Distances
Where:
-
is the
-th ranking.
-
is the rank of element
in ranking
In this case:
Where:
-
is the ranking size.
-
is the number of order-independent combinations.
Where is the set of unordered pairs of distinc elemts in
and
, and:
Where is the value of
in ranking
, opposed to its index
.
Thus:
This formulation is invariant when multiplying the ranks for a constant, while it is affected by summing a constant to the ranks.
OBS! This weighting method does not work when there are values in the ranking. Also switching to a
normalization would not fix the issues.
This formulation is invariant when summing the ranks to a constant, while it is affected by multiplying them for a constant. Also, When there are values in the ranks the distance can still be calculated, by switching to a normal Kendall tau when all the values are the same.
Let's define the sum of the absolute value of the pair-wise difference of rank as:
Then, we can use it for normalizing:
Thus: