trees built with a continuous target variable, the value assigned to a node is simply the average value of the target variable for all rows that end up in the node weighted by the row weights.

For classification trees built with a categorical target variable, the determination of what category to assign to a node is more complex: it is the category that minimizes the misclassification cost for the rows in the node. The calculation of the misclassification cost is somewhat complex. The formula involves the distribution of the categories compared with the distribution in the total (learning) sample, the prior values and the misclassification costs you assign. In the simplest case, every row that is misclassified has a cost of 1 and every row that is correctly classified has a cost of 0. The misclassification cost for every node is displayed in the report generated by DTREG. A misclassification summary table is included near the end of the report.

If you wish, you can specify specific costs for misclassifying one target category as another target category. For example, you might want to assign a greater cost to classifying a heart attack as indigestion than classifying indigestion as a heart attack. These misclassification costs are implemented by generating altered prior values that are used in the calculation. See Breiman, Friedman, et al for a detailed description of how misclassification costs are used.

Stopping Criteria

If no limits were placed on the size of a tree, DTREG theoretically might build a tree so large that every row of the learning dataset ended up in its own terminal node. But doing this would be computationally expensive, and the tree would be so large that it would be difficult or impossible to interpret and display.

Several criteria are used to limit how large a tree DTREG constructs. Once a tree is built, the pruning method described in a following section is used to reduce its size to the optimal number of nodes.

The following criteria are used to limit the size of a tree as it is build:

• Minimum size node to split. On the Design property page, you can specify that nodes containing fewer than a specified number of rows are not to be split.
• Maximum tree depth. On the Design property page, you can specify the maximum number of levels in the tree that are to be constructed.