Question

I have two arrays containing the same elements, but in different orders, and I want to know the extent to which their orders differ.

The method I tried, didn’t work. it was as follows:

For each list I built a matrix which recorded for each pair of elements whether they were above or below each other in the list. I then calculated a pearson correlation coefficient of these two matrices. This worked extremely badly. Here’s a trivial example:

 list 1: 1 2 3 4  list 2: 1 3 2 4 

The method I described above produced matrices like this (where 1 means the row number is higher than the column, and 0 vice-versa):

 list 1:   1 2 3 4 1   1 1 1 2     1 1 3       1 4  list 2:   1 2 3 4  1   1 1 1 2     0 1  3       1 4 

Since the only difference is the order of elements 2 and 3, these should be deemed to be very similar. The Pearson Correlation Coefficient for those two matrices is 0, suggesting they are not correlated at all. I guess the problem is that what I’m looking for is not really a correlation coefficient, but some other kind of similarity measure. Edit distance, perhaps?

Can anyone suggest anything better?

Answer 1

Mean square of differences of indices of each element.

List 1: A B C D E List 2: A D C B E 

Indices of each element of List 1 in List 2 (zero based)

A B C D E 0 3 2 1 4 

Indices of each element of List 1 in List 1 (zero based)

A B C D E 0 1 2 3 4 

Differences:

A  B C D E 0 -2 0 2 0 

Square of differences:

A B C D E   4   4 

Average differentness = 8 / 5.