# 6-4 在线性回归模型中使用梯度下降法

```python
import numpy as np
import matplotlib.pyplot as plt

np.random.seed(666)
x = 2 * np.random.random(size=100)
y = x * 3. + 4. + np.random.normal(size=100)

X = x.reshape(-1, 1)

plt.scatter(x, y)
plt.show()
```

![](http://windmissing.github.io/images/2019/80.png)

## 使用梯度下降法训练

```python
def J(theta, X_b, y):
    try:
        return np.sum((y - X_b.dot(theta))**2) / len(X_b)
    except:
        return float('inf')

def dJ(theta, X_b, y):
    ret = np.empty(len(theta))
    ret[0] = np.sum(X_b.dot(theta) - y)
    for i in range(1, len(theta)):
        ret[i] = (X_b.dot(theta) - y).dot(X_b[:, 1])

    return ret * 2 / len(X_b)

def gradient_descent(X_b, y, initial_theta, eta, n_iters = 1e4, epsilon=1e-8):
    theta = initial_theta
    i_iter = 0
    while i_iter < n_iters:
        gradient = dJ(theta, X_b, y)
        last_theta = theta
        theta = theta - eta * gradient
        if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
            break
        i_iter += 1
    return theta

X_b = np.hstack([np.ones((len(x), 1)), X])
initial_theta = np.zeros(X_b.shape[1])
eta = 0.01

theta = gradient_descent(X_b, y, initial_theta, eta)
theta
```

输出结果：\
array(\[4.02145786, 3.00706277])

## 封装线性回归算法

```python
import numpy as np
from sklearn.metrics import r2_score

class LinearRegression:

    def __init__(self):
        """初始化Linear Regression模型"""
        self.coef_ = None
        self.interception_ = None
        self._theta = None

    def fit_normal(self, X_train, y_train):
        """根据训练数据集X_train, y_train训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], "the size of X_train must be equal to the size of y_train"

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        self._theta = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y_train)

        self.interception_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def fit_gd(self, X_train, y_train, eta=0.01, n_iters = 1e4):
        """根据训练数据集X_train, y_train，使用梯度下降法训练Linear Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], "the size of X_train must be equal to the size of y_train"

        def J(theta, X_b, y):
            try:
                return np.sum((y - X_b.dot(theta))**2) / len(X_b)
            except:
                return float('inf')

        def dJ(theta, X_b, y):
            ret = np.empty(len(theta))
            ret[0] = np.sum(X_b.dot(theta) - y)
            for i in range(1, len(theta)):
                ret[i] = (X_b.dot(theta) - y).dot(X_b[:, 1])

            return ret * 2 / len(X_b)

        def gradient_descent(X_b, y, initial_theta, eta, n_iters = 1e4, epsilon=1e-8):
            theta = initial_theta
            i_iter = 0
            while i_iter < n_iters:
                gradient = dJ(theta, X_b, y)
                last_theta = theta
                theta = theta - eta * gradient
                if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
                    break
                i_iter += 1
            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.zeros(X_b.shape[1])
        self._theta = gradient_descent(X_b, y_train, initial_theta, eta)
        self.interception_ = self._theta[0]
        self.coef_ = self._theta[1:]
        return self

    def predict(self, X_predict):
        """给定待预测数据集X_predict，返回表示X_predict的结果向量"""
        assert self.interception_ is not None and self.coef_ is not None, "must fit before predict"
        assert X_predict.shape[1] == len(self.coef_), "the feature number of X_predict must equal to X_train"

        X_b = np.hstack([np.ones((len(X_predict), 1)), X_predict])
        return X_b.dot(self._theta)

    def score(self, X_test, y_test):
        """根据测试数据集X_test, y_test确定当前模型的准确度"""

        y_predict = self.predict(X_test)
        return r2_score(y_test, y_predict)

    def __repr__(self):
        return "LinearRegression()"
```

## 使用fit\_gd

```python
lin_reg = LinearRegression()
lin_reg.fit_gd(X, y)
```

输入：`lin_reg.coef_`\
输出：array(\[3.00706277])

输入：`lin_reg.interception_`\
输出：4.021457858204859


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