📊 Regression Fit & Overfitting Explorer

About this simulation

Every predictive model in data science is a compromise between fitting the data you have and generalizing to data you haven't seen. This simulation generates a synthetic dataset — points scattered in 3D around a hidden "true" surface with added noise — then fits a polynomial regression surface to a training subset using least-squares. Watching the fitted surface change as you turn the complexity dial up or down makes underfitting and overfitting tangible instead of abstract.

How it works

• Underfitting — a low-degree surface (e.g. a flat plane) is too rigid to capture real curvature, so it misses structure in both the training and test data. Both errors stay high.
• Good fit — a moderate degree tracks the true pattern's shape without chasing noise; training and test error are both low and close together.
• Overfitting — a high-degree surface bends itself around every noisy training point, driving training error toward zero while test error on unseen points climbs — the classic bias–variance tradeoff.
• The regression is solved with the normal-equation method (least squares) over a polynomial basis in the two ground-plane coordinates.

Controls

• Sample size — how many data points are generated; smaller samples overfit more easily.
• Noise level — how much random scatter is added around the true surface.
• Model complexity — the polynomial degree used to fit the surface (1 = plane, 7 = highly flexible).
• True pattern — the hidden function generating the data, for comparison against the fit.
• Train / test split — colors 25% of points orange as a held-out test set and reports separate error metrics.

Did you know?

Practitioners guard against overfitting with techniques like cross-validation, regularization (ridge/lasso) and simply collecting more data — all of which, in effect, trade a little training accuracy for a model that generalizes better.