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.

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.

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: http://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.

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.