Plot Underfitting Overfitting¶
============================ Underfitting vs. Overfitting¶
This example demonstrates the problems of underfitting and overfitting and how we can use linear regression with polynomial features to approximate nonlinear functions. The plot shows the function that we want to approximate, which is a part of the cosine function. In addition, the samples from the real function and the approximations of different models are displayed. The models have polynomial features of different degrees. We can see that a linear function (polynomial with degree 1) is not sufficient to fit the training samples. This is called underfitting. A polynomial of degree 4 approximates the true function almost perfectly. However, for higher degrees the model will overfit the training data, i.e. it learns the noise of the training data. We evaluate quantitatively overfitting / underfitting by using cross-validation. We calculate the mean squared error (MSE) on the validation set, the higher, the less likely the model generalizes correctly from the training data.
Imports for Underfitting vs Overfitting with Polynomial Regression¶
The bias-variance tradeoff in polynomial degree: PolynomialFeatures transforms a single input feature x into a vector [x, x^2, …, x^d], and LinearRegression fits a linear combination of these polynomial bases. With degree 1 (linear), the model cannot capture the cosine curvature – this is underfitting (high bias). With degree 4, the polynomial closely approximates the true cosine function – the sweet spot. With degree 15, the model has enough flexibility to pass through every noisy training point, producing wild oscillations between samples – this is overfitting (high variance). The Pipeline chains the feature transformation and regression into a single estimator.
Cross-validation as the diagnostic tool: cross_val_score with scoring="neg_mean_squared_error" and 10 folds quantifies generalization error for each polynomial degree. The MSE reported in each subplot title (with standard deviation across folds) confirms quantitatively what the plots show visually: degree 1 has high MSE (underfitting), degree 4 has low MSE (good fit), and degree 15 has high MSE again (overfitting despite perfectly fitting training points). This classic demonstration motivates regularization techniques (Ridge, Lasso) that control model complexity without explicitly limiting the polynomial degree.
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
import matplotlib.pyplot as plt
import numpy as np
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import cross_val_score
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures
def true_fun(X):
return np.cos(1.5 * np.pi * X)
np.random.seed(0)
n_samples = 30
degrees = [1, 4, 15]
X = np.sort(np.random.rand(n_samples))
y = true_fun(X) + np.random.randn(n_samples) * 0.1
plt.figure(figsize=(14, 5))
for i in range(len(degrees)):
ax = plt.subplot(1, len(degrees), i + 1)
plt.setp(ax, xticks=(), yticks=())
polynomial_features = PolynomialFeatures(degree=degrees[i], include_bias=False)
linear_regression = LinearRegression()
pipeline = Pipeline(
[
("polynomial_features", polynomial_features),
("linear_regression", linear_regression),
]
)
pipeline.fit(X[:, np.newaxis], y)
# Evaluate the models using crossvalidation
scores = cross_val_score(
pipeline, X[:, np.newaxis], y, scoring="neg_mean_squared_error", cv=10
)
X_test = np.linspace(0, 1, 100)
plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model")
plt.plot(X_test, true_fun(X_test), label="True function")
plt.scatter(X, y, edgecolor="b", s=20, label="Samples")
plt.xlabel("x")
plt.ylabel("y")
plt.xlim((0, 1))
plt.ylim((-2, 2))
plt.legend(loc="best")
plt.title(
"Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
degrees[i], -scores.mean(), scores.std()
)
)
plt.show()