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import numpy as np
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import matplotlib.pyplot as plt
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def least_square_method(m, sampleData):
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x = sampleData[:, 0]
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y = sampleData[:, 1]
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# 标准化x
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x_mean = x.mean()
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x_std = x.std()
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x_normalized = (x - x_mean) / x_std
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# 使用标准化的x构造A矩阵
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A = np.vander(x_normalized, m + 1, increasing=True)
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# 计算拟合系数theta，使用最小二乘法的正规方程
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theta = np.linalg.inv(A.T @ A) @ A.T @ y
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# 计算残差平方和和其他统计量
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residuals = y - A @ theta
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residuals_squared_sum = residuals.T @ residuals
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degrees_of_freedom = len(y) - (m + 1)
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mean_squared_error = residuals_squared_sum / degrees_of_freedom
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covariance_matrix = np.linalg.inv(A.T @ A) * mean_squared_error
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return theta, covariance_matrix, x_mean, x_std
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def compute_curveData(low, high, step, theta, m, x_mean, x_std):
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x_fit = np.arange(low, high, step)
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# 将x_fit标准化
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x_fit_normalized = (x_fit - x_mean) / x_std
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A_fit = np.vander(x_fit_normalized, m + 1, increasing=True)
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y_fit = A_fit @ theta
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return np.column_stack((x_fit, y_fit))
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def draw_dots_and_line(curveData, sampleData, low, high):
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x_fit = curveData[:, 0]
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y_fit = curveData[:, 1]
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x = sampleData[:, 0]
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y = sampleData[:, 1]
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plt.plot(x_fit, y_fit, label='FitCurve')
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plt.scatter(x, y, color='red', label='SampleData')
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plt.xlim(low, high)
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plt.legend()
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plt.show()
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def random_points(n, low, high):
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np.random.seed(0) # For reproducibility
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# x = np.random.uniform(low, high, n)
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# y = np.random.uniform(low, high, n) # This is just an example
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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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return np.column_stack((x, y))
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if __name__ == '__main__':
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sampleData = random_points(20, 0, 1000)
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print(len(sampleData))
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m = 3
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theta, covariance_matrix, x_mean, x_std = least_square_method(m, sampleData)
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curveData = compute_curveData(0, 1000, 1, theta, m, x_mean, x_std)
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draw_dots_and_line(curveData, sampleData, 0, 1000)
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