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"""
===========================================
Ordinary Least Squares and Ridge Regression
===========================================
1. Ordinary Least Squares:
We illustrate how to use the ordinary least squares (OLS) model,
:class:`~sklearn.linear_model.LinearRegression`, on a single feature of
the diabetes dataset. We train on a subset of the data, evaluate on a
test set, and visualize the predictions.
2. Ordinary Least Squares and Ridge Regression Variance:
We then show how OLS can have high variance when the data is sparse or
noisy, by fitting on a very small synthetic sample repeatedly. Ridge
regression, :class:`~sklearn.linear_model.Ridge`, reduces this variance
by penalizing (shrinking) the coefficients, leading to more stable
predictions.
"""
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
# %%
# Data Loading and Preparation
# ----------------------------
#
# Load the diabetes dataset. For simplicity, we only keep a single feature in the data.
# Then, we split the data and target into training and test sets.
from sklearn.datasets import load_diabetes
from sklearn.model_selection import train_test_split
X, y = load_diabetes(return_X_y=True)
X = X[:, [2]] # Use only one feature
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=20, shuffle=False)
# %%
# Linear regression model
# -----------------------
#
# We create a linear regression model and fit it on the training data. Note that by
# default, an intercept is added to the model. We can control this behavior by setting
# the `fit_intercept` parameter.
from sklearn.linear_model import LinearRegression
regressor = LinearRegression().fit(X_train, y_train)
# %%
# Model evaluation
# ----------------
#
# We evaluate the model's performance on the test set using the mean squared error
# and the coefficient of determination.
from sklearn.metrics import mean_squared_error, r2_score
y_pred = regressor.predict(X_test)
print(f"Mean squared error: {mean_squared_error(y_test, y_pred):.2f}")
print(f"Coefficient of determination: {r2_score(y_test, y_pred):.2f}")
# %%
# Plotting the results
# --------------------
#
# Finally, we visualize the results on the train and test data.
import matplotlib.pyplot as plt
fig, ax = plt.subplots(ncols=2, figsize=(10, 5), sharex=True, sharey=True)
ax[0].scatter(X_train, y_train, label="Train data points")
ax[0].plot(
X_train,
regressor.predict(X_train),
linewidth=3,
color="tab:orange",
label="Model predictions",
)
ax[0].set(xlabel="Feature", ylabel="Target", title="Train set")
ax[0].legend()
ax[1].scatter(X_test, y_test, label="Test data points")
ax[1].plot(X_test, y_pred, linewidth=3, color="tab:orange", label="Model predictions")
ax[1].set(xlabel="Feature", ylabel="Target", title="Test set")
ax[1].legend()
fig.suptitle("Linear Regression")
plt.show()
# %%
#
# OLS on this single-feature subset learns a linear function that minimizes
# the mean squared error on the training data. We can see how well (or poorly)
# it generalizes by looking at the R^2 score and mean squared error on the
# test set. In higher dimensions, pure OLS often overfits, especially if the
# data is noisy. Regularization techniques (like Ridge or Lasso) can help
# reduce that.
# %%
# Ordinary Least Squares and Ridge Regression Variance
# ----------------------------------------------------------
#
# Next, we illustrate the problem of high variance more clearly by using
# a tiny synthetic dataset. We sample only two data points, then repeatedly
# add small Gaussian noise to them and refit both OLS and Ridge. We plot
# each new line to see how much OLS can jump around, whereas Ridge remains
# more stable thanks to its penalty term.
import matplotlib.pyplot as plt
import numpy as np
from sklearn import linear_model
X_train = np.c_[0.5, 1].T
y_train = [0.5, 1]
X_test = np.c_[0, 2].T
np.random.seed(0)
classifiers = dict(
ols=linear_model.LinearRegression(), ridge=linear_model.Ridge(alpha=0.1)
)
for name, clf in classifiers.items():
fig, ax = plt.subplots(figsize=(4, 3))
for _ in range(6):
this_X = 0.1 * np.random.normal(size=(2, 1)) + X_train
clf.fit(this_X, y_train)
ax.plot(X_test, clf.predict(X_test), color="gray")
ax.scatter(this_X, y_train, s=3, c="gray", marker="o", zorder=10)
clf.fit(X_train, y_train)
ax.plot(X_test, clf.predict(X_test), linewidth=2, color="blue")
ax.scatter(X_train, y_train, s=30, c="red", marker="+", zorder=10)
ax.set_title(name)
ax.set_xlim(0, 2)
ax.set_ylim((0, 1.6))
ax.set_xlabel("X")
ax.set_ylabel("y")
fig.tight_layout()
plt.show()
# %%
# Conclusion
# ----------
#
# - In the first example, we applied OLS to a real dataset, showing
# how a plain linear model can fit the data by minimizing the squared error
# on the training set.
#
# - In the second example, OLS lines varied drastically each time noise
# was added, reflecting its high variance when data is sparse or noisy. By
# contrast, **Ridge** regression introduces a regularization term that shrinks
# the coefficients, stabilizing predictions.
#
# Techniques like :class:`~sklearn.linear_model.Ridge` or
# :class:`~sklearn.linear_model.Lasso` (which applies an L1 penalty) are both
# common ways to improve generalization and reduce overfitting. A well-tuned
# Ridge or Lasso often outperforms pure OLS when features are correlated, data
# is noisy, or sample size is small.
|
