  Regression evaluation has been performed for decades. Some metrics have been
identified to be robust against shifting and scaling of the data but
considering the different distributions of data is much more difficult to
address (imbalance problem) even though it largely impacts the comparability
between evaluations on different datasets. In classification, it has been
stated repeatedly that performance metrics like the F-Measure and Accuracy are
highly dependent on the class distribution and that comparisons between
different datasets with different distributions are impossible. We show that
the same problem exists in regression. The distribution of odometry parameters
in robotic applications can for example largely vary between different
recording sessions. Here, we need regression algorithms that either perform
equally well for all function values, or that focus on certain boundary regions
like high speed. This has to be reflected in the evaluation metric. We propose
the modification of established regression metrics by weighting with the
inverse distribution of function values $Y$ or the samples $X$ using an
automatically tuned Gaussian kernel density estimator. We show on synthetic and
robotic data in reproducible experiments that classical metrics behave wrongly,
whereas our new metrics are less sensitive to changing distributions,
especially when correcting by the marginal distribution in $X$. Our new
evaluation concept enables the comparison of results between different datasets
with different distributions. Furthermore, it can reveal overfitting of a
regression algorithm to overrepresented target values. As an outcome,
non-overfitting regression algorithms will be more likely chosen due to our
corrected metrics.
