poly-basis-ml 0.2.0

Chebyshev polynomial feature expansion and regression for scikit-learn

poly-basis-ml

Chebyshev polynomial feature expansion and regression for scikit-learn.

Installation

pip install poly-basis-ml

Quick start

from poly_basis_ml import ChebyshevRegressor
from sklearn.datasets import make_friedman1
from sklearn.model_selection import train_test_split

X, y = make_friedman1(n_samples=1000, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)

model = ChebyshevRegressor(complexity=5, alpha=0.1)
model.fit(X_train, y_train)
print(f"R2: {model.score(X_test, y_test):.3f}")
# R2: 0.927

Tuning with GridSearchCV

ChebyshevRegressor is a standard sklearn estimator, so it works directly with GridSearchCV, RandomizedSearchCV, cross_val_score, pipelines, etc.

from sklearn.model_selection import GridSearchCV

param_grid = {"complexity": [3, 5, 7, 9], "alpha": [0.01, 0.1, 1.0, 10.0]}
gs = GridSearchCV(ChebyshevRegressor(), param_grid, cv=5, scoring="r2")
gs.fit(X_train, y_train)
print(gs.best_params_)
# {'alpha': 0.1, 'complexity': 3}
print(f"Best CV R2: {gs.best_score_:.3f}")
# Best CV R2: 0.910
print(f"Test R2:    {gs.score(X_test, y_test):.3f}")
# Test R2:    0.932

Inspecting fitted models

Feature names and coefficient table

When fitted on a DataFrame, original column names are preserved in the polynomial feature names. Array input falls back to x0, x1, etc.

import pandas as pd

X_train_df = pd.DataFrame(X_train, columns=["speed", "temp", "pressure", "humidity", "voltage"])

model = ChebyshevRegressor(complexity=5, alpha=0.1).fit(X_train_df, y_train)

model.get_feature_names_out()[:6]
# array(['speed_T0', 'speed_T1', 'speed_T2', 'speed_T3', 'speed_T4',
#        'speed_T5'], dtype='<U11')

model.coef_table().head(6)
#        feature      coef input_feature
# 0     speed_T0 14.068879         speed
# 1  humidity_T1  4.899597      humidity
# 2     speed_T1  3.218066         speed
# 3      temp_T1  3.163345          temp
# 4   voltage_T1  2.479831       voltage
# 5  pressure_T2  2.413147      pressure

Feature importances

Per-input-feature importances are computed by aggregating absolute coefficient values across all polynomial terms for each original feature (excluding the intercept term), normalised to sum to 1.

for name, imp in zip(X_train_df.columns, model.feature_importances_):
    print(f"  {name:>10s}: {imp:.3f}")
#       speed: 0.245
#        temp: 0.244
#    pressure: 0.128
#    humidity: 0.249
#     voltage: 0.133

Feature mapping

mapping = model.get_feature_mapping()
mapping["temp"]
# ['temp_T1', 'temp_T2', 'temp_T3', 'temp_T4', 'temp_T5']
mapping["speed"]
# ['speed_T0', 'speed_T1', 'speed_T2', 'speed_T3', 'speed_T4', 'speed_T5']

Note: the first input feature retains T0 (the intercept/constant term); remaining features start at T1 to avoid redundant constant columns in the design matrix.

Model tree example

from poly_basis_ml import ChebyshevModelTreeRegressor

X, y = make_friedman1(n_samples=2000, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)

model = ChebyshevModelTreeRegressor(max_depth=3, complexity=3, alpha=0.1, min_samples_leaf=100)
model.fit(X_train, y_train)
print(f"R2:          {model.score(X_test, y_test):.3f}")
# R2:          0.951
print(f"Leaf models: {len(model.leaf_models_)}")
# Leaf models: 7

Key features

• ChebyshevExpander --- standalone sklearn transformer for polynomial feature expansion
• ChebyshevRegressor --- convenience estimator wrapping Chebyshev expansion + Ridge
• ChebyshevModelTreeRegressor --- decision tree with Chebyshev polynomial leaf models
• Bivariate interactions --- optional product, contrast, and additive interaction features

How it works

Features are mapped to [-1, 1] via MinMaxScaler, then expanded into Chebyshev polynomial basis functions with proper intercept handling (one T0 term retained, redundant constant columns stripped). The resulting design matrix is fitted with Ridge regression. For the model tree variant, a decision tree first partitions the data into regions, then each leaf fits a separate ChebyshevRegressor for smooth local approximation.

Main classes

ChebyshevRegressor

Parameter	Default	Description
complexity	5	Chebyshev polynomial degree
alpha	1.0	Ridge regularisation strength
clip_input	True	Clip prediction-time inputs to training range
include_interactions	False	Add bivariate interaction features

ChebyshevModelTreeRegressor

Parameter	Default	Description
max_depth	3	Maximum depth of routing tree
min_samples_leaf	200	Minimum samples per leaf for polynomial fit
complexity	2	Chebyshev degree for leaf models
alpha	10.0	Ridge regularisation for leaf models

Citation

@article{gerber2026revisiting,
  title={Revisiting Chebyshev Polynomial and Anisotropic RBF Models for Tabular Regression},
  author={Gerber, Luciano and Lloyd, Huw},
  year={2026}
}

Licence

MIT