Relative error formula

I'm reading a paper about a regression algorithm and I came across a definition of relative error (p. 15). The authors define relative error as

$$RE = \frac{MAD}{(1/T) \Sigma_i^T |y_i - median(y)|}$$ where MAD is the mean absolute distance

$$MAD= \frac{\Sigma_i^T |y_i - \hat y_i|}{T}$$

And $T$ is the number of test instances (conventionaly designated as $n$).

I get that this formula is used to yield a measure of error that is relative to the size of the value being estimated, which is useful for comparing error across datasets with target values of different magnitudes. What I don't know is why we wouldn't just do

$$\frac{\Sigma_i^T (|y_i - \hat y_i|/| y_i|)}{T}$$ and if not, why we use the median instead of the mean.

• I don't think this formula measures the error relative to the size of the value being estimated. It looks more like an $R^2$-like quantity, which measures the proportion of the variability explained by a model compared to the total variability of the data. Jun 29 '12 at 18:30
• @Aniko I don't know how that would make sense given the context. The authors test their regression algorithm on many different datasets, and then calculate the RE, as per above, for each dataset. Then they average RE across all problems. Jun 29 '12 at 18:59

The median is used because it is a more robust estimate of the center of the distribution. The mean can be heavily influenced by extreme values. The reason that $|y_i|$ is not used is because it does not represent a deviation for a central point.