Fraction of variance unexplained

In statistics, the fraction of variance unexplained (FVU) in the context of a regression task is the fraction of variance of the regressand (dependent variable) "Y" which cannot be explained, i.e., which is not correctly predicted, by the explanatory variables "X".
Formal definition.
Suppose we are given a regression function formula_1 yielding for each formula_2 an estimate formula_3 where formula_4 is the vector of the "i"th observations on all the explanatory variables. We define the fraction of variance unexplained (FVU) as:
where "R"2 is the coefficient of determination and "VAR"err and "VAR"tot are the variance of the residuals and the sample variance of the dependent variable. "SS""err" (the sum of squared predictions errors, equivalently the residual sum of squares), "SS""tot" (the total sum of squares), and "SS""reg" (the sum of squares of the regression, equivalently the explained sum of squares) are given by
Alternatively, the fraction of variance unexplained can be defined as follows:
where MSE("f") is the mean squared error of the regression function "ƒ".
Explanation.
It is useful to consider the second definition to understand FVU. When trying to predict "Y", the most naïve regression function that we can think of is the constant function predicting the mean of "Y", i.e., formula_8. It follows that the MSE of this function equals the variance of "Y"; that is, "SS"err = "SS"tot, and "SS"reg = 0. In this case, no variation in "Y" can be accounted for, and the FVU then has its maximum value of 1.
More generally, the FVU will be 1 if the explanatory variables "X" tell us nothing about "Y" in the sense that the predicted values of "Y" do not covary with "Y". But as prediction gets better and the MSE can be reduced, the FVU goes down. In the case of perfect prediction where formula_9 for all "i", the MSE is 0, "SS"err = 0, "SS"reg = "SS"tot, and the FVU is 0.