I am interested in using the squared log-difference between predictions and observations as a loss function, it appears to me to be a very natural way of dealing with errors where these are expected to scale proportionally with the independent variable and the domain covers multiple orders of magnitude.
This is required in my case since an X% 'miss' of the model prediction should be penalised equally, whether the prediction is small or large (unlike with SSE).
However, it seems that the trade-off of using log residuals is that (at least in the naive implementation e.g.):
$ \mathcal{L}(Y,\hat{Y}) = \ln \left(\frac{Y}{\hat{Y}}\right)^2 $
the loss function is not symmetrical, since relative underestimation is penalised more than relative overestimation (e.g. $ \mathcal{L}(10,9) > \mathcal{L}(10,11) $ ).
My question is, is there a common transformation that can used to skew the profile of the loss function to be symmetrical again, without losing the required behaviour of treating relative errors equally independent of the size of $Y$?
If there is no such transformation, is there any reason not to construct a sort of average log relative error such as:
$ \mathcal{L}_{symmetrical}(Y,\hat{Y}) = \left[ \ln \left(1+ \frac{|Y-\hat{Y}|}{Y} \right) + \ln \left(1-\frac{|Y-\hat{Y}|}{Y} \right) \right]^2 $
which appears to correctly treat overestimates equal to underestimates and not penalise larger independent variables?
