回归分析中的相关度（Corr）和R^2

article/2025/10/8 7:33:21

一、.皮尔逊相关系数（Pearson Correlation Coefficient）

1.可以用来衡量两个数值之间的线性相关程度

2.对应的取值范围为[-1,1],即存在正相关，负相关和不相关

3.计算公式：

$\rho =Cor(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$

$r_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})) }{\sqrt{\sum (x-\bar{x})^2\sum (y-\bar{y})^2}}$

二、R^2

1.定义：决定系数，反应因变量的全部变异能通过回归关系被自变量解释的比例

2.描述：如R^2=0.8，则表示回归关系可以解释为因变量80%的变异。即，如果可以控制自变量保持不变，则因变量的变异程度会减少80%

3.简单线性回归的计算方式：R^2=r*r

4.多元线性回归的计算方式： $R^{2}=\frac{SSR}{SST}=\frac{\sum (\hat{y_{i}}-\bar{y})^2}{\sum (y_{i}-\bar{y})^2}$

$SST=\sum_{i}^{}(y_{i}-\bar{y})^2$

$SSR=\sum_{i}^{}(\hat{y_{i}}-\bar{y})^2$

$SSE=\sum_{i}^{}(y_{i}-\hat{y_{i}})^2$

5.R^2也具有一定的局限性，R^2会随着自变量的增大增大，R^2和样本量具有一定的关系。因此，为了改进R^2的局限性，我们要对R^2进行修正。修正的方法为：

$R^{2}_{adjust}=1-\frac{(1-R^2)(N-1)}{N-P-1}$

其中P为预测值的数量

N为总实例的数量。

3.应用实例代码即结果

import numpy as np
from astropy.units import Ybarn
import mathdef computerCorrelation(X, Y):XBar = np.mean(X)#X的均值YBar = np.mean(Y)#Y的均值SSR = 0#初始化SSRvarX = 0varY = 0for i in range (0, len(X)):diffXXBar = X[i] - XBardiffYYBar = Y[i] - YBarSSR += (diffXXBar  * diffYYBar )varX += diffXXBar**2varY += diffYYBar**2SST = math.sqrt(varX *varY )return SSR/SST
def polyfit (x, y,degree):#degree代表次数results = {}#把结果存储在results中coffs = np.polyfit(x, y, degree)#返回相关系数估计results['polynomfot'] = coffs .tolist()p = np.poly1d(coffs)#返回预测值yhatyhat = p(x)ybar = np.sum(y)/len(y)ssreg = np.sum((yhat-ybar)**2)print('ssreg:', str(ssreg))sstot = np.sum((y-ybar)**2)print('sstot:', str(sstot))results['determinotion'] = ssreg/sstotprint('results:', results)return resultstextX = [1, 3, 5, 7, 9]
textY = [10, 13, 15, 20, 35]
print('r:', computerCorrelation(textX, textY))
print('r^2:', str(computerCorrelation(textX, textY)**2))
print(polyfit(textX, textY, 1), 'determinotion')

运行结果：

r: 0.9136680531834395
r^2: 0.8347893114080164
ssreg: 324.9000000000002
sstot: 389.20000000000005
results: {'polynomfot': [2.8500000000000005, 4.350000000000008], 'determinotion': 0.8347893114080168}
{'polynomfot': [2.8500000000000005, 4.350000000000008], 'determinotion': 0.8347893114080168} determinotion