Symmetric mean squared error
For each entry a Symmetric Mean Squared Error (SMSE) is computed to help judge the goodness of fit.
To compute the SMSE we start by computing the Symmetric Squared Error (SSE) for each data set. The SSE of a data set is computed in this way
where i refers to the data set and j to a given point in data set i. dij stands for the data, pij for the model prediction and wij for the associated weight coefficient. Finally, di and pi in the denominator represent, respectively, the average of all data points (dij) and predictions (pij) in set i:
In the case the sum of the weights wi is equal to zero we simply take the error to be zero. The implication is that this data set does not contribute to the SMSE.
SMSE is then the mean of all SSEi with wi greater than 0.
where n' is the number of data sets with wi greater than 0.
The zerovariate case
In the case of a zero-variate data set the relative error simplifies to
and the connection with the formal definition of relative is more easily seen.