线性回归之决定系数（coefficient of determination）

1. Sum Of Squares Due To Error 

对于第i个观察点, 真实数据的Yi与估算出来的Yi-head的之间的差称为第i个residual, SSE 就是所有观察点的residual的和
2. Total Sum Of Squares

3. Sum Of Squares Due To Regression

通过以上我们能得到以下关于他们三者的关系

决定系数: 判断 回归方程 的拟合程度

(coefficient of determination)决定系数也就是说: 通过回归方程得出的 dependent variable 有 number% 能被 independent variable 所解释. 判断拟合的程度

(Correlation coefficient) 相关系数 : 测试dependent variable 和 independent variable 他们之间的线性关系有多强. 也就是说, independent variable 产生变化时 dependent variable 的变化有多大.

可以反映是正相关还是负相关

参考链接：http://blog.csdn.net/ytdxyhz/article/details/51730995

注意此决定系数不能用来衡量非线性回归的拟合优度

Why Is It Impossible to Calculate a Valid R-squared for Nonlinear Regression?

R-squared is based on the underlying assumption that you are fitting a linear model. If you aren’t fitting a linear model, you shouldn’t use it. The reason why is actually very easy to understand.

For linear models, the sums of the squared errors always add up in a specific manner: SS Regression + SS Error = SS Total.

This seems quite logical. The variance that the regression model accounts for plus the error variance adds up to equal the total variance. Further, R-squared equals SS Regression / SS Total, which mathematically must produce a value between 0 and 100%.

In nonlinear regression, SS Regression + SS Error do not equal SS Total! This completely invalidates R-squared for nonlinear models, and it no longer has to be between 0 and 100%.

参考链接：http://blog.minitab.com/blog/adventures-in-statistics-2/why-is-there-no-r-squared-for-nonlinear-regression

更新：

For cases other than fitting by ordinary least squares, the R² statistic can be calculated as above and may still be a useful measure. If fitting is by weighted least squares or generalized least squares, alternative versions of R² can be calculated appropriate to those statistical frameworks, while the "raw" R² may still be useful if it is more easily interpreted. Values for R² can be calculated for any type of predictive model, which need not have a statistical basis.

参考链接：https://en.wikipedia.org/wiki/Coefficient_of_determination

更新：

https://stats.stackexchange.com/questions/7357/manually-calculated-r2-doesnt-match-up-with-randomforest-r2-for-testing

这篇回答中给了两个信息：

（1）线性回归的R方等于实际值与预测值的相关系数的平方

（2）randomForest is reporting variation explained as opposed to variance explained.