线性模型 #

概述 #

线性模型是机器学习中最基础且广泛使用的模型族，它们假设目标变量与特征之间存在线性关系。

线性模型类型 #

类型 用途 代表模型
回归 预测连续值 线性回归、岭回归、Lasso
分类 预测离散类别 逻辑回归、线性SVM

线性回归 #

基本原理 #

线性回归寻找最优权重 w 和偏置 b，使得预测值 y = Xw + b 最接近真实值。

python 
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error, r2_score
import numpy as np

X = np.array([[1], [2], [3], [4], [5]])
y = np.array([2, 4, 6, 8, 10])

model = LinearRegression()
model.fit(X, y)

print(f"权重: {model.coef_}")
print(f"偏置: {model.intercept_}")
print(f"预测: {model.predict([[6]])}")

多元线性回归 #

python 
from sklearn.datasets import make_regression

X, y = make_regression(n_samples=100, n_features=3, noise=10, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)

model = LinearRegression()
model.fit(X_train, y_train)

y_pred = model.predict(X_test)
print(f"MSE: {mean_squared_error(y_test, y_pred):.4f}")
print(f"R²: {r2_score(y_test, y_pred):.4f}")

正规方程 #

线性回归使用正规方程求解：

text 
w = (X^T X)^(-1) X^T y

岭回归（Ridge） #

L2 正则化 #

岭回归在损失函数中添加 L2 正则化项，防止过拟合。

python 
from sklearn.linear_model import Ridge

X, y = make_regression(n_samples=100, n_features=20, noise=10, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)

ridge = Ridge(alpha=1.0)
ridge.fit(X_train, y_train)

print(f"训练分数: {ridge.score(X_train, y_train):.4f}")
print(f"测试分数: {ridge.score(X_test, y_test):.4f}")

正则化强度 #

python 
import matplotlib.pyplot as plt

alphas = [0.001, 0.01, 0.1, 1, 10, 100]
train_scores = []
test_scores = []

for alpha in alphas:
    ridge = Ridge(alpha=alpha)
    ridge.fit(X_train, y_train)
    train_scores.append(ridge.score(X_train, y_train))
    test_scores.append(ridge.score(X_test, y_test))

plt.semilogx(alphas, train_scores, label='Train')
plt.semilogx(alphas, test_scores, label='Test')
plt.xlabel('Alpha')
plt.ylabel('R² Score')
plt.legend()

Lasso 回归 #

L1 正则化 #

Lasso 添加 L1 正则化项，可以产生稀疏解，实现特征选择。

python 
from sklearn.linear_model import Lasso

X, y = make_regression(n_samples=100, n_features=20, n_informative=5, noise=10, random_state=42)

lasso = Lasso(alpha=0.1)
lasso.fit(X, y)

print(f"非零系数数量: {sum(lasso.coef_ != 0)}")
print(f"系数: {lasso.coef_}")

特征选择效果 #

python 
lasso = Lasso(alpha=0.5)
lasso.fit(X_train, y_train)

selected_features = np.where(lasso.coef_ != 0)[0]
print(f"选择的特征索引: {selected_features}")

ElasticNet #

L1 + L2 正则化 #

ElasticNet 结合 L1 和 L2 正则化的优点。

python 
from sklearn.linear_model import ElasticNet

enet = ElasticNet(alpha=1.0, l1_ratio=0.5)
enet.fit(X_train, y_train)

print(f"训练分数: {enet.score(X_train, y_train):.4f}")
print(f"测试分数: {enet.score(X_test, y_test):.4f}")

参数说明 #

参数 描述
alpha 正则化强度
l1_ratio L1 正则化比例（0-1）
python 
l1_ratio = 0.5
l1_penalty = l1_ratio * alpha
l2_penalty = (1 - l1_ratio) * alpha

逻辑回归 #

二分类 #

逻辑回归用于二分类问题，使用 sigmoid 函数将输出映射到 [0, 1]。

python 
from sklearn.linear_model import LogisticRegression
from sklearn.datasets import make_classification
from sklearn.metrics import accuracy_score, classification_report

X, y = make_classification(n_samples=1000, n_features=10, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)

log_reg = LogisticRegression(max_iter=1000)
log_reg.fit(X_train, y_train)

y_pred = log_reg.predict(X_test)
y_proba = log_reg.predict_proba(X_test)

print(f"准确率: {accuracy_score(y_test, y_pred):.4f}")
print(f"预测概率: {y_proba[:5]}")

多分类 #

python 
from sklearn.datasets import load_iris

iris = load_iris()
X, y = iris.data, iris.target

log_reg = LogisticRegression(multi_class='multinomial', max_iter=1000)
log_reg.fit(X, y)

print(f"类别: {log_reg.classes_}")
print(f"系数形状: {log_reg.coef_.shape}")

正则化 #

python 
log_reg_l1 = LogisticRegression(penalty='l1', solver='saga', max_iter=1000)
log_reg_l2 = LogisticRegression(penalty='l2', solver='lbfgs', max_iter=1000)
log_reg_enet = LogisticRegression(penalty='elasticnet', solver='saga', l1_ratio=0.5, max_iter=1000)

求解器选择 #

求解器 支持正则化 适用场景
lbfgs L2, None 小数据集
liblinear L1, L2 小数据集
saga L1, L2, ElasticNet 大数据集
sag L2, None 大数据集
newton-cg L2, None 多分类

SGD 回归/分类 #

随机梯度下降 #

使用 SGD 优化器，适合大规模数据。

python 
from sklearn.linear_model import SGDRegressor, SGDClassifier

sgd_reg = SGDRegressor(max_iter=1000, tol=1e-3, random_state=42)
sgd_reg.fit(X_train, y_train)

sgd_clf = SGDClassifier(max_iter=1000, tol=1e-3, random_state=42)
sgd_clf.fit(X_train, y_train)

增量学习 #

python 
sgd_reg = SGDRegressor()

for epoch in range(10):
    for X_batch, y_batch in get_batches(X_train, y_train, batch_size=32):
        sgd_reg.partial_fit(X_batch, y_batch)

多项式回归 #

使用 Pipeline #

python 
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import Pipeline

poly_reg = Pipeline([
    ('poly', PolynomialFeatures(degree=2)),
    ('linear', LinearRegression())
])

X = np.array([[1], [2], [3], [4], [5]])
y = np.array([1, 4, 9, 16, 25])

poly_reg.fit(X, y)
print(poly_reg.predict([[6]]))

拟合效果对比 #

python 
degrees = [1, 2, 3, 5, 10]

for degree in degrees:
    poly = Pipeline([
        ('poly', PolynomialFeatures(degree=degree)),
        ('linear', LinearRegression())
    ])
    poly.fit(X_train, y_train)
    print(f"Degree {degree}: R² = {poly.score(X_test, y_test):.4f}")

模型评估 #

回归指标 #

python 
from sklearn.metrics import mean_squared_error, mean_absolute_error, r2_score

y_pred = model.predict(X_test)

mse = mean_squared_error(y_test, y_pred)
rmse = np.sqrt(mse)
mae = mean_absolute_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)

print(f"MSE: {mse:.4f}")
print(f"RMSE: {rmse:.4f}")
print(f"MAE: {mae:.4f}")
print(f"R²: {r2:.4f}")

分类指标 #

python 
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score

y_pred = log_reg.predict(X_test)

print(f"准确率: {accuracy_score(y_test, y_pred):.4f}")
print(f"精确率: {precision_score(y_test, y_pred):.4f}")
print(f"召回率: {recall_score(y_test, y_pred):.4f}")
print(f"F1分数: {f1_score(y_test, y_pred):.4f}")

正则化对比 #

系数对比 #

python 
import matplotlib.pyplot as plt

linear = LinearRegression()
ridge = Ridge(alpha=10)
lasso = Lasso(alpha=0.1)

linear.fit(X_train, y_train)
ridge.fit(X_train, y_train)
lasso.fit(X_train, y_train)

plt.figure(figsize=(10, 5))
plt.plot(linear.coef_, 'o-', label='Linear')
plt.plot(ridge.coef_, 's-', label='Ridge')
plt.plot(lasso.coef_, '^-', label='Lasso')
plt.legend()
plt.xlabel('Feature Index')
plt.ylabel('Coefficient')

对比表 #

模型 正则化 特点
LinearRegression 无 简单，易过拟合
Ridge L2 系数收缩，不稀疏
Lasso L1 系数稀疏，特征选择
ElasticNet L1+L2 结合两者优点

超参数调优 #

交叉验证 #

python 
from sklearn.model_selection import cross_val_score

ridge = Ridge(alpha=1.0)
scores = cross_val_score(ridge, X, y, cv=5, scoring='r2')
print(f"交叉验证分数: {scores}")
print(f"平均分数: {scores.mean():.4f}")

GridSearchCV #

python 
from sklearn.model_selection import GridSearchCV

param_grid = {'alpha': [0.001, 0.01, 0.1, 1, 10, 100]}
ridge_cv = GridSearchCV(Ridge(), param_grid, cv=5)
ridge_cv.fit(X_train, y_train)

print(f"最佳参数: {ridge_cv.best_params_}")
print(f"最佳分数: {ridge_cv.best_score_:.4f}")

实战示例 #

房价预测 #

python 
from sklearn.datasets import fetch_california_housing
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline

housing = fetch_california_housing()
X, y = housing.data, housing.target

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

model = Pipeline([
    ('scaler', StandardScaler()),
    ('ridge', Ridge(alpha=1.0))
])

model.fit(X_train, y_train)
print(f"测试 R²: {model.score(X_test, y_test):.4f}")

癌症分类 #

python 
from sklearn.datasets import load_breast_cancer

cancer = load_breast_cancer()
X, y = cancer.data, cancer.target

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

model = Pipeline([
    ('scaler', StandardScaler()),
    ('log_reg', LogisticRegression(max_iter=1000))
])

model.fit(X_train, y_train)
print(f"测试准确率: {model.score(X_test, y_test):.4f}")

最佳实践 #

1. 数据标准化 #

python 
pipe = Pipeline([
    ('scaler', StandardScaler()),
    ('model', Ridge())
])

2. 选择正则化 #

python 
if n_features > n_samples:
    model = Lasso()
else:
    model = Ridge()

3. 处理共线性 #

python 
ridge = Ridge(alpha=10)

4. 特征选择 #

python 
lasso = Lasso(alpha=0.1)
lasso.fit(X, y)
selected = np.where(lasso.coef_ != 0)[0]

下一步 #

掌握线性模型后，继续学习 决策树 了解非线性模型！

最后更新：2026-04-04