Understanding Decision Trees for Regression: Step-by-Step Explanation
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Decision trees are a popular algorithm for both classification and regression tasks. In this article, we will focus on how a decision tree works for regression, breaking down the process step by step with an example to make it easier to understand. We'll also cover the formula used to calculate the best split by minimizing errors, making this guide both comprehensive and practical.
A regression decision tree is a model that predicts continuous values (like prices, temperatures, etc.) by splitting the data into smaller and smaller subsets based on feature values. At each split, the tree aims to minimize the error, and the final prediction is the average of the target values in the leaf nodes.
Let’s dive into the step-by-step process of building a decision tree for regression with an example dataset.
Consider the following dataset with three input features (x1, x2, x3) and a target value:
x1 | x2 | x3 | Target |
---|---|---|---|
3 | 100 | 70 | 0.1 |
5 | 200 | 80 | 0.2 |
We will use this dataset to understand how the decision tree makes predictions in regression.
The first step in building a regression decision tree is to split the data into regions based on the feature values. The tree evaluates each feature (x1, x2, x3) and splits the data into ranges.
(1-6)
and (6-10)
(1-100)
, (100-300)
, and (300-500)
(1-30)
, (30-60)
, and (60-100)
For each split, the tree calculates the target values within that range.
After splitting the data, the decision tree calculates the average target value in each region. For this example, let’s assume the average target value for each region is calculated as follows:
For simplicity, let’s say the average target value () for a particular region is 3.
The final prediction for a new input is the average of the target values from the regions it falls into. If a new input falls into regions with predictions , , and , the final prediction is:
For example, if the predictions from the regions are 2.5, 3.0, and 3.5, the final prediction will be:
The decision tree selects the best split by minimizing the Sum of Squared Errors (SSE). The split that results in the smallest error is chosen.
The error is calculated using the following formula:
Where:
The tree continues splitting until the error is minimized or it reaches a predefined stopping criterion (like maximum depth or minimum samples per leaf).
Consider a dataset with a feature and a target variable :
2 | 3.0 |
4 | 3.5 |
6 | 5.0 |
8 | 7.5 |
10 | 9.0 |
To determine the optimal split:
In this example, the split at yields a lower total SSE (3.455) compared to the split at (8.455). Therefore, the decision tree would select as the optimal split point to minimize prediction errors.
In a regression decision tree, selecting the optimal split at each node is crucial for accurate predictions. The primary objective is to partition the data into subsets that minimize the Sum of Squared Errors (SSE) within each region.
Steps to Determine the Optimal Split:
Evaluate Potential Splits:
Calculate the Sum of Squared Errors (SSE):
Select the Split with the Minimum SSE:
Illustrative Example:
Consider a dataset with a feature and a target variable :
2 | 3.0 |
4 | 3.5 |
6 | 5.0 |
8 | 7.5 |
10 | 9.0 |
To determine the optimal split:
Potential Split at :
Potential Split at :
In this example, the split at yields a lower total SSE (3.455) compared to the split at (8.455). Therefore, the decision tree would select as the optimal split point to minimize prediction errors.
By systematically evaluating all possible splits and selecting the one that minimizes SSE, regression decision trees effectively partition the data into regions that lead to the most accurate predictions.