#coding:utf-8 import numpy as np import scipy as sp from scipy.optimize import leastsq import matplotlib.pyplot as plt # 目標(biāo)函數(shù) def real_func(x):return np.sin(2*np.pi*x)# 多項式 def fit_func(p, x):f = np.poly1d(p)# print('f=',f)return f(x)# 殘差 def residuals_func(p, x, y):ret = fit_func(p, x) - yreturn ret# 十個點 x = np.linspace(0, 1, 10) x_points = np.linspace(0, 1, 1000) # 加上正態(tài)分布噪音的目標(biāo)函數(shù)的值 y_ = real_func(x) y = [np.random.normal(0, 0.1) + y1 for y1 in y_]def fitting(M=0):"""M 為 多項式的次數(shù)"""# 隨機初始化多項式參數(shù)p_init = np.random.rand(M + 1)# 最小二乘法p_lsq = leastsq(residuals_func, p_init, args=(x, y))print('Fitting Parameters:', p_lsq[0])## 可視化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_1 = fitting(M=1) # M=3 p_lsq_3 = fitting(M=3) # M=9 p_lsq_9 = fitting(M=9)

M分別為0,1,3,9時的多項式系數(shù)。?

?

M=0,即多項式為常數(shù)時?

M=1, 即多項式為一次項時

?M=3，即多項式為三次項時，可看出擬合的比較不錯

M=9時，可看出過擬合了

引入正則化

#加入正則 regularization = 0.0001 def residuals_func_regularization(p, x, y):ret = fit_func(p, x) - yret = np.append(ret, np.sqrt(0.5*regularization*np.square(p))) # L2范數(shù)作為正則化項return ret # 最小二乘法,加正則化項 p_init = np.random.rand(9+1) 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_9[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()

可看出：正則化有效?

?

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