Python知识(7)--最小二乘求解

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这里展示利用python实现的最小二乘的直接求解方法。其求解原理，请参考:最小二乘法拟合非线性函数及其Matlab/Excel 实现

1.一般曲线拟合

代码如下：

# -*- coding:utf-8 -*- '''
Created on Feb 20, 2017
最小二乘拟合给定的函数 fit_fun(x)
已知数据X（2xN）,Y(1xN),可直接计算直线的参数a和b
直接计算的公式：ab = inv(XXt)*X*Yt
@author: Xiankai Chen
@email: xiankai.chen@qq.com
'''
import numpy as np
import matplotlib.pyplot as pltdef fun2ploy(x,n):
'''
数据转化为[x^0,x^1,x^2,...x^n]
'''
lens = len(x)
X = np.ones([1,lens]);
for i in range(1,n):
X = np.vstack((X,np.power(x,i)))
return X def leastseq_byploy(x,y,ploy_dim):
'''
最小二乘求解
'''
#散点图
plt.scatter(x,y,color="r",marker='o',s = 50) X = fun2ploy(x,ploy_dim);
#直接求解
Xt = X.transpose();
XXt=X.dot(Xt);
XXtInv = np.linalg.inv(XXt)
XXtInvX = XXtInv.dot(X)
coef = XXtInvX.dot(y.T) y_est = Xt.dot(coef) return y_est,coefdef fit_fun(x):
'''
要拟合的函数
'''
#return np.power(x,5)
return np.sin(x)
#return 5*x+3if __name__ == '__main__':
data_num = 100;
ploy_dim =10;
noise_scale = 0.2;
## 数据准备
x = np.array(np.linspace(-2*np.pi,2*np.pi,data_num))
y = fit_fun(x)+noise_scale*np.random.rand(1,data_num)
# 最小二乘拟合
[y_est,coef] = leastseq_byploy(x,y,10) #显示拟合结果
org_data = plt.scatter(x,y,color="r",marker='o',s = 50)
est_data = plt.plot(x,y_est,color="b",linewidth= 3) plt.xlabel("X")
plt.ylabel("Y")
plt.title("Fit funtion with leastseq method")
plt.legend(["Noise data","Fited function"]);
plt.show()