Fraction of variance unexplained

You don't need to be Editor-In-Chief to add or edit content to WikiDoc. You can begin to add to or edit text on this WikiDoc page by clicking on the edit button at the top of this page. Next enter or edit the information that you would like to appear here. Once you are done editing, scroll down and click the Save page button at the bottom of the page.

In statistics, the fraction of variance unexplained (or FVU) in the context of a regression task is the amount of variance of the regressand Y which cannot be explained, i.e., which is not correctly predicted, by the explanatory variable X.

Formal definition

Given a regression function f(·) yielding for each y_i, , an estimate , we have:

where R² is the coefficient of determination and

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

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

Explanation

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., . 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, the variations in Y cannot be accounted for, and the FVU then has its maximum value of 1.

The FVU will also be 1 if the explanatory variable X tells us nothing about Y in the sense that the predicted values of Y do not covary with Y. But as prediction gets better and the MSE can be reduced, the FVU does down. In the case of perfect prediction where , the MSE is 0, SS_E = 0, SS_T = SS_E, and the FVU is 0.

See also

• Coefficient of determination
• Correlation
• Regression analysis
• Linear regression