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) which cannot be explained, i.e., which is not correctly predicted, by the explanatory variables .
Formal definition Suppose we are given a regression function yielding for each an estimate where is the vector of the ith observations on all the explanatory variables. We define the fraction of variance unexplained (FVU) as:
where is the coefficient of determination and and are the variance of the residuals and the sample variance of the dependent variable. (the sum of squared predictions errors, equivalently the residual sum of squares), (the total sum of squares), and (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 is the mean squared error of the regression function .
Explanation It is useful to consider the second definition to understand FVU. When trying to predict , the most naive regression function that we can think of is the constant function predicting the mean of , i.e., . It follows that the of this function equals the variance of ; that is, , and . In this case, no variation in 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 tell us nothing about in the sense that the predicted values of do not covary with . But as prediction gets better and the can be reduced, the FVU goes down. In the case of perfect prediction where for all , the is 0, , , and the FVU is 0.
See also Coefficient of determination Correlation Explained sum of squares Lack-of-fit sum of squares Linear regression Regression analysis Mean absolute scaled error
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