线性回归是通过一个或多个自变量与因变量 之间进行建模的回归分析，其特点为一个或多个称为回归系数的模型参数的线性组合。如下图所示，样本点为历史数据，回归曲线要能最贴切的模拟样本点的趋势，将误差降到最小。
用python是实现这个模型
python
import numpy as np
import matplotlib.pyplot as plt
class LinearRegression:
def __init__(self, learning_rate=0.01, epochs=1000):
self.learning_rate = learning_rate
self.epochs = epochs
self.weights = None
self.bias = None
def fit(self, X, y):
"""训练线性回归模型"""
# 获取样本数和特征数
n_samples, n_features = X.shape
# 确保y是一维数组
y = y.ravel()
# 初始化权重和偏置
self.weights = np.zeros(n_features)
self.bias = 0
# 训练过程
for epoch in range(self.epochs):
# 前向传播
y_pred = np.dot(X, self.weights) + self.bias
# 计算损失
loss = (1 / (2 * n_samples)) * np.sum((y_pred - y) ** 2)
# 计算梯度
dw = (1 / n_samples) * np.dot(X.T, (y_pred - y))
db = (1 / n_samples) * np.sum(y_pred - y)
# 确保dw是一维数组
dw = dw.ravel()
# 更新权重和偏置
self.weights -= self.learning_rate * dw
self.bias -= self.learning_rate * db
# 每100个epoch打印一次损失
if (epoch + 1) % 100 == 0:
print(f"Epoch {epoch+1}/{self.epochs}, Loss: {loss:.4f}")
def predict(self, X):
"""预测"""
return np.dot(X, self.weights) + self.bias
def score(self, X, y):
"""计算R²评分"""
y_pred = self.predict(X)
# 确保y是一维数组
y = y.ravel()
y_pred = y_pred.ravel()
# 计算总平方和
ss_total = np.sum((y - np.mean(y)) ** 2)
# 计算残差平方和
ss_residual = np.sum((y - y_pred) ** 2)
# 计算R²
r2 = 1 - (ss_residual / ss_total)
return r2
def generate_data(n_samples=100, noise=0.1):
"""生成线性回归数据"""
# 生成特征
X = np.random.rand(n_samples, 1) * 10
# 生成真实标签
y = 2 * X + 3 + np.random.randn(n_samples, 1) * noise
return X, y
def main():
print("=== 线性回归算法实现 ===")
# 生成数据
X, y = generate_data(n_samples=100, noise=0.5)
print(f"生成数据形状: X={X.shape}, y={y.shape}")
# 划分训练集和测试集
split_idx = int(0.8 * len(X))
X_train, X_test = X[:split_idx], X[split_idx:]
y_train, y_test = y[:split_idx], y[split_idx:]
print(f"训练集大小: {X_train.shape[0]}, 测试集大小: {X_test.shape[0]}")
# 初始化模型
model = LinearRegression(learning_rate=0.01, epochs=1000)
# 训练模型
print("\n开始训练...")
model.fit(X_train, y_train)
# 测试模型
y_pred = model.predict(X_test)
r2 = model.score(X_test, y_test)
print(f"\n测试集R²评分: {r2:.4f}")
# 打印模型参数
print(f"\n模型参数:")
print(f"权重: {model.weights[0]:.4f}")
print(f"偏置: {model.bias:.4f}")
# 可视化结果
plt.figure(figsize=(10, 6))
# 绘制训练数据
plt.scatter(X_train, y_train, color='blue', label='训练数据')
# 绘制测试数据
plt.scatter(X_test, y_test, color='green', label='测试数据')
# 绘制预测线
x_range = np.linspace(0, 10, 100).reshape(-1, 1)
y_range_pred = model.predict(x_range)
plt.plot(x_range, y_range_pred, color='red', label='预测线')
plt.xlabel('X')
plt.ylabel('y')
plt.title('线性回归模型')
plt.legend()
plt.grid(True)
plt.savefig('linear_regression_result.png')
print("\n结果已保存为 linear_regression_result.png")
# 打印预测示例
print("\n预测示例:")
for i in range(5):
x_sample = X_test[i]
y_true = y_test[i][0]
y_pred_sample = model.predict(x_sample.reshape(1, -1))[0]
print(f"X={x_sample[0]:.2f}, 真实值={y_true:.2f}, 预测值={y_pred_sample:.2f}")
print("\n=== 线性回归实现完成 ===")
if __name__ == "__main__":
main()