Underfitting (Intuition) Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Compare two regression models on the same data: Model 1 (linear): training R2=0.40R^2=0.40, test R2=0.38R^2=0.38. Model 2 (cubic): training R2=0.95R^2=0.95, test R2=0.35R^2=0.35. Diagnose each model.

Solution

  1. 1
    Model 1: similar training and test R2R^2 (0.40 vs 0.38) — model generalizes well but fits poorly; this is underfitting (high bias, low variance)
  2. 2
    Model 2: very high training R2R^2 (0.95) but poor test R2R^2 (0.35) — doesn't generalize; this is overfitting (low bias, high variance)
  3. 3
    Recommendation: neither model is optimal; try intermediate complexity (quadratic) which may balance fit and generalization
  4. 4
    Both training AND test errors should be low for the ideal model

Answer

Model 1: underfitting (low R² everywhere). Model 2: overfitting (high train, low test R²). Need intermediate complexity.
Diagnosing over vs. underfitting requires comparing training and test performance. Small training-test gap with low performance = underfitting. Large training-test gap = overfitting. The goal is high performance with a small gap.

About Underfitting (Intuition)

Underfitting occurs when a model is too simple to capture the true pattern in the data, performing poorly on both training data and new data.

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