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import numpy as np
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import matplotlib.pyplot as plt
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def gradient_descent_poly_fit(sampleData, degree, learning_rate=0.001, iterations=50000):
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def error_function(y_pred, y): # 定义误差函数error_function()
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return y_pred - y
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def gradient(degree,error,x_normalized,sampleNum,learning_rate): # 定义梯度函数gradient()
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# 对每个参数计算梯度
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for i in range(degree + 1):
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# 计算损失函数对参数的偏导数（梯度）
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gradient = np.dot(error, x_normalized**i) * 2 / sampleNum
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# 梯度下降更新参数
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theta[i] -= learning_rate * gradient
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return theta
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x = sampleData[:,0]
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y = sampleData[:,1]
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# 初始化参数（多项式系数）为0
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theta = np.zeros(degree + 1)
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sampleNum = len(x) # 样本数量
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# 归一化x以改善数值稳定性
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x_normalized = (x - x.mean()) / x.std()
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# 迭代梯度下降
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for _ in range(iterations):
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# 通过当前参数计算多项式的值
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y_pred = np.polyval(theta[::-1], x_normalized)
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# 计算预测值与真实值之间的误差
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error = error_function(y_pred, y)
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# 计算梯度并更新theta
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theta = gradient(degree,error,x_normalized,sampleNum,learning_rate)
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return theta, x_normalized
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# 设置一个较小的学习率和较多的迭代次数
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learning_rate = 0.001
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iterations = 50000
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degree = 4
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x = np.array([0, 100, 200, 300, 400, 500, 600, 700, 800, 900])
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y = np.array([10, 20, 10, 50, 80, 130, 210, 340, 550, 890])
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sampleData = np.array(list(zip(x, y)))
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# 运行梯度下降算法
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theta, x_normalized = gradient_descent_poly_fit(sampleData, degree, learning_rate, iterations)
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# 用拟合的参数计算多项式的值
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x_fit_normalized = np.linspace(x_normalized.min(), x_normalized.max(), 100)
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y_fit = np.polyval(theta[::-1], x_fit_normalized)
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# 反归一化x_fit
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x_fit = x_fit_normalized * x.std() + x.mean()
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# 绘制原始数据点和拟合的多项式
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plt.scatter(x, y, color='red', label='Sample Data')
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plt.plot(x_fit, y_fit, label='Polynomial Fit')
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plt.legend()
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plt.show()
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# 输出拟合参数
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print(theta)
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# if __name__ == '__main__':
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# LimitNum = 1000
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# sampleData = random_points(20,0, 1000)
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# # 这里为了便于观察使用设计好的数据
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# # x = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
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# # y = np.array([1, 2, 1, 5, 8, 13, 21, 34, 55, 89])
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# # sampleData = np.array(list(zip(x, y))) # 将两个一维数组拼接成二维数组
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# m = 3
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# theta, covariance_matrix = least_square_method(m, sampleData)
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# # print(theta)
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# curveData = compute_curveData(0,1000,1,theta,m)
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# # print("curveData",curveData)
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# draw_dots_and_line(curveData,sampleData,0,1000)
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