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import numpy as np
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import matplotlib.pyplot as plt
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# 示例函数定义，比如二次多项式
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def func(x, theta):
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# 二次多项式: theta[0] + theta[1]*x + theta[2]*x^2
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return theta[0] + theta[1]*x + theta[2]*x**2
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# 修改后的最小二乘法函数
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def least_square_method(func, m, sampleData):
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x = sampleData[:,0]
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y = sampleData[:,1]
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# 构造 A 矩阵，使用 func 函数
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A = np.array([func(xi, np.arange(m + 1)) for xi in x])
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print(A)
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# (2) 多项式阶数m，这里以3阶多项式为例
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# 使用x的幂次来构造A矩阵
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A = np.vander(x, m + 1)
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# 计算拟合系数theta
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theta = np.linalg.inv(A.T @ A) @ A.T @ y
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# 其他步骤保持不变...
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plt.scatter(x, y, color='red')
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x_fit = np.linspace(min(x), max(x), 100)
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y_fit = np.array([func(xi, theta) for xi in x_fit])
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plt.plot(x_fit, y_fit)
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plt.show()
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# 计算残差平方和等...
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residuals = y - np.dot(A, theta)
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residuals_squared_sum = np.dot(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
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# 示例用法
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if __name__ == '__main__':
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sampleData = np.array([[0, 1], [1, 2], [2, 1], [3, 5], [4, 8], [5, 13], [6, 21], [7, 34], [8, 55], [9, 89]])
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m = 2 # 为二次多项式
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theta, covariance_matrix = least_square_method(func, m, sampleData)
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print("Theta:", theta)
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print("Covariance Matrix:", covariance_matrix)
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