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What polynomial feature transformation does
A linear model using two inputs can fit a plane such as w₀ + w₁x₁ + w₂x₂. Expanding the inputs adds terms such as x₁², x₁x₂, and x₂², allowing the model to fit a curved surface or represent an interaction. The resulting model is nonlinear in the original inputs but remains linear in its coefficients: the transformation changes the features supplied to the estimator, not the estimator’s coefficient structure. See scikit-learn’s linear-model guide.
For two inputs [a, b], the full degree-two expansion is [1, a, b, a², ab, b²]. The constant term is included by default. This is a basis expansion: the estimator learns weights for the generated columns.
Choose the expansion that matches your data
Set the maximum degree
PolynomialFeatures generates terms up to the configured maximum degree. Its documented default is degree=2; a tuple can specify a minimum and maximum degree. A higher maximum admits more complex relationships, but also creates more columns and more opportunity to fit noise. Treat degree as a model choice to validate, not as a setting that should automatically be increased. The PolynomialFeatures API documents the options and feature-count warning.
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Decide whether repeated powers are useful
With the default interaction_only=False, the transform includes repeated powers such as a². Set interaction_only=True when you want products of distinct features but not repeated powers: x[0] * x[1] remains, while x[0] ** 2 is excluded. This can suit Boolean inputs, where squaring a Boolean feature adds no information but a product can encode a conjunction. It is a structural choice, not a general guarantee of better performance.
Coordinate the constant column with the estimator intercept
include_bias=True adds a column of ones, which acts as an intercept in a linear model. The documented regression-pipeline example uses that column with fit_intercept=False. If the estimator fits its own intercept, setting include_bias=False avoids adding a redundant constant term. The appropriate combination depends on the estimator’s intercept behavior.
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Build a pipeline for fitting and prediction
A pipeline ensures the feature expansion, any scaling, and the estimator are handled together during fitting and prediction. Here is an illustrative pattern using a degree-two expansion, standardization, and ridge regression:
from sklearn.linear_model import Ridge
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures, StandardScaler
model = Pipeline([
("poly", PolynomialFeatures(degree=2, include_bias=False)),
("scale", StandardScaler()),
("model", Ridge()),
])
model.fit(X_train, y_train)
predictions = model.predict(X_test)This example omits the polynomial bias column; the estimator’s intercept setting should still be chosen deliberately. Keep preprocessing inside the fitted pipeline when using cross-validation so each training partition learns its transformations without using its validation partition. Scikit-learn explains this behavior in its documentation on pipelines and composite estimators.
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Scale and validate the expanded model
Scale when the estimator makes feature scale consequential
Powers of a feature can have very different numeric ranges from the original feature, and generated terms can also differ substantially from one another. Scaling is particularly relevant for penalized linear estimators: a coefficient penalty can treat features unevenly when their scales differ. Scikit-learn’s linear-model guidance explicitly recommends standardizing the feature matrix for TweedieRegressor so its penalty treats features equally. That does not make scaling mandatory for every estimator; check the behavior of the model you use. See the preprocessing guide.
Compare degrees with a consistent validation plan
Evaluate candidate degrees and regularization settings on the same validation splits or cross-validation plan, using the same scoring measure. Choose a validation strategy that reflects how the model will be used—for example, do not randomly mix observations if the intended prediction task requires a time-based split. Compare predictive performance alongside model size and computational cost. High-degree expansions can overfit, so a more complex model should earn its place through validation rather than assumption.
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Manage feature growth and inspect generated terms
The number of generated features grows rapidly with input dimension and degree. Scikit-learn warns that output feature count scales polynomially with the number of inputs and exponentially with degree. More columns increase computation and memory use as well as statistical complexity. If a full expansion is too large, consider a lower degree, interaction_only=True, or selecting terms based on domain knowledge. Regularization can also constrain fitted coefficients, though it does not remove the cost of generating the columns.
For a local smooth curve rather than a single global polynomial basis, scikit-learn’s API points to SplineTransformer as an alternative feature representation.
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To understand which terms were generated, inspect the transformer’s powers_ attribute, which records each term’s exponents across the input features, or use get_feature_names_out() to obtain column names. The API also documents n_features_in_ and n_output_features_ for input and output feature counts. The default dense output order is order='C'; order='F' can make transformation faster but may slow downstream estimators, so keep the default unless profiling supports a change. Confirm option availability and behavior in the documentation for your installed scikit-learn version.
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