Rank-Adapted Singular Value Decomposition

• Activity Number: 617
• Type: Contributed
• Date/Time: Thursday, August 7, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistics in Sports
• Author(s): Kathleen Campbell*+
• Companies: Temple University
• Keywords: Kendall's $\tau$ ; Rank-Adapted Singular Value Decomposition ; SVD ; PCA ; Anderson's Test

Abstract

Rankings of objects, events, and people to determine importance or competitive edge are published in many venues. Frequently, multiple sources publish rankings of the same (or similar) set of objects. Do these rankings agree? No current statistical methodology adequately gauges the association among the sources, nor indicates which sources agree. We propose the Rank-Adapted Singular Value Decomposition, R-A SVD, a new method that uses Kendall's $\tau$ as the underlying correlation method. We begin with P, a matrix of data ranks. The first step is to factor the covariance matrix K = cov(P,method="kendall") = V D^2 V'. V is an orthogonal matrix, which can be used to indicate which sources agree, and D is a diagonal of eigenvalues. By analogy with the SVD, we define U* = PV'inv(D). Anderson's test (1963) identifies the A significantly large eigenvalues from D. We identify clusters of objects, which we interpret as equivalence classes, based on the first A columns of U*. The R-A SVD method allows for comparison of multiple sources of rank data, indicates which sources agree, and in cases of disagreement indicates which objects are equivalent.