统计学习方法[最小二乘法和正则化](李航大神版)
最小二乘法 p11
import numpy as np
import scipy as sp
from scipy.optimize import leastsq
import matplotlib.pyplot as plt
# 目标函数
def real_func(x):
return np.sin(2 * np.pi * x)
# 多项式
def fit_func(p, x):
f = np.poly1d(p)
return f(x)
# 残差
def residuals_func(p, x, y):
return fit_func(p, x) - y
# 十个点
x = np.linspace(0, 1, 10)
# print(x)
x_points = np.linspace(0, 1, 1000)
# print(x_points)
# 加上正态分布噪音的目标函数的值
y_ = real_func(x)
y = [np.random.normal(0, 0.1) + y1 for y1 in y_]
def fitting(M=0):
'''
M 为 多项式函数
:param M: 为 多项式函数
:return:
'''
# 随机初始化多项式参数
p_init = np.random.rand(M + 1)
print("参数初始化:\t", p_init)
# 最小二乘法
p_lsq = leastsq(residuals_func, p_init, args=(x, y))
print("拟合参数Fitting Parameters:\t", p_lsq[0])
# 可视化
plt.plot(x_points, real_func(x_points), label='real')
plt.plot(x_points, fit_func(p_lsq[0], x_points), label='fitted_curve')
plt.plot(x, y, 'bo', label='noise')
plt.legend()
plt.show()
# return p_lsq
# M=0
p_lsq_0 = fitting(M=0)
# M=1
p_lsq_0 = fitting(M=1)
# M=3
p_lsq_0 = fitting(M=3)
# M=9
p_lsq_0 = fitting(M=9)
正则化 p13
import numpy as np
import scipy as sp
from scipy.optimize import leastsq
import matplotlib.pyplot as plt
# 目标函数
def real_func(x):
return np.sin(2 * np.pi * x)
# 多项式
def fit_func(p, x):
f = np.poly1d(p)
return f(x)
# 残差
def residuals_func(p, x, y):
return fit_func(p, x) - y
# 十个点
x = np.linspace(0, 1, 10)
# print(x)
x_points = np.linspace(0, 1, 1000)
# print(x_points)
# 加上正态分布噪音的目标函数的值
y_ = real_func(x)
y = [np.random.normal(0, 0.1) + y1 for y1 in y_]
def fitting(M=0):
'''
M 为 多项式函数
:param M: 为 多项式函数
:return:
'''
# 随机初始化多项式参数
p_init = np.random.rand(M + 1)
print("参数初始化:\t", p_init)
# 最小二乘法
p_lsq = leastsq(residuals_func, p_init, args=(x, y))
print("拟合参数Fitting Parameters:\t", p_lsq[0])
# 可视化
plt.plot(x_points, real_func(x_points), label='real')
plt.plot(x_points, fit_func(p_lsq[0], x_points), label='fitted_curve')
plt.plot(x, y, 'bo', label='noise')
plt.legend()
plt.show()
return p_lsq
regularization = 0.0001
def residuals_func_regularization(p, x, y):
return np.append(fit_func(p, x) - y, np.sqrt(0.5 * regularization * np.square(p))) # L2范数作为正则化项
# 最小二乘法,加正则化项
p_init = np.random.rand(9 + 1)
p_lsq_0 = fitting(9)
p_lsq_regularization = leastsq(residuals_func_regularization, p_init, args=(x, y))
plt.plot(x_points, real_func(x_points), label='real')
plt.plot(x_points, fit_func(p_lsq_0[0], x_points), label="fitted curve")
plt.plot(x_points, fit_func(p_lsq_regularization[0], x_points), label="regularization")
plt.plot(x, y, 'bo', label='noise')
plt.legend()
plt.show()
参考文献
posted on 2019-04-05 12:52 Indian_Mysore 阅读(573) 评论(1) 编辑 收藏 举报