Random Forest Regression – Why Not Predicting Higher Than Training Data?

Question

I've noticed that when building random forest regression models, at least in R, the predicted value never exceeds the maximum value of the target variable seen in the training data. As an example, see the code below. I'm building a regression model to predict mpg based on the mtcars data. I build OLS and random forest models, and use them to predict mpg for a hypothetical car that should have very good fuel economy. The OLS predicts a high mpg, as expected, but random forest does not. I've noticed this in more complex models too. Why is this?

> library(datasets)
> library(randomForest)
> 
> data(mtcars)
> max(mtcars$mpg)
[1] 33.9
> 
> set.seed(2)
> fit1 <- lm(mpg~., data=mtcars) #OLS fit
> fit2 <- randomForest(mpg~., data=mtcars) #random forest fit
> 
> #Hypothetical car that should have very high mpg
> hypCar <- data.frame(cyl=4, disp=50, hp=40, drat=5.5, wt=1, qsec=24, vs=1, am=1, gear=4, carb=1)
> 
> predict(fit1, hypCar) #OLS predicts higher mpg than max(mtcars$mpg)
      1 
37.2441 
> predict(fit2, hypCar) #RF does not predict higher mpg than max(mtcars$mpg)
       1 
30.78899 

Answer 1

As it has been mentioned already in previous answers, random forest for regression / regression trees doesn't produce expected predictions for data points beyond the scope of training data range because they cannot extrapolate (well). A regression tree consists of a hierarchy of nodes, where each node specifies a test to be carried out on an attribute value and each leaf (terminal) node specifies a rule to calculate a predicted output. In your case the testing observation flow through the trees to leaf nodes stating, e.g., "if x > 335, then y = 15", which are then averaged by random forest.

Here is an R script visualizing the situation with both random forest and linear regression. In random forest's case, predictions are constant for testing data points that are either below the lowest training data x-value or above the highest training data x-value.

library(datasets)
library(randomForest)
library(ggplot2)
library(ggthemes)

# Import mtcars (Motor Trend Car Road Tests) dataset
data(mtcars)

# Define training data
train_data = data.frame(
    x = mtcars$hp,  # Gross horsepower
    y = mtcars$qsec)  # 1/4 mile time

# Train random forest model for regression
random_forest <- randomForest(x = matrix(train_data$x),
                              y = matrix(train_data$y), ntree = 20)
# Train linear regression model using ordinary least squares (OLS) estimator
linear_regr <- lm(y ~ x, train_data)

# Create testing data
test_data = data.frame(x = seq(0, 400))

# Predict targets for testing data points
test_data$y_predicted_rf <- predict(random_forest, matrix(test_data$x)) 
test_data$y_predicted_linreg <- predict(linear_regr, test_data)

# Visualize
ggplot2::ggplot() + 
    # Training data points
    ggplot2::geom_point(data = train_data, size = 2,
                        ggplot2::aes(x = x, y = y, color = "Training data")) +
    # Random forest predictions
    ggplot2::geom_line(data = test_data, size = 2, alpha = 0.7,
                       ggplot2::aes(x = x, y = y_predicted_rf,
                                    color = "Predicted with random forest")) +
    # Linear regression predictions
    ggplot2::geom_line(data = test_data, size = 2, alpha = 0.7,
                       ggplot2::aes(x = x, y = y_predicted_linreg,
                                    color = "Predicted with linear regression")) +
    # Hide legend title, change legend location and add axis labels
    ggplot2::theme(legend.title = element_blank(),
                   legend.position = "bottom") + labs(y = "1/4 mile time",
                                                      x = "Gross horsepower") +
    ggthemes::scale_colour_colorblind()