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Fraction of variance unexplained

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For broader coverage of this topic, seeExplained variation.

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Instatistics, thefraction of variance unexplained (FVU) in the context of aregression task is the fraction of variance of theregressand (dependent variable)Y which cannot be explained, i.e., which is not correctly predicted, by theexplanatory variablesX.

Formal definition

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Suppose we are given a regression function $f {\displaystyle f}$ yielding for each $y_{i}$ an estimate ${\widehat {y}}_{i}=f(x_{i})$ where $x_{i}$ is the vector of thei^th observations on all the explanatory variables.^[1]^: 181 We define the fraction of variance unexplained (FVU) as:

{\begin{aligned}{\text{FVU}}&={{\text{VAR}}_{\text{err}} \over {\text{VAR}}_{\text{tot}}}={{\text{SS}}_{\text{err}}/N \over {\text{SS}}_{\text{tot}}/N}={{\text{SS}}_{\text{err}} \over {\text{SS}}_{\text{tot}}}\left(=1-{{\text{SS}}_{\text{reg}} \over {\text{SS}}_{\text{tot}}},{\text{ only true in some cases such as linear regression}}\right)\\[6pt]&=1-R^{2}\end{aligned}}

whereR² is thecoefficient of determination andVAR_err andVAR_tot are the variance of the residuals and the sample variance of the dependent variable.SS_err (the sum of squared predictions errors, equivalently theresidual sum of squares),SS_tot (thetotal sum of squares), andSS_reg (the sum of squares of the regression, equivalently theexplained sum of squares) are given by

{\begin{aligned}{\text{SS}}_{\text{err}}&=\sum _{i=1}^{N}\;(y_{i}-{\widehat {y}}_{i})^{2}\\{\text{SS}}_{\text{tot}}&=\sum _{i=1}^{N}\;(y_{i}-{\bar {y}})^{2}\\{\text{SS}}_{\text{reg}}&=\sum _{i=1}^{N}\;({\widehat {y}}_{i}-{\bar {y}})^{2}{\text{ and}}\\{\bar {y}}&={\frac {1}{N}}\sum _{i=1}^{N}\;y_{i}.\end{aligned}}

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

{\text{FVU}}={\frac {\operatorname {MSE} (f)}{\operatorname {var} [Y]}}

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

Explanation

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It is useful to consider the second definition to understand FVU. When trying to predictY, the most naive regression function that we can think of is the constant function predicting the mean ofY, i.e., $f(x_{i})={\bar {y}}$ . It follows that the MSE of this function equals the variance ofY; that is,SS_err =SS_tot, andSS_reg = 0. In this case, no variation inY can be accounted for, and the FVU then has its maximum value of 1.

More generally, the FVU will be 1 if the explanatory variablesX tell us nothing aboutY in the sense that the predicted values ofY do notcovary withY. But as prediction gets better and the MSE can be reduced, the FVU goes down. In the case of perfect prediction where ${\hat {y}}_{i}=y_{i}$ for alli, the MSE is 0,SS_err = 0,SS_reg =SS_tot, and the FVU is 0.

References

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^Achen, C. H. (1990). "'What Does "Explained Variance" Explain?: Reply".Political Analysis.2 (1):173–184.doi:10.1093/pan/2.1.173.

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