python-最小二乘法-拟合非线性函数

需要补充知识：

　　最小二乘法——非线性拟合推导

 

import numpy as np
import matplotlib.pyplot as plt


plt.rcParams['font.sans-serif'] = ['SimHei']
plt.rcParams['axes.unicode_minus'] = False
# 处理中文乱码


def init_fx_data():
    # f(x)=x^(cosx-0.5)
    X=np.arange(0,15,0.1) # [0,15) step=0.05 即：300个横坐标
    I=[np.cos(x)-0.5 for x in X] # index指数
    Y=np.float_power(X,I)
    noise=[(np.random.randint(low=-15,high=15)/100) for x in X]
    Y+=noise
    return X,Y


def LS_Gaussian(xs,ys,n):
    X,Y=[],[]
    for i in range(0,n+1):
        x_row=[]
        for j in range(0,n+1):
            sum=0.0
            for x in xs:
                sum+=x**(i+j)
            x_row.append(sum)
        X.append(x_row)
    for i in range(0,n+1):
        sum=0.0
        for j in range(len(xs)):
            sum+=(xs[j]**(i))*ys[j]
        Y.append(sum)
    A=np.linalg.solve(np.array(X),np.array((Y)))
    # 求解XA=Y
    # 可以写高斯消元，这里调用numpy自带函数
    return A


def draw_figure(xs,ys,A,n):
    result_figure=plt.figure().add_subplot(111)
    X,Y=np.arange(min(xs),max(xs),0.01),[]
    for i in range(0, len(X)):
        y=0.0
        for k in range(0,n+1):
            y+=A[k]*X[i]**k
        Y.append(y)
    result_figure.plot(X,Y,color='r',linestyle='-',marker='',label='多项式拟合曲线')
    result_figure.plot(xs,ys,color='b',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(n))
    plt.legend(loc="best") # 添加默认图例到合适位置
    plt.show()


if __name__ == '__main__':
    X,Y=init_fx_data()
    order=36 #拟合的多项式项数
    A=LS_Gaussian(X,Y,order)
    draw_figure(X,Y,A,order)