---
title: 线性回归
tags: linear-regression
categories: machinelearning
mathjax: true
abbrlink: 52662
date: 2025-01-19 16:46:51
---

### 线性回归简介
>用于预测一个连续的目标变量（因变量），与一个或多个特征（自变量）之间存在线性关系。

假设函数：  
$$y = w_1x_1 + w_2x_2 + \cdot\cdot\cdot+w_nx_n$$
- $y$ 是目标变量（因变量），即我们希望预测的值。
- $x1​,x2​,…,xn$​ 是特征变量（自变量），即输入的值。
### 损失函数

为了找到最佳的线性模型，我们需要通过最小化损失函数来优化模型参数。在线性回归中，常用的损失函数是 **均方误差（MSE）**：
$$J(\theta) = \frac{1}{2N} \sum_{i=1}^{N} (y_i - f_\theta(x_i))^2$$
- N 是样本的数量。
- $y_i$​ 是第 i 个样本的真实值。
- $f_\theta(x_i)$ 是模型预测的第 i 个样本的值。

### 线性回归优化

- 梯度下降法
    ```python
    from sklearn.datasets import fetch_california_housing
    from sklearn.model_selection import train_test_split
    from sklearn.preprocessing import StandardScaler
    from sklearn.linear_model import SGDRegressor
    from sklearn.metrics import mean_squared_error

    # 1. 获取数据集
    housing = fetch_california_housing()

    # 2. 数据集处理
    # 2.1 分割数据集
    X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=0.25)

    # 3. 特征工程
    # 3.1 标准化
    transfer = StandardScaler()
    X_train = transfer.fit_transform(X_train)
    X_test = transfer.transform(X_test)  # 使用 transform() 而不是 fit_transform()

    # 4.机器学习- 梯度下降法
    estimater = SGDRegressor(max_iter=1000, eta0=0.01)
    estimater.fit(X_train, y_train)
    print(f"SGD模型的偏置是：{estimater.intercept_}")
    print(f"SGD模型的系数是：{estimater.coef_}")

    # 5. 模型评估
    y_pred = estimater.predict(X_test)
    print(f"SGD模型预测值：{y_pred}")
    mse = mean_squared_error(y_test, y_pred)
    print(f"SGD模型均方误差:{mse}")
    ```

- 正规方程
    ```python
    from sklearn.datasets import fetch_california_housing
    from sklearn.model_selection import train_test_split
    from sklearn.preprocessing import StandardScaler
    from sklearn.linear_model import LinearRegression
    from sklearn.metrics import mean_squared_error

    # 1. 获取数据集
    housing = fetch_california_housing()

    # 2. 数据集处理
    # 2.1 分割数据集
    X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=0.25)

    # 3. 特征工程
    # 3.1 标准化
    transfer = StandardScaler()
    X_train = transfer.fit_transform(X_train)
    X_test = transfer.fit_transform(X_test)

    # 4.机器学习- 线性回归
    estimater = LinearRegression()
    estimater.fit(X_train, y_train)
    print(f"模型的偏置是：{estimater.intercept_}")
    print(f"模型的系数是：{estimater.coef_}")

    # 5. 模型评估
    y_pred = estimater.predict(X_test)
    print(f"模型预测值：{y_pred}")
    mse = mean_squared_error(y_test, y_pred)
    print(f"模型均方误差:{mse}")
    ```

- 岭回归
    ```python
    from sklearn.datasets import fetch_california_housing
    from sklearn.model_selection import train_test_split
    from sklearn.preprocessing import StandardScaler
    from sklearn.linear_model import Ridge, RidgeCV
    from sklearn.metrics import mean_squared_error

    # 1. 获取数据集
    housing = fetch_california_housing()

    # 2. 数据集处理
    # 2.1 分割数据集
    X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=0.25)

    # 3. 特征工程
    # 3.1 标准化
    transfer = StandardScaler()
    X_train = transfer.fit_transform(X_train)
    X_test = transfer.transform(X_test)  # 使用 transform() 而不是 fit_transform()

    # 4.机器学习- 岭回归 使用了Ridge的alpha的搜索
    # estimater = Ridge(alpha=1.0)
    estimater = RidgeCV(alphas=[0.001, 0.01, 0.1, 1, 10, 100])
    estimater.fit(X_train, y_train)
    print(f"Ridge模型的偏置是：{estimater.intercept_}")
    print(f"Ridge模型的系数是：{estimater.coef_}")

    # 查看最佳 alpha
    print(f"最佳 alpha 值是：{estimater.alpha_}")

    # 5. 模型评估
    y_pred = estimater.predict(X_test)
    print(f"Ridge模型预测值：{y_pred}")
    mse = mean_squared_error(y_test, y_pred)
    print(f"Ridge模型均方误差:{mse}")
    ```

这样每个代码块的缩进保持一致，便于阅读和理解。如果有其他优化需求，随时告诉我！


![](/img/machinelearning/linear.png)

![](/img/machinelearning/fitting.png)
### 模型保存和加载
```python
from sklearn.datasets import fetch_california_housing
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.linear_model import Ridge, RidgeCV
from sklearn.metrics import mean_squared_error
import joblib

def save_model():
    # 1. 获取数据集
    housing = fetch_california_housing()
    # 2. 数据集处理
    # 2.1 分割数据集
    X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=0.25)
    # 3. 特征工程
    # 3.1 标准化
    transfer = StandardScaler()
    X_train = transfer.fit_transform(X_train)
    X_test = transfer.transform(X_test)  # 使用 transform() 而不是 fit_transform()
    # 4. 机器学习 - 岭回归 使用了Ridge的alpha的搜索
    estimater = RidgeCV(alphas=[0.001, 0.01, 0.1, 1, 10, 100])
    estimater.fit(X_train, y_train)
    print(f"Ridge模型的偏置是：{estimater.intercept_}")
    print(f"Ridge模型的系数是：{estimater.coef_}")
    # 保存模型
    joblib.dump(estimater, 'ridge_model.pkl')
    # 查看最佳 alpha
    print(f"最佳 alpha 值是：{estimater.alpha_}")
    # 5. 模型评估
    y_pred = estimater.predict(X_test)
    mse = mean_squared_error(y_test, y_pred)
    print(f"Ridge模型均方误差:{mse}")

def load_model():
    # 1. 获取数据集
    housing = fetch_california_housing()
    # 2. 数据集处理
    # 2.1 分割数据集
    X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=0.25)
    # 3. 特征工程
    # 3.1 标准化
    transfer = StandardScaler()
    X_train = transfer.fit_transform(X_train)
    X_test = transfer.transform(X_test)  # 使用 transform() 而不是 fit_transform()
    # 加载模型
    estimater = joblib.load('ridge_model.pkl')
    print(f"Ridge模型的偏置是：{estimater.intercept_}")
    print(f"Ridge模型的系数是：{estimater.coef_}")
    # 查看最佳 alpha
    print(f"最佳 alpha 值是：{estimater.alpha_}")
    # 5. 模型评估
    y_pred = estimater.predict(X_test)
    mse = mean_squared_error(y_test, y_pred)
    print(f"Ridge模型预测值：{y_pred}")
    print(f"Ridge模型均方误差:{mse}")

print("训练并保存模型：")
save_model()
print("加载模型")
load_model()
```