Difference between coefficient of determination and explained variance I'm trying to evaluate how good my model predictions w.r.t. the true values (paired). I came across two method coefficient of determination (https://en.wikipedia.org/wiki/Coefficient_of_determination) and explained variance (http://scikit-learn.org/stable/modules/model_evaluation.html#explained-variance-score) to do this but I can't really see what the difference between the two are and the pros and cons. The formulas seem about the same...
Can someone explain the difference and pros and cons?
 A: Let $X$ be a random vector, and $Y$ a random variable that is modeled by a normal distribution with center $\mu+\Psi^\textrm{T}X$, in this case, the proportion of randomness $\rho_C^2$ equals the squared coefficient of determination, $R^2$.
Note the strong model assumptions: the center of the $Y$ distribution must be a linear function of $X$, and for any given $x$, the $Y$ distribution must be normal. In other situations, it is generally not justified to interpret $R^2$ as proportion of explained variance. What Does Explained Variance Explain?
A: They are the same. Using a bit of algebra, we can see that they are equivalent.
$R^2 = 1 - \dfrac{Sum\ of\ squared\ errors}{Total\ sum\ of\ squares}$
$Sum\ of\ Squared\ Errors = \sum(Y - \hat Y)^2$
$Total\ Sum\ of\ Squares = \sum(Y- \bar Y)^2$
$R^2 = 1 -\frac{\sum(Y - \hat Y)^2}{\sum(Y- \bar Y)^2}$
According to your source,
$ Explained\ variance\ score\ = 1 - \dfrac{Var(Y- \hat Y)}{Var(Y)}$
$Explained\ variance\ score\ = 1 - \dfrac{\cfrac{\sum[(Y - \hat Y ) - E(Y - \hat Y)]^2}{n -1}}{\dfrac{\sum(Y- \bar Y)^2}{n-1}}$
$\sum[(Y - \hat Y ) - E(Y - \hat Y)]^2 = \sum(Y - \hat Y)^2$, i.e. SSE
So,
$Explained\ variance\ score\ = 1 - \dfrac{\dfrac{\sum(Y - \hat Y )^2}{n -1}}{\dfrac{\sum(Y- \bar Y)^2}{n-1}}$
$Explained\ variance\ score\ =  1 -  \dfrac{(n-1)\sum(Y - \hat Y)^2}{(n-1)\sum(Y - \bar Y)^2}$
$Explained\ variance\ score\ = 1 - \dfrac{\sum(Y - \hat Y)^2}{\sum(Y - \bar Y)^2} = R^2$
Why are there two different names for the same thing? Because, statistics. Maybe there is some deeper reason, but often different terms for the same thing arise because of disciplinary differences.
