Feature engineering: deriving a feature that makes the model's job easier

The problem, from first principles

Imagine labeling points as "inside" or "outside" a circle of radius 3, based only on their (x, y) coordinates. Individually, x alone tells you almost nothing about the label (a point at x=2 could be inside the circle or far outside it, depending on y), and the same goes for y alone, 04-data-leakage's own correlation check would find both features nearly USELESS on their own, even though the label is entirely determined by x and y together. A linear model, in particular, genuinely cannot represent "inside a circle" as a straight-line decision boundary in raw (x, y) space, no matter how it's trained, the true boundary is curved.

But compute ONE new number, sqrt(x^2 + y^2) (distance from the origin), and the exact same problem becomes trivial: "inside the circle" is now just "this single number is less than 3," a problem any linear model handles instantly. That's feature engineering: deriving a new feature from existing ones specifically because it captures the actual structure of the problem in a form the model can use directly, rather than hoping the model discovers that structure on its own from raw, poorly-shaped inputs.

From theory to code

Theory gives two concrete, common engineered features: distance from the origin (sqrt(x^2 + y^2), useful whenever "how far from center" matters more than raw coordinates) and a ratio between two raw features (useful whenever the RELATIONSHIP between two quantities matters more than either alone).

Implement radius_feature(x, y) and ratio_feature(numerator, denominator) against that reasoning. The signatures and docstrings are already in the editor.

Constraints

• Both functions are vectorized: they accept arrays and return arrays of the same shape.
• radius_feature always returns non-negative values.

Hints

Open one at a time. Each gives away a little more than the last.