用python写多项式拟合_多项式最小二乘法拟合的python代码实现
最近学习李航《统计学习方法》，在github上找到了这本书对应的源码，决定自己跟着敲一敲代码，也感谢代码的贡献者，链接如下：https://github.com/fengdu78/lihang-codegithub.com
第一章有一个关于使用最小二乘法演示过拟合的示列，但最小二乘法的实现直接使用了scipy库中的leastsq，于是自己网上找了关于最小二乘法的矩阵求解，据此进行了简单的代码实现。关于多项式系数的最小二乘法推导如下(线代杀我 )：
实现代码如下：
def leastsq_mutifunc(x, y, m):
"""多项式最小二乘法实现:param x:输入:param y:目标输出:param m:多项式阶数:return:多项式系数"""
x = np.array(x)
y = np.array(y)
assert m <= x.shape[0], f"the number of m({m}) need less than x's size({x.shape[0]})"
assert x.shape[0] == y.shape[0], f"the size of x({x.shape[0]}) must equal to y's size({y.shape[0]}"
x_mat = np.zeros((x.shape[0], m+1))
for i in range(x.shape[0]):
x_mat_h = np.zeros((1, m+1))
for j in range(m+1):
x_mat_h[0][j] = x[i] ** (m-j)
x_mat[i] = x_mat_h
theta = np.dot(np.dot(np.linalg.inv(np.dot(x_mat.T, x_mat)), x_mat.T), y.T)
return theta
(多项式阶数)m = 1:
m = 3:
m = 9:
整体代码如下：
import numpy as np
import scipy as sp
from scipy.optimize import leastsq
import matplotlib.pyplot as plt
# target function
def real_func(x):
return np.sin(2 * np.pi * x)
# fit function
def fit_func(p, x):
"""
numpy.poly1d([1,2,3]) -> 1(x*x) + 2(x) + 3
"""
f = np.poly1d(p)
return f(x)
def residuals_func(p, x, y):
res = fit_func(p, x) - y
return res
def leastsq_mutifunc(x, y, m):
"""
多项式最小二乘法实现
:param x:输入
:param y:目标输出
:param m:多项式阶数
:return:多项式系数
"""
x = np.array(x)
y = np.array(y)
assert m <= x.shape[0], f"the number of m({m}) need less than x's size({x.shape[0]})"
assert x.shape[0] == y.shape[0], f"the size of x({x.shape[0]}) must equal to y's size({y.shape[0]}"
x_mat = np.zeros((x.shape[0], m+1))
for i in range(x.shape[0]):
x_mat_h = np.zeros((1, m+1))
for j in range(m+1):
x_mat_h[0][j] = x[i] ** (m-j)
x_mat[i] = x_mat_h
theta = np.dot(np.dot(np.linalg.inv(np.dot(x_mat.T, x_mat)), x_mat.T), y.T)
return theta
def fitting(x, y, M=0):
"""
:param M: 拟合目标函数的多项式次数
:return: 多项式系数
"""
# 初始化多项式系数
p_init = np.random.rand(M + 1)
# 最小二乘法
my_p_lsq = leastsq_mutifunc(x, y, m=M)
p_lsq = leastsq(residuals_func, p_init, args=(x, y))
print(f'fitting parameters: {p_lsq[0]}, my fitting parameters: {my_p_lsq}')
# 可视化
x_points = np.linspace(np.min(x), np.max(x), int(x.shape[0] * 10000))
plt.plot(x_points, real_func(x_points), label='real curve')
plt.plot(x_points, fit_func(p_lsq[0], x_points), label='fitted curve')
plt.plot(x_points, fit_func(my_p_lsq, x_points), label='my fitted curve')
plt.plot(x, y, 'bo', label='noise curve')
plt.legend()
plt.show()
return
if __name__ == '__main__':
x = np.linspace(0, 1, 10)
y_ = real_func(x)
# noise
y = [np.random.normal(0, 0.1) + y1 for y1 in y_]
p_lsq_0 = fitting(x, y, M=9)
# theta = leastsq_mutifunc(x, y, 4)
# print(theta)