最小二乘法擬合多項式原理以及c++實現

轉載請註明出處：http://blog.csdn.net/lsh_2013/article/details/46697625

最小二乘法（又稱最小平方法）是一種數學優化技術。它通過最小化誤差的平方和尋找數據的最佳函數匹配。

c++實現代碼如下：

#include <iostream>
#include <vector>
#include <cmath>
using namespace std;

//最小二乘擬合相關函數定義
double sum(vector<double> Vnum, int n);
double MutilSum(vector<double> Vx, vector<double> Vy, int n);
double RelatePow(vector<double> Vx, int n, int ex);
double RelateMutiXY(vector<double> Vx, vector<double> Vy, int n, int ex);
void EMatrix(vector<double> Vx, vector<double> Vy, int n, int ex, double coefficient[]);
void CalEquation(int exp, double coefficient[]);
double F(double c[],int l,int m);
double Em[6][4];

//主函數，這裏將數據擬合成二次曲線
int main(int argc, char* argv[])
{
	double arry1[5]={0,0.25,0,5,0.75};
	double arry2[5]={1,1.283,1.649,2.212,2.178};
	double coefficient[5];
	memset(coefficient,0,sizeof(double)*5);
	vector<double> vx,vy;
	for (int i=0; i<5; i++)
	{
		vx.push_back(arry1[i]);
		vy.push_back(arry2[i]);
	}
	EMatrix(vx,vy,5,3,coefficient);
	printf("擬合方程爲：y = %lf + %lfx + %lfx^2 \n",coefficient[1],coefficient[2],coefficient[3]);
	return 0;
}
//累加
double sum(vector<double> Vnum, int n)
{
	double dsum=0;
	for (int i=0; i<n; i++)
	{
		dsum+=Vnum[i];
	}
	return dsum;
}
//乘積和
double MutilSum(vector<double> Vx, vector<double> Vy, int n)
{
	double dMultiSum=0;
	for (int i=0; i<n; i++)
	{
		dMultiSum+=Vx[i]*Vy[i];
	}
	return dMultiSum;
}
//ex次方和
double RelatePow(vector<double> Vx, int n, int ex)
{
	double ReSum=0;
	for (int i=0; i<n; i++)
	{
		ReSum+=pow(Vx[i],ex);
	}
	return ReSum;
}
//x的ex次方與y的乘積的累加
double RelateMutiXY(vector<double> Vx, vector<double> Vy, int n, int ex)
{
	double dReMultiSum=0;
	for (int i=0; i<n; i++)
	{
		dReMultiSum+=pow(Vx[i],ex)*Vy[i];
	}
	return dReMultiSum;
}
//計算方程組的增廣矩陣
void EMatrix(vector<double> Vx, vector<double> Vy, int n, int ex, double coefficient[])
{
	for (int i=1; i<=ex; i++)
	{
		for (int j=1; j<=ex; j++)
		{
			Em[i][j]=RelatePow(Vx,n,i+j-2);
		}
		Em[i][ex+1]=RelateMutiXY(Vx,Vy,n,i-1);
	}
	Em[1][1]=n;
	CalEquation(ex,coefficient);
}
//求解方程
void CalEquation(int exp, double coefficient[])
{
	for(int k=1;k<exp;k++) //消元過程
	{
		for(int i=k+1;i<exp+1;i++)
		{
			double p1=0;

			if(Em[k][k]!=0)
				p1=Em[i][k]/Em[k][k];

			for(int j=k;j<exp+2;j++) 
				Em[i][j]=Em[i][j]-Em[k][j]*p1;
		}
	}
	coefficient[exp]=Em[exp][exp+1]/Em[exp][exp];
	for(int l=exp-1;l>=1;l--)   //回代求解
		coefficient[l]=(Em[l][exp+1]-F(coefficient,l+1,exp))/Em[l][l];
}
//供CalEquation函數調用
double F(double c[],int l,int m)
{
	double sum=0;
	for(int i=l;i<=m;i++)
		sum+=Em[l-1][i]*c[i];
	return sum; 
}