Fraction of variance unexplained: Difference between revisions

In [[statistics]], the '''fraction of variance unexplained (FVU)''' in the context of a [[Regression analysis|regression task]] is the amountfraction of variance of the [[regressand]] ''Y'' which cannot be explained, i.e., which is not correctly predicted, by the [[explanatory variable]]s ''X''.

It is useful to consider the second definition to get the idea behind 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_E'' = ''SS_T'', and ''SS_R'' = 0. In this case, theno variationsvariation in ''Y'' cannotcan be accounted for, and the FVU then has its maximum value of 1.

TheMore generally, the FVU will also be 1 if the explanatory variablevariables ''X'' tellstell 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>, the MSE is 0, ''SS_E'' = 0, ''SS_TR = SS_RT'', and the FVU is 0.