多项式拟合的cpp实现

当我们拥有一组散点图数据时，通常更愿意看到其走势。

对现有数据进行拟合，并输出拟合优度是常用的方法之一。

拟合结果正确性的验证，可以使用excel自带的功能。

下面是c++代码的实现：

#ifndef __Fit_h__
#define __Fit_h__

#include <vector>

template<size_t Degree>
class CFit
{
public:
	CFit(std::vector<double>& xArr,std::vector<double>& yArr):m_xArr(xArr),m_yArr(yArr),m_ssr(0.0),m_sse(0.0),m_rmse(0.0){m_bFitYet = false;}

	~CFit(){}
protected:
	//- 高斯消除
	template<size_t realDegree>
	static void GaussianElimination(double (&matrix)[realDegree+1][realDegree+2]
	,double (&coeArr)[realDegree+1])
	{
		int i,j,k;
		for (i = 0; i< realDegree; i++ )					//loop to perform the gauss elimination
		{
			for (k = i+1; k < (realDegree+1); k++)
			{
				double t = matrix[k][i]/matrix[i][i];
				for (j=0;j<=(realDegree+1);j++)
					matrix[k][j] -= t*matrix[i][j];			//make the elements below the pivot elements equal to zero or elimnate the variables
			}
		}
		for (i = realDegree; i >= 0; i--)					//back-substitution
		{													//x is an array whose values correspond to the values of x,y,z..
			coeArr[i] = matrix[i][realDegree+1];    //make the variable to be calculated equal to the rhs of the last equation
			for (j=0;j<(realDegree+1);j++)
				if (j!=i)									//then subtract all the lhs values except the coefficient of the variable whose value                                   is being calculated
					coeArr[i] -= matrix[i][j]*coeArr[j];
			coeArr[i] = coeArr[i]/matrix[i][i];            //now finally divide the rhs by the coefficient of the variable to be calculated
		}
	}

	///
	/// \brief 根据x获取拟合方程的y值
	/// \return 返回x对应的y值
	///
	template<typename T>
	double getY(const T x) const
	{
		double ans(0);
		for (size_t i=0;i<(Degree+1);++i)
		{
			ans += m_coefficientArr[i]*pow((double)x,(int)i);
		}
		return ans;
	}

	///
	/// \brief 计算均值
	/// \return 均值
	///
	template <typename T>
	static T Mean(const std::vector<T>& v)
	{
		return Mean(&v[0],v.size());
	}
	template <typename T>
	static T Mean(const T* v,size_t length)
	{
		T total(0);
		for (size_t i=0;i<length;++i)
		{
			total += v[i];
		}
		return (total / length);
	}

	template<typename T>
	void calcError(const T* x
		,const T* y
		,size_t length
		,double& r_ssr
		,double& r_sse
		,double& r_rmse
		)
	{
		T mean_y = Mean<T>(y,length);
		T yi(0);

		for (size_t i=0; i<length; ++i)
		{
			yi = getY(x[i]);
			r_ssr += ((yi-mean_y)*(yi-mean_y));//计算回归平方和
			r_sse += ((yi-y[i])*(yi-y[i]));//残差平方和
		}
		r_rmse = sqrt(r_sse/(double(length)));
	}

	/**
	* @brief    根据两组数据进行一元多项式拟合
	* @author
	* @param    [in]	int N,数据个数
				[in]	const std::vector<double>& xArr,横坐标数据
				[in]	const std::vector<double>& yArr,纵坐标数据
	* @param    [out]	double (&coefficientArr)[Degree+1],拟合结果.一元多项式系数,从低到高
	* @return   none
	* @note     none
	*/
	static void PolynomialFit(int N,const std::vector<double>& xArr,const std::vector<double>& yArr,double (&coefficientArr)[Degree+1])
	{
		int i = 0,j = 0,k = 0;
		//const int realDegree = Degree -1;
		double X[2*Degree+1] = {0};              //Array that will store the values of sigma(xi),sigma(xi^2),sigma(xi^3)....sigma(xi^2n)
		for (i=0;i<2*Degree+1;i++)
		{
			for (j=0;j<N;j++)
				X[i] += pow(xArr[j],i);              //consecutive positions of the array will store N,sigma(xi),sigma(xi^2),sigma(xi^3)....sigma(xi^2n)
		}
		double Y[Degree+1] = {0};                //Array to store the values of sigma(yi),sigma(xi*yi),sigma(xi^2*yi)...sigma(xi^n*yi)
		for (i=0;i<Degree+1;i++)
		{
			for (j=0;j<N;j++)
				Y[i] += pow(xArr[j],i)*yArr[j];		//consecutive positions will store sigma(yi),sigma(xi*yi),sigma(xi^2*yi)...sigma(xi^n*yi)
		}

		double B[Degree+1][Degree+2] = {0};			//B is the Normal matrix(augmented) that will store the equations
		for (i=0;i<=Degree;i++)
		{
			for (j=0;j<=Degree;j++)
			{
				B[i][j] = X[i+j];					//Build the Normal matrix by storing the corresponding coefficients at the right positions except the last column of the matrix
			}
			B[i][Degree+1] = Y[i];				//load the values of Y as the last column of B(Normal Matrix but augmented)
		}

		GaussianElimination<Degree>(B,coefficientArr);
	}
public:
	void PolyFit()
	{
		if (m_xArr.size() == m_yArr.size())
		{
			PolynomialFit(static_cast<int>(m_xArr.size()),m_xArr,m_yArr,m_coefficientArr);
			m_bFitYet = true;
			calcError(&m_xArr[0],&m_yArr[0],static_cast<int>(m_xArr.size()),m_ssr,m_sse,m_rmse);
		}
	}
	//- 一元多项式计算
	double UnaryPolynomialCalc(double dX)
	{
		double dY = 0.0;
		for (size_t ulDegree = 0; ulDegree <= Degree; ++ulDegree)
		{
			dY += pow(dX,(double)ulDegree) * m_coefficientArr[ulDegree];
		}
		return m_bFitYet ? dY : 0.0;
	}

	///
	/// \brief 剩余平方和
	/// \return 剩余平方和
	///
	double getSSE(){return m_sse;}
	///
	/// \brief 回归平方和
	/// \return 回归平方和
	///
	double getSSR(){return m_ssr;}
	///
	/// \brief 均方根误差
	/// \return 均方根误差
	///
	double getRMSE(){return m_rmse;}
	///
	/// \brief 确定系数，系数是0~1之间的数，是数理上判定拟合优度(goodness-of-fit)的一个量
	/// \return 确定系数
	///
	double getR_square(){return 1-(m_sse/(m_ssr+m_sse));}

	///
	/// \brief 根据阶次获取拟合方程的系数，
	/// 如getFactor(2),就是获取y=a0+a1*x+a2*x^2+……+apoly_n*x^poly_n中a2的值
	/// \return 拟合方程的系数
	///
	double getFactor(size_t i)
	{
		return (i <= Degree) ? m_coefficientArr[i] : 0.0;
	}

private:
	double m_coefficientArr[Degree+1];
	const std::vector<double>& m_xArr;
	const std::vector<double>& m_yArr;
	bool m_bFitYet;//- 一元多项式计算时多项式拟合是否完成 [1/6/2017 wWX210786]

	double m_ssr;                 ///<回归平方和
	double m_sse;                 ///<(剩余平方和)
	double m_rmse;                ///<RMSE均方根误差
};

#endif // __Fit_h__

　　使用起来也很方便：

        double y[] = {7,16,6,18,6,6,10,8};
		double x[] = {-109.71,-101.81,-103.83,-99.89,-90,-112.17,-93.5,-96.13};
		std::vector<double> xArr(std::begin(x),std::end(x));
		std::vector<double> yArr(std::begin(y),std::end(y));
		typedef CFit<4> LineFit;
		LineFit objPolyfit(xArr,yArr);
		objPolyfit.PolyFit();
		std::wstring coeArr[] = {L"",L"x",L"x²",L"x\u00b3",L"x "};
		CString info(_T("y = "));
		for (int i=1;i>=0;i--)
			info.AppendFormat(_T("+( %f%s )"),objPolyfit.m_coefficientArr[i],coeArr[i].c_str());

		std::wcout << info.GetString() << "\n";

		//std::wcout << "斜率 = " << objPolyfit.getFactor(1) << "\n";
		//std::wcout << "截距 = " << objPolyfit.getFactor(0) << "\n";
		std::wcout << "goodness-of-fit = "<< objPolyfit.getR_square() << "\n";