决定系数（coefficient ofdetermination），有的书上翻译为判定系数，也称为拟合优度。
决定系数反应了y的波动有多少百分比能被x的波动所描述，即表征依变数Y的变异中有多少百分比,可由控制的自变数X来解释.
表达式：R方=SSR/sst=1-SSE/SST
其中：SST=SSR SSE，SST(total sum of squares)为总平方和，SSR(regression sum of squares)为回归平方和，SSE(error sum of squares) 为残差平方和。
回归平方和：SSR(Sum of Squares forregression) = ESS (explained sum of squares)
残差平方和：SSE（Sum of Squares for Error） = RSS(residual sum of squares)
总离差平方和：SST(Sum of Squares fortotal) = TSS(total sum of squares)
SSE SSR=SST RSS ESS=TSS
意义：拟合优度越大，自变量对因变量的解释程度越高，自变量引起的变动占总变动的百分比高。观察点在回归直线附近越密集。取值范围：0-1.
举例：
假设有10个点，如下图：
用R来实现如何求线性方程和R2：
# 线性回归的方程
mylr = function(x,y){
plot(x,y)
x_mean = mean(x)
y_mean = mean(y)
xy_mean = mean(x*y)
xx_mean = mean(x*x)
yy_mean = mean(y*y)
m = (x_mean*y_mean - xy_mean)/(x_mean^2 - xx_mean)
b = y_mean - m*x_mean
f = m*x b# 线性回归方程
lines(x,f)
sst = sum((y-y_mean)^2)
sse = sum((y-f)^2)
ssr = sum((f-y_mean)^2)
result = c(m,b,sst,sse,ssr)
names(result) = c('m','b','sst','sse','ssr')
return(result)
}
x = c(60,34,12,34,71,28,96,34,42,37)
y = c(301,169,47,178,365,126,491,157,202,184)
f = mylr(x,y)
f['m']
f['b']
f['sse'] f['ssr']
f['sst']
R2= f['ssr']/f['sst']
最后方程为：f(x)=5.3x-15.5
R2为99.8，说明x对y的解释程度非常高。
本期课程就到这里哦，感谢大家耐心观看！每日更新，敬请关注！
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