📊 Model Evaluation and Validation

About this simulation

A polynomial curve is fitted, live in your browser, to a noisy scatter of data points on the "Data & Fit" board. Behind the scenes it uses ordinary least-squares regression solved with Gaussian elimination — the same core idea behind fitting far more complex models. To the right, a bar chart plots training error against held-out validation error for every complexity level, and below sits a spinning k-fold cross-validation wheel showing which slice of data is currently held out for testing.

How it works

• Model complexity — the polynomial degree. Low degree underfits (high bias, misses the pattern); high degree overfits (low bias, but wiggles to chase noise and generalizes poorly).
• Cross-validation folds (k) — the dataset is split into k roughly equal slices ("folds"). Each fold takes a turn as the held-out validation set while the model trains on the rest.
• Held-out fold — pick which slice is currently used for validation; the curve refits on the remaining k−1 folds only.
• Data noise — how much random scatter is added around the true underlying function.
• The bar chart's blue bars (train error) keep shrinking as complexity rises, while the red bars (validation error) form a U-shape — the classic signature of the bias-variance tradeoff. The lowest point of the red curve is the model complexity least likely to overfit.

Controls

• Model complexity — polynomial degree of the fitted curve (1–10).
• Cross-validation folds (k) — number of folds the data is split into (3–8).
• Held-out fold — which fold is currently the validation set.
• Data noise — amount of random noise added to the training data.
• Auto-cycle folds — automatically rotates through every fold, exactly like running a full k-fold cross-validation pass.
• ↺ Regenerate data — draws a fresh noisy sample. Drag to rotate the scene, scroll to zoom.

Did you know?

k-fold cross-validation (commonly k=5 or k=10) is preferred over a single train/test split because every data point gets to be validated exactly once, giving a much more stable estimate of how a model will perform on truly unseen data.