Statistics criterion for evaluating the reordering of elements

Question

Mathematics

There are two permutation strings, for example, 9 5 4 1 2 3 6 8 7 and 9 7 4 1 3 3 6 2 5. For example, this is the order of some experiments sorted by quality. Is there any specialized statistic to evaluate how much or not the order in two rows has changed a lot?

Answer 1

Distance between permutations . Over this distance, you can already enter statistics . In general, take a look at this article.

Answer 2

As a distance, you can take something like the sum of the modules of the differences in the position of each element in these permutations. That is, if A, B are the original permutations, and A*, B* are their inverses, then dist(A,B)=sum_k |A*(k)-B*(k)|.
For permutations 9 5 4 1 2 3 6 8 7 and 9 7 4 1 3 8 6 2 5 it will be 0+3+1+0+7+0+7+2+0=20.

Answer 3

I think that Kendal Tau is the best, aka Bubble-Sort Distance - the number of inversions between these two lists (see 0agr.ru/wiki/index.php/Inversion_Count )
In general, you can also take a look at this list: 0agr.ru/ wiki/index.php/Rank_Correlation