Distances

A ranking definition

Let be the ranking of weighted elements:

• is the label of the -th item in
• is the weighte of the -th item in
• indicates that has elements
• indicates that the rank is based on the elements weight, and ties are allowed
• indicates that each label can be present only once in

Then:

• Let be the index of label in , such that .
  Thus .
• Let denote the set of labels in : ,
  and denote the -th label in rank .
• Let denote the set of weights in : ,
  and denote the -th weight in rank .

Distance between rankings

A distance between two ranking and , respectively of and elements, can be calculated only when all the labels of one rank are ranked in the other:

In this case, .

Kendall tau (): consider only the order

![Kdef]

Where:

Weighted Kendall tau (): consider the weights

...

Earth Mover's Distance (EMD)

...