<\/span>Transforming the target for fun and\u00a0profit<\/span><\/h2>\nSo we have seen that not everything is as it seems or as we might have expected by doing some rather academical analysis. But can we somehow use this knowledge in our use case of predicting the market value of a used car? Yeah, this is the point where we close the circle to the beginning of the story. We have already argued that the RMSE might not be the right error measure to minimize. Log-transforming the target and still minimizing the RMSE gave us an approximation to the result we would have gotten if we had minimized the MAE, which quite likely is a more appropriate error measure in this use case than the RMSE. This is a neat trick if our regression method only allows minimizing the MSE or if it is too slow or unstable when minimizing the MAE directly. A word of caution again, this only works if the residual distribution approximates a log-normal distribution. So far we have only seen that the target distribution, not the residual distribution, is quite log-normal but since we are dealing with positive numbers, and also taking into account the fact that a car seller might be more inclined to start with a higher price, this justifies the assumption that the residual distribution (in case of a multivariate model) will also approximate a log-normal distribution.<\/p>\n
Well, the MAE surely is quite nice, but how about minimizing some relative measure like the MAPE? Assuming that our regression method does not support minimizing it directly, does the log-transformation do any good here? Intuitively, since we know that multiplicative, and thus relative, relationships become additive in log-space, we might expect it to be advantageous and indeed it does help. But before we do some experiments, let’s first look at some other relative error measure, namely the Root Mean Square Percentage Error (RMSPE), for example.<\/p>\n