Fraction of variance unexplained

Formal definition

Suppose we are given a regression function ${\displaystyle f}$ yielding for each ${\displaystyle y_i}$ an estimate ${\displaystyle {\hat{y}}_i=f(x_i)}$ where ${\displaystyle x_i}$ is the vector of the i^th observations on all the explanatory variables. We define the fraction of variance unexplained (FVU) as:

${\displaystyle \begin{array}{cc}\text{FVU}& =\frac{{\text{VAR}}_{\text{err}}}{{\text{VAR}}_{\text{tot}}}=\frac{{\text{SS}}_{\text{err}}/N}{{\text{SS}}_{\text{tot}}/N}=\frac{{\text{SS}}_{\text{err}}}{{\text{SS}}_{\text{tot}}}\left(=1-\frac{{\text{SS}}_{\text{reg}}}{{\text{SS}}_{\text{tot}}},\text{ only true in some cases such as linear regression}\right)\\ & =1-R^2\end{array}}$

where R² 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

${\displaystyle \begin{array}{cc}{\text{SS}}_{\text{err}}& =\sum _{i=1}^N(y_i-{\hat{y}}_i{)}^2\\ {\text{SS}}_{\text{tot}}& =\sum _{i=1}^N(y_i-\overline{y}{)}^2\\ {\text{SS}}_{\text{reg}}& =\sum _{i=1}^N({\hat{y}}_i-\overline{y}{)}^2\text{ and}\\ \overline{y}& =\frac{1}{N}\sum _{i=1}^N{y}_i.\end{array}}$

Alternatively, the fraction of variance unexplained can be defined as follows:

${\displaystyle \text{FVU}=\frac{\mathrm{MSE}(f)}{\mathrm{var}[Y]}}$

where MSE(f) is the mean squared error of the regression function ƒ.