Explain MAAPE (Mean Arctangent Absolute Percentage Error) in simple terms (intermittent demand forecasting)

Question

n order to measure the accuracy of highly intermitted demand time series, I recently discovered a new accuracy measure, that overcomes the problem of zero values and values close to zero, when comparing a test forecast to the actual values. This is pretty useful, when it comes to forecast intermittent demand.

I am able to understand the simple calculation of measures like RMSE and MAPE, however, when it comes to MAAPE I do struggle, to understand the math behind it.

I found this paper, which is explaining it in very theoretical terms: https://www.sciencedirect.com/science/article/pii/S0169207016000121

In the abstract it sums up the meaning of MAAPE like this:

In essence, MAAPE is a slope as an angle, while MAPE is a slope as a ratio, considering a triangle with adjacent and opposite sides that are equal to an actual value and the difference between the actual and forecast values, respectively.

However, I could not find any simple example of MAAPE's calculation. The easiest way, to explain it to a customer would be some easy to understand visualizations or even a calculation done in excel.

Answer 1

I don't think there is much to understand here. In cases where your error can be sometimes very large, you don't want your overall average to be skewed by these freak events

if $\alpha=\arctan\left(x\right)$ then:

• $\alpha\approx x$ when $x$ is small
• $\alpha\to\pi/2\approx 1.57$ when $x\to+\infty$.

So if you have errors $E_i=\left(A_i -F_i\right)/A_i$ for actual ($A_i$) and forecast ($F_i$) at time $i$, transforming your errors using $\arctan$ and averaging the result (assuming $i=1\dots N$):

$$ MAAPE=\frac{1}{N}\sum_{i=1}^N \arctan(\left|E_i\right|) $$

Will simply give you a measure that is approximately equal to normal MAPE for small errors, and is only marginally increased for large ones. Arctan is not special in this respect. One could choose a sigmoid, for example.

Excell support ARCTAN https://support.microsoft.com/en-us/office/atan-function-50746fa8-630a-406b-81d0-4a2aed395543