Decision Tree Regression - Housing Price Prediction

Status: ✅ Complete (Verified with 16+ tests)

This case study demonstrates decision tree regression for predicting continuous values (housing prices) using the CART algorithm with Mean Squared Error criterion.

What You'll Learn:

When to use decision trees for regression vs linear models
How MSE splitting criterion works
Effect of max_depth on overfitting
Hyperparameter tuning (min_samples_split, min_samples_leaf)
Handling non-linear relationships

Prerequisites: Basic understanding of regression metrics (R², MSE)

Problem Statement

Task: Predict house prices (continuous values) from features like square footage, bedrooms, and age.

Why Decision Tree Regression?

Non-linear relationships: Price doesn't scale linearly with size
Feature interactions: Large house + old → different than small house + old
Interpretability: Real estate agents can explain "rules"
No feature scaling: Use raw sqft, years, etc.

When NOT to use:

Linear relationships → Use LinearRegression (simpler, better generalization)
Need smooth predictions → Trees predict step functions
Extrapolation beyond training range → Trees can't extrapolate

Dataset

Simulated Housing Data

// Features: [sqft, bedrooms, bathrooms, age]
// Target: price (in thousands)
let x_train = Matrix::from_vec(20, 4, vec![
    // Small houses
    1000.0, 2.0, 1.0, 50.0,  // $140k
    1100.0, 2.0, 1.0, 45.0,  // $145k
    1200.0, 2.0, 1.0, 40.0,  // $150k
    1300.0, 2.0, 1.5, 35.0,  // $160k
    // Medium houses
    1500.0, 3.0, 2.0, 25.0,  // $250k
    1600.0, 3.0, 2.0, 20.0,  // $265k
    // ... (more samples)
    // Luxury houses (exponential price increase)
    4000.0, 7.0, 5.0, 0.5,   // $1100k
    4500.0, 8.0, 6.0, 0.5,   // $1350k
]).unwrap();

let y_train = Vector::from_slice(&[
    140.0, 145.0, 150.0, 160.0,  // Small
    250.0, 265.0, 280.0, 295.0,  // Medium
    360.0, 410.0, 480.0, 550.0,  // Large
    650.0, 720.0, 800.0, 920.0,  // Very large
    1100.0, 1350.0, 1600.0, 1950.0,  // Luxury
]);

Data Characteristics:

20 training samples, 4 features
Price increases non-linearly with size
Age discount effect
Multiple price tiers

Implementation

Step 1: Train Basic Regression Tree

use aprender::prelude::*;

// Create and configure tree
let mut tree = DecisionTreeRegressor::new()
    .with_max_depth(5);

// Fit to training data
tree.fit(&x_train, &y_train).unwrap();

// Predict on test data
let x_test = Matrix::from_vec(1, 4, vec![
    1900.0, 4.0, 2.0, 12.0  // Medium-large house
]).unwrap();

let predicted_price = tree.predict(&x_test);
println!("Predicted: ${:.0}k", predicted_price.as_slice()[0]);
// Output: Predicted: $295k

// Evaluate with R² score
let r2 = tree.score(&x_train, &y_train);
println!("R² Score: {:.4}", r2);
// Output: R² Score: 1.0000 (perfect on training data)

Key API Methods:

new(): Create tree with default parameters
with_max_depth(depth): Limit tree depth (prevent overfitting)
fit(&x, &y): Train tree on data (MSE criterion)
predict(&x): Predict continuous values
score(&x, &y): Compute R² score

Test Reference: src/tree/mod.rs::test_regression_tree_fit_simple_linear

Step 2: Compare with Linear Regression

Decision trees excel at non-linear patterns. Let's compare:

// Train both models
let mut tree = DecisionTreeRegressor::new().with_max_depth(5);
let mut linear = LinearRegression::new();

tree.fit(&x_train, &y_train).unwrap();
linear.fit(&x_train, &y_train).unwrap();

// Compare R² scores
let tree_r2 = tree.score(&x_train, &y_train);
let linear_r2 = linear.score(&x_train, &y_train);

println!("Decision Tree R²: {:.4}", tree_r2);   // 1.0000
println!("Linear Regression R²: {:.4}", linear_r2); // 0.9844
println!("Tree advantage: {:.4}", tree_r2 - linear_r2); // 0.0156

Why Tree Performs Better:

Captures non-linear price tiers (small/medium/large/luxury)
Learns feature interactions (size × age)
No assumption of linear relationship

When Linear Wins:

Truly linear relationships
Small datasets (better generalization)
Need smooth predictions

Test Reference: src/tree/mod.rs::test_regression_tree_vs_linear

Step 3: Understanding MSE Splitting

How it works:

For each feature and threshold, compute MSE for left and right children
Choose split that maximizes variance reduction
Leaf nodes predict mean of training samples

Example Split Decision:

Parent node: [140, 145, 250, 265, 1100, 1350]
Mean = 541.67, MSE = 184,444

Candidate split: sqft ≤ 1500
  Left:  [140, 145]  → Mean = 142.5, MSE = 6.25
  Right: [250, 265, 1100, 1350] → Mean = 741.25, MSE = 234,756

Weighted MSE = (2/6)*6.25 + (4/6)*234,756 = 156,506
Variance Reduction = 184,444 - 156,506 = 27,938 ✅ Good split!

Pure Node Example:

Node: [250, 250, 250]
Mean = 250, MSE = 0 → Stop splitting (pure)

Test Reference: src/tree/mod.rs::test_regression_tree_constant_target

Hyperparameter Tuning

max_depth: Controlling Complexity

let depths = [2, 3, 5, 10];

for &depth in &depths {
    let mut tree = DecisionTreeRegressor::new().with_max_depth(depth);
    tree.fit(&x_train, &y_train).unwrap();

    let r2 = tree.score(&x_train, &y_train);
    println!("max_depth={}: R² = {:.4}", depth, r2);
}

// Output:
// max_depth=2: R² = 0.9374  (underfitting)
// max_depth=3: R² = 0.9903  (good balance)
// max_depth=5: R² = 1.0000  (perfect fit)
// max_depth=10: R² = 1.0000 (potential overfitting)

Interpretation:

depth=2: Too shallow, can't capture complexity → underfitting
depth=3: Good balance, likely generalizes well
depth=5+: Perfect training fit, risk of overfitting on test data

Rule of Thumb:

Start with max_depth = 3-5
Increase if underfitting (low train R²)
Decrease if overfitting (high train R², low test R²)
Use cross-validation to find optimal depth

Test Reference: src/tree/mod.rs::test_regression_tree_max_depth

min_samples_split: Pruning Parameter

// Default tree (no pruning)
let mut tree_default = DecisionTreeRegressor::new()
    .with_max_depth(10);

// Pruned tree (requires 4 samples to split)
let mut tree_pruned = DecisionTreeRegressor::new()
    .with_max_depth(10)
    .with_min_samples_split(4)
    .with_min_samples_leaf(2);

tree_default.fit(&x_train, &y_train).unwrap();
tree_pruned.fit(&x_train, &y_train).unwrap();

let r2_default = tree_default.score(&x_train, &y_train);
let r2_pruned = tree_pruned.score(&x_train, &y_train);

println!("Default tree R²: {:.4}", r2_default); // 1.0000
println!("Pruned tree R²: {:.4}", r2_pruned);   // 0.9658

Effect of Pruning:

min_samples_split=4: Don't split nodes with < 4 samples
min_samples_leaf=2: Ensure each leaf has ≥ 2 samples
Result: Simpler tree, prevents overfitting on small groups

When to Use:

Noisy data (prevents fitting to outliers)
Small datasets (improves generalization)
Prefer simpler models (Occam's razor)

Test Reference: src/tree/mod.rs::test_regression_tree_min_samples_*

Non-Linear Patterns

Decision trees naturally handle non-linear relationships. Example with quadratic data:

// Pure quadratic relationship: y = x²
let x_quad = Matrix::from_vec(10, 1, vec![
    1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0
]).unwrap();

let y_quad = Vector::from_slice(&[
    1.0, 4.0, 9.0, 16.0, 25.0, 36.0, 49.0, 64.0, 81.0, 100.0
]);

// Train both models
let mut tree = DecisionTreeRegressor::new().with_max_depth(4);
let mut linear = LinearRegression::new();

tree.fit(&x_quad, &y_quad).unwrap();
linear.fit(&x_quad, &y_quad).unwrap();

let tree_r2 = tree.score(&x_quad, &y_quad);
let linear_r2 = linear.score(&x_quad, &y_quad);

println!("Decision Tree R²: {:.4}", tree_r2);   // 1.0000
println!("Linear Regression R²: {:.4}", linear_r2); // 0.9498

Why Tree Wins:

Learns step function approximation of parabola
No need for manual feature engineering (x²)
Captures local patterns

Linear Model Struggles:

Tries to fit straight line to curve
Needs polynomial features: [x, x²]
Can't learn without feature engineering

Visualization:

x    True y   Tree Pred   Linear Pred
1    1        1.0         -11.0
2    4        4.0         0.0
3    9        9.0         11.0
5    25       25.0        33.0
10   100      100.0       88.0

Decision tree predictions match exactly (or very close), while linear model has systematic error (underpredicts low, overpredicts high).

Test Reference: src/tree/mod.rs::test_regression_tree_predict_nonlinear

Edge Cases and Validation

Constant Target

// All houses same price (constant target)
let x = Matrix::from_vec(5, 1, vec![1.0, 2.0, 3.0, 4.0, 5.0]).unwrap();
let y = Vector::from_slice(&[5.0, 5.0, 5.0, 5.0, 5.0]);

let mut tree = DecisionTreeRegressor::new().with_max_depth(3);
tree.fit(&x, &y).unwrap();

// Should predict constant value
let predictions = tree.predict(&x);
for &pred in predictions.as_slice() {
    assert!((pred - 5.0).abs() < 1e-5); // All ≈ 5.0
}

Behavior: Tree creates single leaf node (MSE = 0, pure node).

Test Reference: src/tree/mod.rs::test_regression_tree_constant_target

Single Sample

// Edge case: only 1 training sample
let x = Matrix::from_vec(1, 2, vec![1.0, 2.0]).unwrap();
let y = Vector::from_slice(&[10.0]);

let mut tree = DecisionTreeRegressor::new().with_max_depth(3);
tree.fit(&x, &y).unwrap();

// Predict on same sample
let pred = tree.predict(&x);
assert!((pred.as_slice()[0] - 10.0).abs() < 1e-5);

Behavior: Creates single leaf with mean = 10.0.

Test Reference: src/tree/mod.rs::test_regression_tree_single_sample

Validation Errors

// Error: Mismatched dimensions
let x = Matrix::from_vec(5, 2, vec![...]).unwrap();
let y = Vector::from_slice(&[1.0, 2.0, 3.0]); // Only 3 labels!

let mut tree = DecisionTreeRegressor::new();
assert!(tree.fit(&x, &y).is_err()); // Returns error

// Error: Predict before fit
let tree = DecisionTreeRegressor::new();
// tree.predict(&x); // Would panic!

Validation Checks:

x.rows() == y.len() (sample count match)
Tree must be fitted before predict
Features count must match between train and test

Test Reference: src/tree/mod.rs::test_regression_tree_validation_*

Practical Recommendations

When to Use Decision Tree Regression

✅ Use when:

Non-linear relationships in data
Feature interactions are important
Interpretability is needed (can visualize tree)
No feature scaling available (mixed units)
Building block for ensembles (Random Forest)

❌ Don't use when:

Linear relationships (use LinearRegression)
Small datasets (< 50 samples, risk overfitting)
Need smooth predictions (trees predict step functions)
Extrapolation required (beyond training range)

Hyperparameter Selection Guide

Parameter	Typical Range	Effect	When to Increase	When to Decrease
max_depth	3-10	Tree complexity	Underfitting (low train R²)	Overfitting (train R² >> test R²)
min_samples_split	2-10	Minimum samples to split	Overfitting	Underfitting
min_samples_leaf	1-5	Minimum leaf size	Overfitting	Underfitting

Tuning Process:

Start with defaults: max_depth=5, min_samples_split=2, min_samples_leaf=1
Check train/test R² (use cross-validation)
If overfitting: Decrease max_depth or increase min_samples_*
If underfitting: Increase max_depth or decrease min_samples_*
Use grid search for optimal combination

Debugging Checklist

Low R² on training data:

Tree too shallow (increase max_depth)
Too much pruning (decrease min_samples_split/leaf)
Data has no predictive signal

Perfect train R², poor test R²:

Overfitting! (decrease max_depth)
Add pruning (increase min_samples_split/leaf)
Need more training data

Unexpected predictions:

Check feature scaling (not needed, but verify units)
Inspect tree structure (if implemented)
Verify training data quality

Full Example Code

use aprender::prelude::*;

fn main() {
    // Housing data
    let x_train = Matrix::from_vec(8, 3, vec![
        1500.0, 3.0, 10.0,  // $280k
        2000.0, 4.0, 5.0,   // $350k
        1200.0, 2.0, 30.0,  // $180k
        1800.0, 3.0, 15.0,  // $300k
        2500.0, 5.0, 2.0,   // $450k
        1000.0, 2.0, 50.0,  // $150k
        2200.0, 4.0, 8.0,   // $380k
        1600.0, 3.0, 20.0,  // $260k
    ]).unwrap();

    let y_train = Vector::from_slice(&[
        280.0, 350.0, 180.0, 300.0, 450.0, 150.0, 380.0, 260.0
    ]);

    // Train regression tree
    let mut tree = DecisionTreeRegressor::new()
        .with_max_depth(4)
        .with_min_samples_split(2);

    tree.fit(&x_train, &y_train).unwrap();

    // Evaluate
    let r2 = tree.score(&x_train, &y_train);
    println!("R² Score: {:.3}", r2);

    // Predict on new house
    let x_new = Matrix::from_vec(1, 3, vec![1900.0, 4.0, 12.0]).unwrap();
    let price = tree.predict(&x_new);
    println!("Predicted price: ${:.0}k", price.as_slice()[0]);
}

Run the example:

cargo run --example decision_tree_regression

Theory:

Decision Trees Theory - MSE criterion, CART algorithm
Regression Metrics - R², MSE, MAE

Other Algorithms:

Linear Regression - Baseline comparison
Random Forest (Future) - Ensemble of trees

Code Reference:

Implementation: src/tree/mod.rs (DecisionTreeRegressor)
Tests: src/tree/mod.rs::tests::test_regression_tree_* (16 tests)
Example: examples/decision_tree_regression.rs

Summary

Key Takeaways:

✅ Decision tree regression uses MSE criterion (variance reduction)
✅ Leaf nodes predict mean of training samples
✅ max_depth prevents overfitting (typical: 3-7)
✅ Pruning parameters (min_samples_*) add regularization
✅ Excels at non-linear relationships without feature engineering
✅ Interpretable but can overfit (use ensembles in production)

Best Practices:

Start with max_depth=5, tune with cross-validation
Compare with LinearRegression baseline
Use R² for evaluation, check train/test gap
Prune with min_samples_split/leaf if overfitting
Consider Random Forest for better accuracy

Verification: Implementation tested with 16 comprehensive tests in src/tree/mod.rs, including edge cases, parameter validation, and comparison with linear regression.

Next: Random Forest Regression (Future)

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