1. 已知附件中提供的数据集linearRegression_data.txt, 完成以下线性回归任务： (1) 采用参数估计的方法建立线性回归模型（可以采用标准线性回归、局部加权线性回归、岭回归中的一种）；(10分) (2) 采用梯度下降的方法建立线性回归模型；(10分) (3) 比较 (1) 与 (2) 两种方法所建立回归模型的泛化性能。(10分)python代码

以下是Python代码实现，使用了标准线性回归和梯度下降两种方法：

import numpy as np
import matplotlib.pyplot as plt

# 加载数据
data = np.loadtxt('linearRegression_data.txt', delimiter=',')
X, y = data[:, :-1], data[:, -1]

# 标准线性回归
class LinearRegression:
    def __init__(self):
        self.w = None

    def fit(self, X, y):
        X = np.insert(X, 0, 1, axis=1)
        self.w = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(y)

    def predict(self, X):
        X = np.insert(X, 0, 1, axis=1)
        return X.dot(self.w)

# 局部加权线性回归
class LocallyWeightedLinearRegression:
    def __init__(self, k=1.0):
        self.k = k

    def fit(self, X, y):
        X = np.insert(X, 0, 1, axis=1)
        m, n = X.shape
        self.w = np.zeros((m, m))
        for i in range(m):
            distances = np.linalg.norm(X - X[i], axis=1)
            weights = np.exp(-distances ** 2 / (2 * self.k ** 2))
            W = np.diag(weights)
            self.w[i] = np.linalg.inv(X.T.dot(W).dot(X)).dot(X.T).dot(W).dot(y)

    def predict(self, X):
        X = np.insert(X, 0, 1, axis=1)
        return np.array([X[i].dot(self.w[i]) for i in range(len(X))])

# 岭回归
class RidgeRegression:
    def __init__(self, alpha=1.0):
        self.alpha = alpha
        self.w = None

    def fit(self, X, y):
        X = np.insert(X, 0, 1, axis=1)
        m, n = X.shape
        self.w = np.linalg.inv(X.T.dot(X) + self.alpha * np.identity(n)).dot(X.T).dot(y)

    def predict(self, X):
        X = np.insert(X, 0, 1, axis=1)
        return X.dot(self.w)

# 梯度下降
class GradientDescent:
    def __init__(self, alpha=0.01, max_iter=1000, tol=1e-4):
        self.alpha = alpha
        self.max_iter = max_iter
        self.tol = tol
        self.w = None

    def fit(self, X, y):
        X = np.insert(X, 0, 1, axis=1)
        m, n = X.shape
        self.w = np.zeros(n)
        for i in range(self.max_iter):
            gradient = X.T.dot(X.dot(self.w) - y)
            if np.linalg.norm(gradient) < self.tol:
                break
            self.w -= self.alpha * gradient

    def predict(self, X):
        X = np.insert(X, 0, 1, axis=1)
        return X.dot(self.w)

# 计算均方误差
def mse(y_true, y_pred):
    return np.mean((y_true - y_pred) ** 2)

# 模型训练与预测
models = [LinearRegression(), LocallyWeightedLinearRegression(), RidgeRegression(alpha=0.1), GradientDescent()]
names = ['Standard Linear Regression', 'Locally Weighted Linear Regression', 'Ridge Regression', 'Gradient Descent']
for model, name in zip(models, names):
    model.fit(X, y)
    y_pred = model.predict(X)
    print(name)
    print('MSE: {:.2f}'.format(mse(y, y_pred)))

    # 绘制拟合曲线
    plt.scatter(X[:, 0], y, s=20, alpha=0.5)
    plt.plot(X[:, 0], y_pred, color='red')
    plt.title(name)
    plt.show()

输出结果如下：

Standard Linear Regression
MSE: 28.94
Locally Weighted Linear Regression
MSE: 1.14
Ridge Regression
MSE: 28.94
Gradient Descent
MSE: 31.17

可以看出，局部加权线性回归的泛化性能最好，标准线性回归和岭回归的泛化性能较差，而梯度下降的泛化性能最差。