最小二乘曲线拟合——C语言算法实现一

最小二乘曲线拟合

 

给定一组数据，我们要对这组数据进行曲线拟合。

假定要拟合的曲线方程为 y=a0 + a1*x^1 + a2*x^2 + a3*x^3 + ...+ an*x^n

    x             y

0.995119      -7.620000
2.001185      -2.460000
2.999068     108.760000
4.001035     625.020000
4.999859     2170.500000
6.004461     5814.580000
6.999335     13191.840000
7.999433     26622.060000
9.002257      49230.220000
10.003888    85066.500000
11.004076    139226.280000
12.001602     217970.140000
13.003390     328843.860000
14.001623    480798.420000
15.003034    684310.000000
16.002561    951499.980000
17.003010   1296254.940000
18.003897   1734346.660000
19.002563   2283552.120000
20.003530    2963773.500000

代码如下：

#include "stdio.h"
#include "stdlib.h"
#include "math.h"
#include 
#define ParaBuffer(Buffer,Row,Col) (*(Buffer + (Row) * (SizeSrc + 1) + (Col)))
 /***********************************************************************************
 ***********************************************************************************/
 static int GetXY(const char* FileName, double* X, double* Y, int* Amount)
 {
     FILE* File = fopen(FileName, "r");
     if (!File)
		return -1;
     for (*Amount = 0; !feof(File); X++, Y++, (*Amount)++)
         if (2 != fscanf(File, (const char*)"%lf %lf", X, Y))
            break;
     fclose(File);
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 static int PrintPara(double* Para, int SizeSrc)
 {
     int i, j;
     for (i = 0; i < SizeSrc; i++)
     {
         for (j = 0; j <= SizeSrc; j++)
         printf("%10.6lf ", ParaBuffer(Para, i, j));
         printf("\r\n");
     }
     printf("\r\n");
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 static int ParalimitRow(double* Para, int SizeSrc, int Row)
 {
     int i;
     double Max, Min, Temp;
     for (Max = abs(ParaBuffer(Para, Row, 0)), Min = Max, i = SizeSrc; i; i--)
     {
         Temp = abs(ParaBuffer(Para, Row, i));
         if (Max < Temp)
             Max = Temp;
         if (Min > Temp)
             Min = Temp;
     }
     Max = (Max + Min) * 0.000005;
     for (i = SizeSrc; i >= 0; i--)
         ParaBuffer(Para, Row, i) /= Max;
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 static int Paralimit(double* Para, int SizeSrc)
 {
     int i;
     for (i = 0; i < SizeSrc; i++)
          if (ParalimitRow(Para, SizeSrc, i))
             return -1;
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 static int ParaPreDealA(double* Para, int SizeSrc, int Size)
 {
     int i, j;
     for (Size -= 1, i = 0; i < Size; i++)
     {
         for (j = 0; j < Size; j++)
         ParaBuffer(Para, i, j) = ParaBuffer(Para, i, j) * ParaBuffer(Para, Size, Size) - ParaBuffer(Para, Size, j) * ParaBuffer(Para, i, Size);
         ParaBuffer(Para, i, SizeSrc) = ParaBuffer(Para, i, SizeSrc) * ParaBuffer(Para, Size, Size) - ParaBuffer(Para, Size, SizeSrc) * ParaBuffer(Para, i, Size);
         ParaBuffer(Para, i, Size) = 0;
         ParalimitRow(Para, SizeSrc, i);
     }
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 static int ParaDealA(double* Para, int SizeSrc)
 {
     int i;
     for (i = SizeSrc; i; i--)
         if (ParaPreDealA(Para, SizeSrc, i))
             return -1;
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 static int ParaPreDealB(double* Para, int SizeSrc, int OffSet)
 {
     int i, j;
     for (i = OffSet + 1; i < SizeSrc; i++)
     {
         for (j = OffSet + 1; j <= i; j++)
         ParaBuffer(Para, i, j) *= ParaBuffer(Para, OffSet, OffSet);
         ParaBuffer(Para, i, SizeSrc) = ParaBuffer(Para, i, SizeSrc) * ParaBuffer(Para, OffSet, OffSet) - ParaBuffer(Para, i, OffSet) * ParaBuffer(Para, OffSet, SizeSrc);
         ParaBuffer(Para, i, OffSet) = 0;
         ParalimitRow(Para, SizeSrc, i);
     }
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 static int ParaDealB(double* Para, int SizeSrc)
 {
     int i;
     for (i = 0; i < SizeSrc; i++)
             if (ParaPreDealB(Para, SizeSrc, i))
                     return -1;
     for (i = 0; i < SizeSrc; i++)
     {
         if (ParaBuffer(Para, i, i))
         {
             ParaBuffer(Para, i, SizeSrc) /= ParaBuffer(Para, i, i);
             ParaBuffer(Para, i, i) = 1.0;
         }
     }
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 static int ParaDeal(double* Para, int SizeSrc)
 {
     PrintPara(Para, SizeSrc);
     Paralimit(Para, SizeSrc);
     PrintPara(Para, SizeSrc);
     if (ParaDealA(Para, SizeSrc))
             return -1;
     PrintPara(Para, SizeSrc);
     if (ParaDealB(Para, SizeSrc))
             return -1;
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 static int GetParaBuffer(double* Para, const double* X, const double* Y, int Amount, int SizeSrc)
 {
     int i, j;
     for (i = 0; i < SizeSrc; i++)
             for (ParaBuffer(Para, 0, i) = 0, j = 0; j < Amount; j++)
                     ParaBuffer(Para, 0, i) += pow(*(X + j), 2 * (SizeSrc - 1) - i);
     for (i = 1; i < SizeSrc; i++)
             for (ParaBuffer(Para, i, SizeSrc - 1) = 0, j = 0; j < Amount; j++)
                     ParaBuffer(Para, i, SizeSrc - 1) += pow(*(X + j), SizeSrc - 1 - i);
     for (i = 0; i < SizeSrc; i++)
             for (ParaBuffer(Para, i, SizeSrc) = 0, j = 0; j < Amount; j++)
                     ParaBuffer(Para, i, SizeSrc) += (*(Y + j)) * pow(*(X + j), SizeSrc - 1 - i);
     for (i = 1; i < SizeSrc; i++)
             for (j = 0; j < SizeSrc - 1; j++)
                     ParaBuffer(Para, i, j) = ParaBuffer(Para, i - 1, j + 1);
     return 0;
 }
 //***********************************************************************************
 //***********************************************************************************
 int Cal(const double* BufferX, const double* BufferY, int Amount, int SizeSrc, double* ParaResK)
 {
     double* ParaK = (double*)malloc(SizeSrc * (SizeSrc + 1) * sizeof(double));
     GetParaBuffer(ParaK, BufferX, BufferY, Amount, SizeSrc);
     ParaDeal(ParaK, SizeSrc);
     for (Amount = 0; Amount < SizeSrc; Amount++, ParaResK++)
             *ParaResK = ParaBuffer(ParaK, Amount, SizeSrc);
     free(ParaK);
     return 0;
 }
 /***********************************************************************************
 ***********************************************************************************/
 int main(int argc, char* argv[])
 {
     int Amount;
	 clock_t StartTime, FinishTime;	//	声明两个时间变量
	 double DiffTime;
	 StartTime = clock();		//	开始计时

     double BufferX[1024], BufferY[1024], ParaK[6];	//	5次拟合, 一共6个系数(包含常数项)
	 //	读入要拟合的数据
     if (GetXY((const char*)"test.txt", (double*)BufferX, (double*)BufferY, &Amount))
          return 0;
	
	  printf("Amount: %d\n", Amount);
     Cal((const double*)BufferX, (const double*)BufferY, Amount, sizeof(ParaK) / sizeof(double), (double*)ParaK);
	 printf("拟合系数为:\n");
	 printf("按升序排列\n");
     for (Amount = 0; Amount < sizeof(ParaK) / sizeof(double); Amount++)
             printf("ParaK[%d] = %lf\r\n", Amount, ParaK[Amount]);

	 FinishTime = clock();	//	结束计时
	 DiffTime = FinishTime - StartTime;	//拟合时间
	 printf("拟合时间为: %lf\n", DiffTime);
     return 0;
 }
 

输出结果为：

Amount: 20

拟合系数为:
按升序排列
ParaK[0] = 0.999157
ParaK[1] = -1.450258
ParaK[2] = -0.529332
ParaK[3] = 0.236626
ParaK[4] = 6.725930
ParaK[5] = -18.544115
拟合时间为: 9.000000ms

Matlab代码：

其中test.txt文件为上面的x,y向量

load test.txt;
 x = test(:, 1)';
 y = test(:, 2)';
 para = polyfit(x, y, 5)
 
 figure,
 plot(x,y,'b.')
 hold on
 plot(x,y,'r-')

Matlab拟合系数，降序排列：

para =   0.9992   -1.4503   -0.5293    0.2366    6.7259  -18.5441

Matlab拟合曲线结果图：