一元线性回归模型系数、方差估计、检验回归效果显著性,b的置信区间，Y约为X的指数函数时，求Y关于x的回归方程

$y=\hat a + \hat b x$
$S_{xy}=(x_1y_1+x_2y_2+...+x_ny_n)-\frac{(x_1+x_2+...+x_n)(y_1+y_2+...+y_n)}{n}$
$S_{xx}=(x_1^2+x_2^2+...+x_n^2)-\frac{(x_1+x_2+...+x_n)^2}{n}$
$\hat b=\frac{S_{xy}}{S_{xx}}$
$\hat a =\frac{y_1+y_2+...+y_n}{n}-\frac{(x_1+x_2+...+x_n)}{n} \hat b$
$\hat{\sigma^2}=\frac{S_{yy}-S_{xy}\hat{b}}{n-2}$

python代码实现：

import numpy as np 
import math
def Sxy(x,y):
    sum = 0
    sumx = 0
    sumy = 0
    for i in range(len(x)):
        sum+=x[i]*y[i]
        sumx+=x[i]
        sumy+=y[i]
    return sum-sumx*sumy/len(x)

def a(x,y,b):
    sumy = 0
    sumx = 0
    for i in range(len(x)):
        sumy += y[i]
        sumx += x[i]
    sumy/=len(x)
    sumx/=len(x)
    return sumy - sumx * b
    

	
x = [100,110,120,130,140,150,160,170,180,190]
y = [45,51,54,61,66,70,74,78,85,89]
sxy = Sxy(x,y)
print('sxy:',sxy)
sxx = Sxy(x,x)
print('sxx:',sxx)
b = sxy/sxx
print('b:',b)
a = a(x,y,b)
print('a:',a)
print('所有一元回归函数为Y=',a,'x+',b)

检验回归效果显著性
1）设$H_0:b=0,H_1:b\not =0$
2）判断$\frac{|\hat b|}{\hat \sigma}\sqrt{S_{xx}}与t_{\frac{\alpha}{2}(n-2)}则回归效果显著$

import numpy as np 
import math
def Sxy(x,y):
    sum = 0
    sumx = 0
    sumy = 0
    for i in range(len(x)):
        sum+=x[i]*y[i]
        sumx+=x[i]
        sumy+=y[i]
    return sum-sumx*sumy/len(x)

def a(x,y,b):
    sumy = 0
    sumx = 0
    for i in range(len(x)):
        sumy += y[i]
        sumx += x[i]
    sumy/=len(x)
    sumx/=len(x)
    return sumy - sumx * b
    

	
x = [100,110,120,130,140,150,160,170,180,190]
y = [45,51,54,61,66,70,74,78,85,89]
sxy = Sxy(x,y)
print('sxy:',sxy)
sxx = Sxy(x,x)
print('sxx:',sxx)
b = sxy/sxx
print('b:',b)
a = a(x,y,b)
print('a:',a)
print('所有一元回归函数为Y=',a,'x+',b)
syy = Sxy(y,y)
print('syy:',syy)
sigma2 = (syy-b*sxy)/(len(x)-2) 
print('sigma2:',sigma2)
print(abs(b)/math.sqrt(sigma2)*math.sqrt(sxx))

求系数b的置信区间
置信区间为$(\hat b-t_{\frac{\alpha}{2}(n-2)}\frac{\hat a}{S_{xx}},\hat b+t_{\frac{\alpha}{2}(n-2)}\frac{\hat a}{S_{xx}})$

来源：CSDN

作者：Philtell

链接：https://blog.csdn.net/CCCrunner/article/details/103820808