'''

最小二乘法拟合函数曲线f(x)

1、拟合多项式为：y = a0 + a1*x + a2*x^2 + ... + ak*x^k

2、求每个点到曲线的距离之和：Loss = ∑(yi - (a0 + a1*x + a2*x^2 + ... + ak*x^k))^2

3、最优化Loss函数，即求Loss对a0,a1,...ak的偏导数为0

3.1、数学解法——求解线性方程组：

整理最优化的偏导数矩阵为：X：含有xi的系数矩阵，A：含有ai的系数矩阵，Y：含有yi的系数矩阵

求解：XA=Y中的A矩阵

3.2、迭代解法——梯度下降法：

计算每个系数矩阵A[k]的梯度，并迭代更新A[k]的梯度

A[k] = A[k] - (learn_rate * gradient)

'''

import numpy as np

import matplotlib.pyplot as plt

plt.rcParams['font.sans-serif'] = ['SimHei']

plt.rcParams['axes.unicode_minus'] = False

'''

高斯列主消元算法

'''

# 得到增广矩阵

def get_augmented_matrix(matrix, b):

row, col = np.shape(matrix)

matrix = np.insert(matrix, col, values=b, axis=1)

return matrix

# 取出增广矩阵的系数矩阵（第一列到倒数第二列）

def get_matrix(a_matrix):

return a_matrix[:, :a_matrix.shape[1] - 1]

# 选列主元，在第k行后的矩阵里，找出最大值和其对应的行号和列号

def get_pos_j_max(matrix, k):

max_v = np.max(matrix[k:, :])

pos = np.argwhere(matrix == max_v)

i, _ = pos[0]

return i, max_v

# 矩阵的第k行后，行交换

def exchange_row(matrix, r1, r2, k):

matrix[[r1, r2], k:] = matrix[[r2, r1], k:]

return matrix

# 消元计算(初等变化)

def elimination(matrix, k):

row, col = np.shape(matrix)

for i in range(k + 1, row):

m_ik = matrix[i][k] / matrix[k][k]

matrix[i] = -m_ik * matrix[k] + matrix[i]

return matrix

# 回代求解

def backToSolve(a_matrix):

matrix = a_matrix[:, :a_matrix.shape[1] - 1] # 得到系数矩阵

b_matrix = a_matrix[:, -1] # 得到值矩阵

row, col = np.shape(matrix)

x = [None] * col # 待求解空间X

# 先计算上三角矩阵对应的最后一个分量

x[-1] = b_matrix[col - 1] / matrix[col - 1][col - 1]

# 从倒数第二行开始回代x分量

for _ in range(col - 1, 0, -1):

i = _ - 1

sij = 0

xidx = len(x) - 1

for j in range(col - 1, i, -1):

sij += matrix[i][j] * x[xidx]

xidx -= 1

x[xidx] = (b_matrix[i] - sij) / matrix[i][i]

return x

# 求解非齐次线性方程组：ax=b

def solve_NLQ(a, b):

a_matrix = get_augmented_matrix(a, b)

for k in range(len(a_matrix) - 1):

# 选列主元

max_i, max_v = get_pos_j_max(get_matrix(a_matrix), k=k)

# 如果A[ik][k]=0，则矩阵奇异退出

if a_matrix[max_i][k] == 0:

print('矩阵A奇异')

return None, []

if max_i != k:

a_matrix = exchange_row(a_matrix, k, max_i, k=k)

# 消元计算

a_matrix = elimination(a_matrix, k=k)

# 回代求解

X = backToSolve(a_matrix)

return a_matrix, X

'''

最小二乘法多项式拟合曲线

'''

# 生成带有噪点的待拟合的数据集合

def init_fx_data():

# 待拟合曲线f(x) = sin2x * [(x^2 - 1)^3 + 0.5]

xs = np.arange(-1, 1, 0.01) # 200个点

ys = [((x ** 2 - 1) ** 3 + 0.5) * np.sin(x * 2) for x in xs]

ys1 = []

for i in range(len(ys)):

z = np.random.randint(low=-10, high=10) / 100 # 加入噪点

ys1.append(ys[i] + z)

return xs, ys1

# 计算最小二乘法当前的误差

def last_square_current_loss(xs, ys, A):

error = 0.0

for i in range(len(xs)):

y1 = 0.0

for k in range(len(A)):

y1 += A[k] * xs[i] ** k

error += (ys[i] - y1) ** 2

return error

# 迭代解法：最小二乘法+梯度下降法

def last_square_fit_curve_Gradient(xs, ys, order, iternum=1000, learn_rate=0.001):

A = [0.0] * (order + 1)

for r in range(iternum + 1):

for k in range(len(A)):

gradient = 0.0

for i in range(len(xs)):

y1 = 0.0

for j in range(len(A)):

y1 += A[j] * xs[i]**j

gradient += -2 * (ys[i] - y1) * xs[i]**k # 计算A[k]的梯度

A[k] = A[k] - (learn_rate * gradient) # 更新A[k]的梯度

# 检查误差变化

if r % 100 == 0:

error = last_square_current_loss(xs=xs, ys=ys, A=A)

print('最小二乘法+梯度下降法：第{}次迭代，误差下降为：{}'.format(r, error))

return A

# 数学解法：最小二乘法+求解线性方程组

def last_square_fit_curve_Gauss(xs, ys, order):

X, Y = [], []

# 求解偏导数矩阵里，含有xi的系数矩阵X

for i in range(0, order + 1):

X_line = []

for j in range(0, order + 1):

sum_xi = 0.0

for xi in xs:

sum_xi += xi ** (j + i)

X_line.append(sum_xi)

X.append(X_line)

# 求解偏导数矩阵里，含有yi的系数矩阵Y

for i in range(0, order + 1):

Y_line = 0.0

for j in range(0, order + 1):

sum_xi_yi = 0.0

for k in range(len(xs)):

sum_xi_yi += (xs[k] ** i * ys[k])

Y_line = sum_xi_yi

Y.append(Y_line)

a_matrix, A = solve_NLQ(np.array(X), Y) # 高斯消元：求解XA=Y的A

# A = np.linalg.solve(np.array(X), np.array(Y)) # numpy API 求解XA=Y的A

error = last_square_current_loss(xs=xs, ys=ys, A=A)

print('最小二乘法+求解线性方程组，误差下降为：{}'.format(error))

return A

# 可视化多项式曲线拟合结果

def draw_fit_curve(xs, ys, A, order):

fig = plt.figure()

ax = fig.add_subplot(111)

fit_xs, fit_ys = np.arange(min(xs) * 0.8, max(xs) * 0.8, 0.01), []

for i in range(0, len(fit_xs)):

y = 0.0

for k in range(0, order + 1):

y += (A[k] * fit_xs[i] ** k)

fit_ys.append(y)

ax.plot(fit_xs, fit_ys, color='g', linestyle='-', marker='', label='多项式拟合曲线')

ax.plot(xs, ys, color='m', linestyle='', marker='.', label='曲线真实数据')

plt.title(s='最小二乘法拟合多项式N={}的函数曲线f(x)'.format(order))

plt.legend()

plt.show()

if __name__ == '__main__':

order = 10 # 拟合的多项式项数

xs, ys = init_fx_data() # 曲线数据

# 数学解法：最小二乘法+求解线性方程组

A = last_square_fit_curve_Gauss(xs=xs, ys=ys, order=order)

# 迭代解法：最小二乘法+梯度下降

# A = last_square_fit_curve_Gradient(xs=xs, ys=ys, order=order, iternum=10000, learn_rate=0.001)

draw_fit_curve(xs=xs, ys=ys, A=A, order=order) # 可视化多项式曲线拟合结果