JasonWho_sysu 于 2018-11-18 23:30:03 发布
这篇内容详细证明了$R^2$等于相关系数平方的数学推导，通过展示$R^2 = [corr(y, hat{y})]^2$的等式成立，涉及协方差、方差和均值的概念。最后，通过一系列代数操作验证了等式的正确性。" 118464948,11209044,Java编程选择题解析及答案,"['Java基础', '编程知识', '字符串操作', '数据类型'] R 2 = [ c o r r ( y , y ^ ) ] 2 R^2=[corr(y,\hat{y})]^2 R2=[corr(y,y^)]2
Proof:
R 2 = S S E S S T = ∑ i = 1 n ( y i ^ − y ˉ ) 2 ∑ i = 1 n ( y i − y ˉ ) 2 R^2=\dfrac{SSE}{SST}=\dfrac{\sum\limits_{i=1}^n(\hat{y_i}-\bar{y})^2}{\sum\limits_{i=1}^n(y_i-\bar{y})^2} R2=SSTSSE=i=1∑n(yi−yˉ)2i=1∑n(yi^−yˉ)2
[ c o r r ( y , y ^ ) ] 2 = [ C o v ( y , y ^ ) V a r ( y ) V a r ( y ^ ) ] 2 = ( C o v ( y , y ^ ) ) 2 V a r ( y ) V a r ( y ^ ) [corr(y,\hat{y})]^2=[\dfrac{Cov(y,\hat{y})}{\sqrt{Var(y)Var(\hat{y})}}]^2=\dfrac{(Cov(y,\hat{y}))^2}{Var(y)Var(\hat{y})} [corr(y,y^)]2=[Var(y)Var(y^)
Cov(y,y^)]2=Var(y)Var(y^)(Cov(y,y^))2
C o v ( y , y ^ ) = 1 n ∑ i = 1 n ( y i − y ˉ ) ( y i ^ − y ^ ˉ ) Cov(y,\hat{y})=\dfrac{1}{n}\sum\limits_{i=1}^n(y_i-\bar{y})(\hat{y_i}-\bar{\hat{y}}) Cov(y,y^)=n1i=1∑n(yi−yˉ)(yi^−y^ˉ)
