Definition
The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, yi = a + b1x1i + b2x2i + ... + εi, where yi is the i th observation of the response variable, xji is the i th observation of the j th explanatory variable, a and bi are coefficients, i indexes the observations from 1 to n, and εi is the i th value of the error term. In general, the greater the ESS, the better the estimated model performs.
If and are the estimated coefficients, then
is the i th predicted value of the response variable. The ESS is the sum of the squares of the differences of the predicted values and the mean value of the response variable:
In general: total sum of squares = explained sum of squares + residual sum of squares.
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