What are the gaps in the predicted error rates: MAPE and WMAPE?

Question

I know that MAPE and WMAPE as indicators of forecast errors have certain advantages. But what are the gaps? Someone says:

  For MAPE:
 "Combinations with very small or zero volumes can cause large skew in results"

 And for WMAPE:
 "Combinations with large weights can skew the results in their favor" 

I can’t understand if anyone can explain the two statements about the weakness of the two indicators? Thank you

Answer 1

For MAPE, the average absolute percentage error [1], suppose we denote the actual value with A and the predicted value with P. You have a series of data at a point in time from 1 to n, then

MAPE = 100/n * ( Sum of |(A(t) - P(t))/A(t)| ), for t in 1..n where A(t) is the actual value at time t, P(t) is the predicted value at time t. 

Since A (t) is in the denominator, whenever you have a very small or almost zero A (t), this division is similar to that divided by zero, which creates very large changes in the Absolute Percentage Error. Combinations of such large changes to a large extent lead to large distortions.

For WMAPE, the weighted average absolute percentage error,

  Sum of |(A(t) - P(t))/A(t)| * W(t) WMPAE = -------------------------------------, for t in 1..n Sum of W(t) where W(t) is the weight you associate with the prediction at time t. 

Since this is a measured measure, it does not have the same problems as MAPE, for example, excessive skew due to very small or zero volumes.

However, the weight coefficient will indicate the subjective value that we want to put on each prediction [2].

For example, given the release of the date, we can assign the weight so that the higher the weight, the higher the value that we put at a later stage in the data. In this case, we could notice that even when the MAE is at a reasonable threshold, system performance may be inadequate when analyzing this particular function.

Thus, the advantage of recent data distorts the results.

 [1] http://en.wikipedia.org/wiki/Mean_absolute_percentage_error [2] http://ir.ii.uam.es/rue2012/papers/rue2012-cleger-tamayo.pdf 

Answer 2

There is another error metric:

 WAPE = 100/n * Sum(|(A(t) - P(t)|)/sum(A(t)), for t in 1..n where A(t) is the actual value at time t, P(t) is the predicted value at time t. 

It is not sensitive to large distortions.