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{{Merge to|Coefficient of determination|date=July 2013}}
{{Unreferenced|date=February 2007}}
In [[statistics]], the '''fraction of variance unexplained (FVU)''' in the context of a [[Regression analysis|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 variable]]s ''X''.

For a more general concept, see [[explained variation]].

==Formal definition==
Suppose we are given a regression function ''&fnof;'' yielding for each ''y<sub>i</sub>'', 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''N'', an estimate <math>\widehat{y}_i = f(x_i)</math> where <math>x_i</math> is the vector of the ''i''<sup>th</sup> observations on all the explanatory variables. We define the fraction of variance unexplained (FVU) as:

:<math>\begin{align}
\text{FVU} & = {VAR_{\rm err} \over VAR_{\rm tot}} = {SS_{\rm err}/n \over SS_{\rm tot}/n} = {SS_{\rm err} \over SS_{\rm tot}} = 1-{SS_{\rm reg} \over SS_{\rm tot}} \\[6pt]
 & = 1 - R^2,
\end{align}</math>

where ''R''<sup>2</sup> is the [[coefficient of determination]] and ''VAR''<sub>err</sub> and ''VAR''<sub>tot</sub> are the variance of the estimated residuals and the sample variance of the dependent variable. ''SS''<sub>''err''</sub> (the sum of squared predictions errors, equivalently the [[residual sum of squares]]), ''SS''<sub>''tot''</sub> (the [[total sum of squares]]), and ''SS''<sub>''reg''</sub> (the sum of squares of the regression, equivalently the [[explained sum of squares]]) are given by

:<math>\begin{align} 
SS_{\rm err} & = \sum_{i=1}^N\;(y_i - \widehat{y_i})^2\\
SS_{\rm tot} & = \sum_{i=1}^N\;(y_i-\bar{y})^2 \\
SS_{\rm 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{align}</math>

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

:<math> \text{FVU} = \frac{\text{MSE}(f)}{\text{var}[Y]},</math>

where MSE(''f'') is the [[mean squared error]] of the regression function&nbsp;''&fnof;''.

==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., <math>f(x_i)=\bar{y}</math>. It follows that the MSE of this function equals the variance of ''Y''; that is, ''SS''<sub>err</sub> = ''SS''<sub>tot</sub>, and ''SS''<sub>reg</sub> = 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 [[covariance|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 <math>\hat{y}_i = y_i</math> for all ''i'', the MSE is 0, ''SS''<sub>err</sub> = 0, ''SS''<sub>reg</sub> = ''SS''<sub>tot</sub>, and the FVU is 0.

==See also==
* [[Coefficient of determination]]
* [[Correlation]]
* [[Regression analysis]]
* [[Linear regression]]

{{DEFAULTSORT:Fraction Of Variance Unexplained}}
[[Category:Estimation theory]]
[[Category:Parametric statistics]]
[[Category:Regression analysis]]
[[Category:Statistical ratios]]
[[Category:Least squares]]