Stats Metric for how dynamic/changing a market is

Question

Note: I am not a stats expert.

I have the following hypothetical example: The image below describes the five highest sold car brands (monthly) in 2014 versus in 2015 (only 5 months).

In 2014, the market was relatively "stable". A few changes happened in the top 5 rankings of brands. And only 2 new brands (Jeep and VW) made it to the top 5 list.

In 2015, the ranking of brands was constantly changing from month to month and new brands jumped to the top and disrupted the market. The market was very "dynamic".

I want to have a mathematical metric that describes how dynamic/stable a market is, where 0.0 is highest stable (stagnant) and 1.0 is very dynamic. In my example: the 2014 market will be around 0.2 while the 2015 market will be around 0.8.

By "dynamic", I mean "the ranking is aggressively changing from month to month and new brands disrupt the dominance of previous top brands".

Is there such a metric? if not, any suggestions for a formula?

Thank you!

Answer 1

If you compare two consecutive months, you have two rankings of the car brands. Thus, you can calculate the Kendall tau distance betwen the rankings (if you want the distance to be in the unit interval, you will need to normalize by the maximum possible distance).

Do this between all consecutive pairs of months, and you will get a time series of these distances. Then you can start looking at things like "the distance in 2015 was consistently higher than in 2014" or "there is an upward trend" (corresponding to increasing dynamics in the market).

This SO post may be helpful if you want to implement this in R: Kendall tau distance (a.k.a bubble-sort distance) between permutations in base R. In addition, this Math.SE post gives you the average distance you would expect, as a kind of a benchmark: https://math.stackexchange.com/questions/1842062/how-to-find-the-average-kendalls-distance-between-2-rankings.

You will need to think of something to deal with new market entrants, and/or brands exiting the market, but that should be doable, e.g., by simply setting them to a rank $N+1$ and dealing with ties in ranks from multiple not-yet-entered brands.