最强总结！8个线性回归核心点！！

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error

np.random.seed(0)
X1 = 2 * np.random.rand(100, 1)
X2 = 3 * np.random.rand(100, 1)
X3 = 4 * np.random.rand(100, 1)
Y = 4 + 3*X1 + 5*X2 + 2*X3 + np.random.randn(100, 1)

# 使用sklearn进行多元线性回归拟合
X = np.concatenate((X1, X2, X3), axis=1)
model = LinearRegression()
model.fit(X, Y)
Y_pred = model.predict(X)

# 计算均方误差
mse = mean_squared_error(Y, Y_pred)
print("Mean Squared Error:", mse)

# 可视化
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

ax.scatter(X1, X2, Y, color='b', label='Actual Data')
ax.scatter(X1, X2, Y_pred, color='r', label='Predicted Data')

# 绘制平面
xx1, xx2 = np.meshgrid(np.linspace(0, 2, 100), np.linspace(0, 3, 100))
yy = model.intercept_ + model.coef_[0][0]*xx1 + model.coef_[0][1]*xx2
ax.plot_surface(xx1, xx2, yy, alpha=0.5, color='y')

ax.set_xlabel('X1')
ax.set_ylabel('X2')
ax.set_zlabel('Y')
ax.set_title('3D Plot of Multivariate Linear Regression')

plt.legend()
plt.show()

import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error

# 生成样本数据
np.random.seed(0)
n_samples = 100
n_features = 5
X = np.random.rand(n_samples, n_features)  # 生成自变量
coefficients = np.random.rand(n_features)  # 真实的系数
intercept = 2.5  # 截距
epsilon = 0.1 * np.random.randn(n_samples)  # 生成误差项
Y = intercept + np.dot(X, coefficients) + epsilon  # 生成因变量

# 分割训练集和测试集
X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size=0.2, random_state=42)

# 定义函数，用于评估模型的性能
def evaluate_model(X_train, X_test, Y_train, Y_test):
    model = LinearRegression()
    model.fit(X_train, Y_train)
    Y_pred = model.predict(X_test)
    mse = mean_squared_error(Y_test, Y_pred)
    return mse, Y_pred

# 模型评估
all_features_mse, _ = evaluate_model(X_train, X_test, Y_train, Y_test)
print("MSE using all features:", all_features_mse)

# 选择不同的自变量组合进行建模
selected_features = [[0, 1], [1, 2, 3], [2, 3, 4]]  # 不同的自变量组合
mse_values = []
for i, features in enumerate(selected_features):
    X_train_selected = X_train[:, features]
    X_test_selected = X_test[:, features]
    mse, Y_pred = evaluate_model(X_train_selected, X_test_selected, Y_train, Y_test)
    mse_values.append(mse)
    
    # 可视化预测结果
    plt.figure(figsize=(8, 6))
    plt.scatter(Y_test, Y_pred, label=f'Features: {features}')
    plt.plot([Y_test.min(), Y_test.max()], [Y_test.min(), Y_test.max()], 'k--', lw=2)  # 对角线
    plt.xlabel('True Values')
    plt.ylabel('Predictions')
    plt.title(f'True vs Predicted Values using Features: {features}')
    plt.legend()
    plt.show()

# 打印不同自变量组合的 MSE 值
for i, features in enumerate(selected_features):
    print(f"MSE using selected features {features}: {mse_values[i]}")

# MSE using selected features [0, 1]: 0.07382629999218492
# MSE using selected features [1, 2, 3]: 0.0296225705109124
# MSE using selected features [2, 3, 4]: 0.03780369819446803

import numpy as np
import matplotlib.pyplot as plt
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression, Lasso, Ridge
from sklearn.pipeline import make_pipeline
from sklearn.metrics import mean_squared_error

# 生成多项式数据集
np.random.seed(0)
X = np.sort(5 * np.random.rand(50, 1), axis=0)
y = np.sin(X).ravel() + np.random.normal(0, 0.1, X.shape[0])

# 将特征进行多项式转换
degrees = [1, 4, 15]  # 不同的多项式阶数
models = [('No Regularization', 'r', '-'), 
          ('L1 Regularization (Lasso)', 'g', '--'), 
          ('L2 Regularization (Ridge)', 'b', '-.')]

plt.figure(figsize=(14, 5))

for i, (name, color, linestyle) in enumerate(models):
    ax = plt.subplot(1, len(models), i + 1)
    plt.setp(ax, xticks=(), yticks=())

    # 将特征进行多项式转换
    polynomial_features = PolynomialFeatures(degree=degrees[i], include_bias=False)
    if name == 'L1 Regularization (Lasso)':
        model = make_pipeline(polynomial_features, Lasso(alpha=0.001))
    elif name == 'L2 Regularization (Ridge)':
        model = make_pipeline(polynomial_features, Ridge(alpha=0.1))
    else:
        model = make_pipeline(polynomial_features, LinearRegression())

    # 拟合模型
    model.fit(X, y)
    y_pred = model.predict(X)
    mse = mean_squared_error(y, y_pred)

    # 绘制拟合曲线
    plt.plot(X, y, color='cornflowerblue', linewidth=2, label="Ground Truth")
    plt.plot(X, y_pred, color=color, linestyle=linestyle, linewidth=2,
             label="{} (MSE: {:.2f})".format(name, mse))
    plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(degrees[i], mse, 0.01), fontsize=14)
    plt.xlabel("x", fontsize=14)
    plt.ylabel("y", fontsize=14)
    plt.legend(loc="best", fontsize=10)

plt.show()

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error

# 生成样本数据
np.random.seed(0)
X = 2 * np.random.rand(100, 1)
Y = 4 + 3 * X + np.random.randn(100, 1)  # 加入随机噪声

# 拟合线性回归模型
model = LinearRegression()
model.fit(X, Y)
Y_pred = model.predict(X)

# 残差分布检验
residuals = Y - Y_pred
plt.figure(figsize=(12, 4))

plt.subplot(1, 2, 1)
plt.hist(residuals, bins=20)
plt.title('Residuals Histogram')
plt.xlabel('Residuals')
plt.ylabel('Frequency')

plt.subplot(1, 2, 2)
plt.scatter(Y_pred, residuals)
plt.axhline(y=0, color='r', linestyle='--')
plt.title('Residuals vs Predictions')
plt.xlabel('Predictions')
plt.ylabel('Residuals')

plt.show()

# 计算均方误差
mse = mean_squared_error(Y, Y_pred)
print("Mean Squared Error:", mse)
# Mean Squared Error: 0.9924386487246479

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

# 生成样本数据
np.random.seed(0)
X = 2 * np.random.rand(100, 1)
Y = 4 + 3 * X + np.random.randn(100, 1)  # 加入随机噪声

# 拟合线性回归模型
model = LinearRegression()
model.fit(X, Y)

# 进行预测
new_X = np.array([[1.5]])  # 新的自变量值
predicted_Y = model.predict(new_X)
print("Predicted Y for new X:", predicted_Y[0][0])

# 进行推断
print("Intercept (β₀):", model.intercept_[0])
print("Coefficient (β₁):", model.coef_[0][0])

# Predicted Y for new X: 8.67485234349876
# Intercept (β₀): 4.222151077447231
# Coefficient (β₁): 2.968467510701019