决定系数（coefficient ofdetermination），有的教材上翻译为判定系数，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：R2=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。

注：（不同书命名不同）

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：

我们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']复制

R2为99.8，说明x对y的解释程度非常高。

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