一元線性迴歸模型求通解

原創

xhyan523

2019-07-03 00:17

一元非線性迴歸模型求解：

一元線性迴歸方程求解

c語言求解過程

// 求線性迴歸方程：Y = a + bx
// dada[rows*2]數組：X, Y；rows：數據行數；a, b：返回迴歸係數
// SquarePoor[4]：返回方差分析指標: 迴歸平方和，剩餘平方和，迴歸平方差，剩餘平方差
// 返回值：0求解成功，-1錯誤
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(迴歸平方和)
        SquarePoor[1] += ((*p - Lxy) * (*p - Lxy)); // Q(剩餘平方和)
    }
    SquarePoor[2] = SquarePoor[0];                  // 迴歸方差
    SquarePoor[3] = SquarePoor[1] / (rows - 2);     // 剩餘方差
    return 0;
}

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("迴歸方程式:    Y = %.5lf", Answer[0]);
    for (i = 1; i < cols; i ++)
        printf(" + %.5lf*X%d", Answer[i], i);
    printf(" ");
    printf("迴歸顯著性檢驗: ");
    printf("迴歸平方和：%12.4lf  迴歸方差：%12.4lf ", SquarePoor[0], SquarePoor[2]);
    printf("剩餘平方和：%12.4lf  剩餘方差：%12.4lf ", SquarePoor[1], SquarePoor[3]);
    printf("離差平方和：%12.4lf  標準誤差：%12.4lf ", SquarePoor[0] + SquarePoor[1], sqrt(SquarePoor[3]));
    printf("F   檢  驗：%12.4lf  相關係數：%12.4lf ", SquarePoor[2] /SquarePoor[3],
           sqrt(SquarePoor[0] / (SquarePoor[0] + SquarePoor[1])));
    printf("剩餘分析: ");
    printf("      觀察值      估計值      剩餘值    剩餘平方 ");
    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;
}

結果