A new distance between rankings.

Area of Science:

• Statistics
• Information Science

Background:

• Spearman's footrule distance (F-distance) is commonly used to measure differences between rankings.
• The F-distance's lack of invariance to object labeling limits its practical applications.

Purpose of the Study:

• To address the labeling dependency of the F-distance.
• To introduce a novel ranking distance metric that is invariant to labeling.
• To provide a robust alternative to existing ranking distance measures.

Main Methods:

• Analysis of the properties of the Spearman's footrule distance.
• Development of a new distance metric for rankings.
• Demonstration of invariance to indexing (labeling).
• Incorporation of importance weights into the new distance metric.

Main Results:

• The F-distance was shown to be sensitive to labeling, a significant drawback.
• A new ranking distance metric, invariant under indexing, was successfully developed.
• The proposed metric offers an improvement over the F-distance and Kemeny's distance in terms of labeling invariance.
• The new distance metric effectively handles importance weights.

Conclusions:

• The proposed labeling-invariant ranking distance is a valuable alternative for applications where object labeling varies.
• This new metric enhances the reliability and applicability of ranking comparisons in diverse scenarios.
• The method's adaptability with importance weights further broadens its utility.