PolynomialFeatures

Generates a new feature matrix consisting of all polynomial combinations of the input features up to a given degree. This enables fitting polynomial relationships with a linear model.

Feature Expansion

For input features $[x_1, x_2]$ with degree $d = 2$ and bias:

$$\phi(x) = [1,\; x_1,\; x_2,\; x_1^2,\; x_1 x_2,\; x_2^2]$$

The number of output features is:

$$\text{output features} = \binom{p + d}{d} \quad \text{(with bias)}, \qquad \binom{p + d}{d} - 1 \quad \text{(without bias)}$$

where $p$ is the number of input features and $d$ is the degree. For example, $p = 2,\, d = 3$ produces $\binom{5}{3} = 10$ features.

Interaction Only

When interaction_only = true, only interaction features are generated (no powers $x_j^k$ for $k > 1$). For $[x_1, x_2]$ with $d = 2$, this yields $[1,\; x_1,\; x_2,\; x_1 x_2]$.

Usage with Linear Models

Polynomial features combined with a linear model create a polynomial regression pipeline:

auto pipe = Skigen::make_pipeline(
    Skigen::PolynomialFeatures<>(3),
    Skigen::LinearRegression<>()
);

This fits $y = w_0 + w_1 x + w_2 x^2 + w_3 x^3$ while keeping the model linear in the parameters.

Mirrors sklearn.preprocessing.PolynomialFeatures.

Constructor

Skigen::PolynomialFeatures<Scalar> poly(int degree = 2,
                                         bool include_bias = true,
                                         bool interaction_only = false);

Parameter	Default	Description
degree	2	Maximum polynomial degree
include_bias	true	Include a bias column of ones
interaction_only	false	Only interaction features (no powers)

Methods

Method	Description
fit(X)	Compute output feature structure
transform(X)	Generate polynomial features
fit_transform(X)	Fit and transform in one call

Fitted Attributes

Accessor	Type	Description
n_output_features()	Eigen::Index	Number of output features

Example

#include <Skigen/Preprocessing>

// X has 2 features → degree=2 generates: 1, x1, x2, x1², x1·x2, x2²
Skigen::PolynomialFeatures poly(2);
auto X_poly = poly.fit_transform(X);