### Joannes Vermorel

Metrics available to assess the performance of a forecast are many:

- Mean Absolute Error (MAE)
- Mean Squared Error (MSE)
- Mean Absolute Percentage Error (MAPE)
- Pinball Loss Function
- …

*In this post, we will try to address the question of the ‘best’ forecasting metric. It turns out to be simpler than most practioner would expect.*

Among those, MAE and MAPE are probably the most widely used metrics by practicitioners both in retail and manufacturing. Let’s start by having a look at graphs for those two metrics.

*Plot of the Mean Absolute Error. X = real (forecast is 1). Y = error.*

The behavior of the MAE is resonably straightforward. The one tricky aspect - from a mathematical viewpoint - is that the function is not differentiable everywhere (not for x=1 in the example here above).

*Plot of the Mean Absolute Percentage Error. X = real (forecast is 1). Y = error.*

The MAPE, however, is a lot more convoluted. Indeed, the behavior between *over* and *under* forecasts is very different: the under forecast error is capped to 1 whereas the over forecast error tends to infinity toward zero.

This later aspect in particular tends to wreak havoc when combined with out-of-stock (OOS) events. Indeed, OOS generate very low actual sales values, hence potentially very high MAPE values.

*In practice, we suggest to think twice before opting for MAPE, as interpreting results is likely to a be a small challenge in itself.*

### The best metric should be expressed in Dollars or Euros

From a mathematical perspective, some metrics (such as L2) are considered as **more practical** for statistical analysis (because of being differentiable for example), however, we believe that this **viewpoint is moot** when facing real business situations.

The **one and only unit to be used** to assess the performance of a forecast should be **money**. Forecasts are always wrong, and the only reasonable way to quantify the error consists of assessing how much money the delta between forecast and reality did cost to the company.

### Modeling business costs

In practice, defining such an ad-hoc cost function requires a careful examination of the business, triggering questions such as:

- How much does inventory cost?
- How much inventory obsolescence should be expected?
- How much does stock-out cost?
- …

As far **company politics** are concerned, modeling the forecast error as, say, a percentage, hence ignoring all those troublesome questions, has the one advantage of being **neutral** - leaving the rest of the company with the burden of actually translating the forecast into a course of action.

The process of establishing a sensible cost function is not rocket-science, however, it forces, within the company, the entity in charge of the forecasts, to write down explicitely all those costs. By doing so, **choices are made**, not beneficiting every division of the company, but clearly beneficiting the company itself.