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先看看一元线性回归函数代码:
// 求线性回归方程:Y = a + bx
// dada[rows*2]数组:X, Y;rows:数据行数;a, b:返回回归系数
// SquarePoor[4]:返回方差分析指标: 回归平方和,剩余平方和,回归平方差,剩余平方差
// 返回值:0求解成功,-1错误
int LinearRegression(double
*data, int rows, double
*a, double
*b, double
*SquarePoor)
{
int m;
double
*p, Lxx =
0.0, Lxy =
0.0, xa =
0.0, ya =
0.0;
if (data ==
0
|| a ==
0
|| b ==
0
|| rows <
1)
return
-1;
for (p = data, m =
0; m < rows; m ++)
{
xa +=
*p ++;
ya +=
*p ++;
}
xa /= rows; // X平均值
ya /= rows; // Y平均值
for (p = data, m =
0; m < rows; m ++, p +=
2)
{
Lxx += ((*p - xa) * (*p - xa)); // Lxx = Sum((X - Xa)平方)
Lxy += ((*p - xa) * (*(p +
1) - ya)); // Lxy = Sum((X - Xa)(Y - Ya))
}
*b = Lxy / Lxx; // b = Lxy / Lxx
*a = ya -
*b * xa; // a = Ya - b*Xa
if (SquarePoor ==
0)
return
0;
// 方差分析
SquarePoor[0] = SquarePoor[1] =
0.0;
for (p = data, m =
0; m < rows; m ++, p ++)
{
Lxy =
*a +
*b *
*p ++;
SquarePoor[0] += ((Lxy - ya) * (Lxy - ya)); // U(回归平方和)
SquarePoor[1] += ((*p - Lxy) * (*p - Lxy)); // Q(剩余平方和)
}
SquarePoor[2] = SquarePoor[0]; // 回归方差
SquarePoor[3] = SquarePoor[1] / (rows -
2); // 剩余方差
return
0;
}
为了理解代码,把几个与代码有关的公式写在下面(回归理论和公式推导就免了,网上搜索到处是,下面的公式图片也是网上搜的,有些公式图形网上没找到或者不合适,可参见后面多元回归中的公式):
1、回归方程式:![](http://www.fjtu.com.cn/fjnu/courseware/0422/course/_source/web/lesson/char9/9.files/image087.gif)
2、回归系数: ![](http://source.mastvu.ah.cn/gaoxiao/ku/dbsf/gllysltj/chap9/jiangjie/91.files/image037.gif)
其中:
![](http://source.mastvu.ah.cn/gaoxiao/ku/dbsf/gllysltj/chap9/jiangjie/91.files/image031.gif)
![](http://source.mastvu.ah.cn/gaoxiao/ku/dbsf/gllysltj/chap9/jiangjie/91.files/image032.gif)
3、回归平方和:![](http://source.mastvu.ah.cn/gaoxiao/ku/dbsf/gllysltj/chap9/jiangjie/91.files/image082.gif)
4、剩余平方和:![](http://source.mastvu.ah.cn/gaoxiao/ku/dbsf/gllysltj/chap9/jiangjie/91.files/image081.gif)
实例计算:
double data1[12][2] = {
// X Y
{187.1, 25.4},
{179.5, 22.8},
{157.0, 20.6},
{197.0, 21.8},
{239.4, 32.4},
{217.8, 24.4},
{227.1, 29.3},
{233.4, 27.9},
{242.0, 27.8},
{251.9, 34.2},
{230.0, 29.2},
{271.8, 30.0}
};
void Display(double
*dat, double
*Answer, double
*SquarePoor, int rows, int cols)
{
double v, *p;
int i, j;
printf("回归方程式: Y = %.5lf", Answer[0]);
for (i =
1; i < cols; i ++)
printf(" + %.5lf*X%d", Answer, i);
printf("
");
printf("回归显著性检验: ");
printf("回归平方和:%12.4lf 回归方差:%12.4lf ", SquarePoor[0], SquarePoor[2]);
printf("剩余平方和:%12.4lf 剩余方差:%12.4lf ", SquarePoor[1], SquarePoor[3]);
printf("离差平方和:%12.4lf 标准误差:%12.4lf ", SquarePoor[0] + SquarePoor[1], sqrt(SquarePoor[3]));
printf("F 检 验:%12.4lf 相关系数:%12.4lf ", SquarePoor[2] /SquarePoor[3],
sqrt(SquarePoor[0] / (SquarePoor[0] + SquarePoor[1])));
printf("剩余分析: ");
printf(" 观察值 估计值 剩余值 剩余平方 ");
for (i =
0, p = dat; i < rows; i ++, p ++)
{
v = Answer[0];
for (j =
1; j < cols; j ++, p ++)
v +=
*p * Answer[j];
printf("%12.2lf%12.2lf%12.2lf%12.2lf ", *p, v, *p - v, (*p - v) * (*p - v));
}
system("pause");
}
int main()
{
double Answer[2], SquarePoor[4];
if (LinearRegression((double*)data1, 12, &Answer[0], &Answer[1], SquarePoor) ==
0)
Display((double*)data1, Answer, SquarePoor, 12, 2);
return
0;
}
运行结果:
上面的函数和例子程序不仅计算了回归方程式,还计算了显著性检验指标,例如F检验指标,我们可以在统计F分布表上查到F0.01(1,10)=10.04(注:括号里的1,10分别为回归平方和和剩余平方和所拥有的自由度),小于计算的F检验值25.94,可以认为该回归例子高度显著。
如果使用图形界面,可以根据原始数据和计算结果绘制各种图表,如散点图、趋势图、控制图等。很多非线性方程可以借助数学计算,转化为直线方程进行回归分析。
同一元线性回归相比,多元线性回归分析代码可就复杂多了,必须求解线性方程,因此本代码中包含一个可独立使用的线性方程求解函数:
void FreeData(double
**dat, double
*d, int count)
{
int i, j;
free(d);
for (i =
0; i < count; i ++)
free(dat);
free(dat);
}
// 解线性方程。data[count*(count+1)]矩阵数组;count:方程元数;
// Answer[count]:求解数组 。返回:0求解成功,-1无解或者无穷解
int LinearEquations(double
*data, int count, double
*Answer)
{
int j, m, n;
double tmp, **dat, *d = data;
dat = (double**)malloc(count *
sizeof(double*));
for (m =
0; m < count; m ++, d += (count +
1))
{
dat[m] = (double*)malloc((count +
1) *
sizeof(double));
memcpy(dat[m], d, (count +
1) *
sizeof(double));
}
d = (double*)malloc((count +
1) *
sizeof(double));
for (m =
0; m < count -
1; m ++)
{
// 如果主对角线元素为0,行交换
for (n = m +
1; n < count && dat[m][m] ==
0.0; n ++)
{
if ( dat[n][m] !=
0.0)
{
memcpy(d, dat[m], (count +
1) *
sizeof(double));
memcpy(dat[m], dat[n], (count +
1) *
sizeof(double));
memcpy(dat[n], d, (count +
1) *
sizeof(double));
}
}
// 行交换后,主对角线元素仍然为0,无解,返回-1
if (dat[m][m] ==
0.0)
{
FreeData(dat, d, count);
return
-1;
}
// 消元
for (n = m +
1; n < count; n ++)
{
tmp = dat[n][m] / dat[m][m];
for (j = m; j <= count; j ++)
dat[n][j] -= tmp * dat[m][j];
}
}
for (j =
0; j < count; j ++)
d[j] =
0.0;
// 求得count - 1的元
Answer[count -
1] = dat[count -
1][count] / dat[count -
1][count -
1];
// 逐行代入求各元
for (m = count -
2; m >=
0; m --)
{
for (j = count -
1; j > m; j --)
d[m] += Answer[j] * dat[m][j];
Answer[m] = (dat[m][count] - d[m]) / dat[m][m];
}
FreeData(dat, d, count);
return
0;
}
// 求多元回归方程:Y = B0 + B1X1 + B2X2 + ...BnXn
// data[rows*cols]二维数组;X1i,X2i,...Xni,Yi (i=0 to rows-1)
// rows:数据行数;cols数据列数;Answer[cols]:返回回归系数数组(B0,B1...Bn)
// SquarePoor[4]:返回方差分析指标: 回归平方和,剩余平方和,回归平方差,剩余平方差
// 返回值:0求解成功,-1错误
int MultipleRegression(double
*data, int rows, int cols, double
*Answer, double
*SquarePoor)
{
int m, n, i, count = cols -
1;
double
*dat, *p, a, b;
if (data ==
0
|| Answer ==
0
|| rows <
2
|| cols <
2)
return
-1;
dat = (double*)malloc(cols * (cols +
1) *
sizeof(double));
dat[0] = (double)rows;
for (n =
0; n < count; n ++) // n = 0 to cols - 2
{
a = b =
0.0;
for (p = data + n, m =
0; m < rows; m ++, p += cols)
{
a +=
*p;
b += (*p *
*p);
}
dat[n +
1] = a; // dat[0, n+1] = Sum(Xn)
dat[(n +
1) * (cols +
1)] = a; // dat[n+1, 0] = Sum(Xn)
dat[(n +
1) * (cols +
1) + n +
1] = b; // dat[n+1,n+1] = Sum(Xn * Xn)
for (i = n +
1; i < count; i ++) // i = n+1 to cols - 2
{
for (a =
0.0, p = data, m =
0; m < rows; m ++, p += cols)
a += (p[n] * p);
dat[(n +
1) * (cols +
1) + i +
1] = a; // dat[n+1, i+1] = Sum(Xn * Xi)
dat[(i +
1) * (cols +
1) + n +
1] = a; // dat[i+1, n+1] = Sum(Xn * Xi)
}
}
for (b =
0.0, m =
0, p = data + n; m < rows; m ++, p += cols)
b +=
*p;
dat[cols] = b; // dat[0, cols] = Sum(Y)
for (n =
0; n < count; n ++)
{
for (a =
0.0, p = data, m =
0; m < rows; m ++, p += cols)
a += (p[n] * p[count]);
dat[(n +
1) * (cols +
1) + cols] = a; // dat[n+1, cols] = Sum(Xn * Y)
}
n = LinearEquations(dat, cols, Answer); // 计算方程式
// 方差分析
if (n ==
0
&& SquarePoor)
{
b = b / rows; // b = Y的平均值
SquarePoor[0] = SquarePoor[1] =
0.0;
p = data;
for (m =
0; m < rows; m ++, p ++)
{
for (i =
1, a = Answer[0]; i < cols; i ++, p ++)
a += (*p * Answer); // a = Ym的估计值
SquarePoor[0] += ((a - b) * (a - b)); // U(回归平方和)
SquarePoor[1] += ((*p - a) * (*p - a)); // Q(剩余平方和)(*p = Ym)
}
SquarePoor[2] = SquarePoor[0] / count; // 回归方差
if (rows - cols > 0.0)
SquarePoor[3] = SquarePoor[1] / (rows - cols); // 剩余方差
else
SquarePoor[3] = 0.0;
}
free(dat);
return n;
}
为了理解代码,同样贴几个主要公式在下面,其中回归平方和和剩余平方和公式和一元回归相同:
1、回归方程式: , ![](http://www.advancedtechnic.com/ud/images/image1481.gif)
2、回归系数方程组:
![](http://www.advancedtechnic.com/ud/images/image1512.gif)
3、F检验:![](http://www.advancedtechnic.com/ud/images/image1792.gif)
4、相关系数: ,其中,Syy是离差平方和(回归平方和与剩余平方和之和)。该公式其实就是U/(U+Q)的平方根(没找到这个公式的图)。
5、回归方差:U / m,m为回归方程式中自变量的个数(没找到图)。
6、剩余方差:Q / (n - m - 1),n为观察数据的样本数,m同上(没找到图)。
7、标准误差:也叫标准误,就是剩余方差的平方根(没找到图)。
下面是一个多元回归的例子:
double data[15][5] = {
// X1 X2 X3 X4 Y
{ 316, 1536, 874, 981, 3894 },
{ 385, 1771, 777, 1386, 4628 },
{ 299, 1565, 678, 1672, 4569 },
{ 326, 1970, 785, 1864, 5340 },
{ 441, 1890, 785, 2143, 5449 },
{ 460, 2050, 709, 2176, 5599 },
{ 470, 1873, 673, 1769, 5010 },
{ 504, 1955, 793, 2207, 5694 },
{ 348, 2016, 968, 2251, 5792 },
{ 400, 2199, 944, 2390, 6126 },
{ 496, 1328, 749, 2287, 5025 },
{ 497, 1920, 952, 2388, 5924 },
{ 533, 1400, 1452, 2093, 5657 },
{ 506, 1612, 1587, 2083, 6019 },
{ 458, 1613, 1485, 2390, 6141 },
};
void Display(double
*dat, double
*Answer, double
*SquarePoor, int rows, int cols)
{
double v, *p;
int i, j;
printf("回归方程式: Y = %.5lf", Answer[0]);
for (i =
1; i < cols; i ++)
printf(" + %.5lf*X%d", Answer, i);
printf("
");
printf("回归显著性检验: ");
printf("回归平方和:%12.4lf 回归方差:%12.4lf ", SquarePoor[0], SquarePoor[2]);
printf("剩余平方和:%12.4lf 剩余方差:%12.4lf ", SquarePoor[1], SquarePoor[3]);
printf("离差平方和:%12.4lf 标准误差:%12.4lf ", SquarePoor[0] + SquarePoor[1], sqrt(SquarePoor[3]));
printf("F 检 验:%12.4lf 相关系数:%12.4lf ", SquarePoor[2] / SquarePoor[3],
sqrt(SquarePoor[0] / (SquarePoor[0] + SquarePoor[1])));
printf("剩余分析: ");
printf(" 观察值 估计值 剩余值 剩余平方 ");
for (i =
0, p = dat; i < rows; i ++, p ++)
{
v = Answer[0];
for (j =
1; j < cols; j ++, p ++)
v +=
*p * Answer[j];
printf("%12.2lf%12.2lf%12.2lf%12.2lf ", *p, v, *p - v, (*p - v) * (*p - v));
}
system("pause");
}
int main()
{
double Answer[5], SquarePoor[4];
if (MultipleRegression((double*)data, 15, 5, Answer, SquarePoor) ==
0)
Display((double*)data, Answer, SquarePoor, 15, 5);
return
0;
}
运行结果见下图,同上面,查F分布表,F检验远远大于F0.005(4,10)的7.34,可以说是极度回归显著。
![](http://p.blog.csdn.net/images/p_blog_csdn_net/maozefa/16-2.gif)
如果要根据回归方程进行预测和控制,还应该计算很多指标,如偏相关指标,t分布检验指标等,不过,本文只是介绍2个函数,并不是完整的回归分析程序,因此没必要计算那些指标。
其实,一元线性回归是多元线性回归的一个特例,完全可以使用同一个函数,如前面的例子:
if (MultipleRegression((double*)data1, 12, 2, Answer, SquarePoor) == 0)
Display((double*)data, Answer, SquarePoor, 12, 2);
其运行结果是一样的,可能以前我为了DOS下的运行速度,单独写了一个函数,因为毕竟多元回归分析很少用到,而一元回归是经常使用的。 |
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