Ratio is Not Rational

We often use ratios as KPIs when we train models for classification or for other tasks. However, if you think about it, ratio is a poor way for capturing the connection between two variables.

* Note - I am referring to variables with positive values only.

Let's say you want to compare the sales of products. You decided to take the sales in 2019 and divide them by the sales of 2018. Sometimes this ratio is referred to as the lift in sales or as the sales growth. One problem with this KPI is that it is not bounded. The smaller the sales in 2018 - the higher the ratio. When the sales are 0 - the ratio is infinite. Another problem is the symmetry (or the lack of one) of the KPI. The smaller the sales in 2019 - the closer the KPI will be to 0 - very different than the behvior of the KPI when the sales of 2018 drops to 0.

Meet the natural growth ratio, denoted by (x-y)/(x+y). This ratio have some very nice characteristics:

1. The ratio is (almost) bounded - to make it bounded, there is only one case that needs to be addressed - when x+y=0. In this case, we will set the KPI to 0. This will make the KPI bounded between -1 and 1.
2. The ratio is symmetric - the KPI is symmetric around 0. if you draw the function Z=(X-Y)/(X+Y) and play with the values of X and Y you will get a mirror image on the negative axis of Z
3. This ratio will obtain the sames rank as X/Y. This means that if X1/Y1 > X2/Y2, then also (X1-Y1)/(X1+Y1)>(X2-Y2)/(X2+Y2)

This is the scaling of (x-y)/(x+y) vs x/y, you can see that the natural growth KPI will asymptotically reach 1, when x/y reaches infinity.

These characteristics make the natural growth KPI much better than the usual ratio most of us are used to calculate.

So, next time you are planning to compare variables, be rational and don't use a simple ratio.