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

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.

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