# How should I tune parameters for a balanced comparison of relative vs. absolute growth?

Here's the situation I'm in:

I need to create a measurement to compare different companies by how much they grew in the last year, both in website visits and employees, as part of a larger scoring system that takes many factors into account. Each of the individual scores will get z-scored and the final score is an average of all z-scores.

The issue is that I need to find a way to balance relative and absolute growth in this score. One one hand, a company that went from 100 to 150 employees has grown much more impressively than one that went from 1 to 2, so purely relative growth doesn't work. On the other, a company going from 25 to 100 employees in one year is much more impressive than one that went from 1000 to 1100, so purely absolute growth doesn't cut it either.

It's also not helpful to have huge outliers so this aspect of the score doesn't overshadow others, so some sort of diminishing returns factor would be interesting.

I keep trying different methods to balance the two, but I feel like I'm flying blind. At one point I landed on (relative * absolute^2 )^0.1 which seemed to work okay, but honestly I have little idea if it even makes any sense.

I'm proficient in Python, which I'm working in, but I haven't used much pure math in years, so I'm rusty in my intuitive understanding of what I'm doing here. I keep thinking that maybe I should visualize this function somehow so that I can fine tune the weighting, but I can't quite get it clearly defined in my head how I would do that.

So anyway, that's my problem. Can anyone help shed some light on where I could begin to work or things I should read up on?

If you are after an analytical formula, you are right to simply choose some form that "sounds good" then try them out and see how good they do. Sometimes your best forms will seem a little hard to interpret. This is normal in the field of ML and you have to balance that with your numeric fitness. That's called model transparency. What you have ended up with is effectively a linear mixture model in the log domain ($$\exp(a*\log(x)+b*\log(y))$$), which is not uncommon.