Understanding Random Forests for Regression: Beyond the Linear Equation

Random forests for regression, while powerful, operate with a fundamentally different approach compared to linear regression. This article explores the key differences, focusing on the interpretabilty, model representation, and the mathematical underpinnings of these methods.

Linear Regression: A Classical Approach

Linear regression is a widely used method for predicting a continuous target variable. Its simplicity and interpretability make it a preferred choice in many applications. The output of a linear regression model can be expressed as an equation of the form:

y b_0 b_1x_1 b_2x_2 … b_nx_n

where b_0, b_1, b_2, …, b_n are the coefficients that are determined during the training process. These coefficients indicate the strength and direction of the relationship between each feature and the target variable, making it straightforward to interpret the model.

Random Forest Regression: A Robust Ensemble Method

Random forests for regression are a form of ensemble learning that combines multiple decision trees to improve the prediction accuracy and robustness. This method does not yield a simple equation like linear regression. Instead, it generates predictions based on an ensemble of decision trees. Here's a deeper dive into the key aspects of random forests for regression:

Output

Random forest regression works by creating multiple decision trees, each trained on a random subset of the data. Each tree makes a prediction based on the input features, and the final output is the average of all tree predictions. This aggregation process captures complex, non-linear relationships in the data.

Interpretability

One of the challenges of random forest regression is interpretability. Unlike linear regression, it does not directly provide an explicit formula for predictions. Instead, each tree in the ensemble can be seen as a path within a decision tree, where the final output is the average of the predictions from all the trees. This makes it more difficult to interpret the model in a straightforward manner.

Feature Importance

Despite the lack of a simple equation, random forests offer insights into feature importance. These insights can help identify which features have the most influence on the model's predictions. This is particularly useful in understanding the underlying data patterns without having to fully interpret the model as a whole.

Mathematical Representation

Mathematically, it is possible to represent a single decision tree in terms of indicator variables, where the output can be expressed as a sum of products of these variables. For a random forest, this would involve summing the output of many such decision trees.

For instance, a partial decision function for a single tree on two features might look like this:

2.7Ix2Iy-1 3.1Ix2Iy-1 …

To represent an entire random forest, you would sum many of these decision functions. While this could be simplified with some effort, the resulting equations would not be simple or quickly interpretable like a linear regression equation.

Technically, you could write down the decision function learned by a random forest, but doing so would not be simple or satisfying to look at. Instead, the model is often treated as a black-box at this level of granularity. The decision function would look more like a large program or algorithm consisting of nested IF-THEN/ELSE statements, with each unique path eventually leading to a separate linear regression equation.

However, it is worth noting that the random forest of regressors allows you to follow multiple paths simultaneously, then average the outputs of the linear regressions at the end of each path to produce a final output.

In conclusion, random forests for regression do not yield a simple equation like linear regression. Instead, they operate as a complex model that captures non-linear relationships without providing an explicit formula for predictions. While this approach is beneficial for capturing intricate patterns in the data, it sacrifices some interpretability.