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Decision Tree - Regression

Decision tree builds regression or classification models in the form of a tree structure. It brakes down a dataset into smaller and smaller subsets while at the same time an associated decision tree is incrementally developed. The final result is a tree with decision nodes and leaf nodes. A decision node (e.g., Outlook) has two or more branches (e.g., Sunny, Overcast and Rainy), each representing values for the attribute tested. Leaf node (e.g., Hours Played) represents a decision on the numerical target. The topmost decision node in a tree which corresponds to the best predictor called root node. Decision trees can handle both categorical and numerical data. 


Decision Tree Algorithm

The core algorithm for building decision trees called ID3 by J. R. Quinlan which employs a top-down, greedy search through the space of possible branches with no backtracking. The ID3 algorithm can be used to construct a decision tree for regression by replacing Information Gain with Standard Deviation Reduction.
Standard Deviation
A decision tree is built top-down from a root node and involves partitioning the data into subsets that contain instances with similar values (homogenous). We use standard deviation to calculate the homogeneity of a numerical sample. If the numerical sample is completely homogeneous its standard deviation is zero.

a) Standard deviation for one attribute:

b) Standard deviation for two attributes:

Standard Deviation Reduction
The standard deviation reduction is based on the decrease in standard deviation after a dataset is split on an attribute. Constructing a decision tree is all about finding attribute that returns the highest standard deviation reduction (i.e., the most homogeneous branches).
Step 1: The standard deviation of the target is calculated. 

Standard deviation (Hours Played) = 9.32

Step 2: The dataset is then split on the different attributes. The standard deviation for each branch is calculated. The resulting standard deviation is subtracted from the standard deviation before the split. The result is the standard deviation reduction. 

Step 3: The attribute with the largest standard deviation reduction is chosen for the decision node. 

Step 4a: Dataset is divided based on the values of the selected attribute.

Step 4b: A branch set with standard deviation more than 0 needs further splitting. 
In practice, we need some termination criteria. For example, when standard deviation for the branch becomes smaller than a certain fraction (e.g., 5%) of standard deviation for the full dataset OR when too few instances remain in the branch (e.g., 3).

Step 5: The process is run recursively on the non-leaf branches, until all data is processed.
When the number of instances is more than one at a leaf node we calculate the average as the final value for the target.
Try to invent a new algorithm to construct a decision tree from data using MLR instead of average at the leaf node.