Showing posts with label eigenvalue. Show all posts
Showing posts with label eigenvalue. Show all posts

Wednesday, July 11, 2007

Plotting A Divided Court



On July 5, 2007 the Washington Post published the table above in the article “Parsing a Divided Court.” This is a similarity matrix. Displayed for each pair of Supreme Court Justices are the percentages of time that they agreed in non-unanimous rulings. We can use the method of principal co-ordinates by Gower "Some distance properties of latent roots and vector methods used in multivariate analysis" (1966, Biometrika, 53, pp. 325-338) to obtain points in Euclidean 2-space whose distances approximately mirror these similarity scores (distance^2 = 2(1 – similarity)). Justices with high similarity scores are plotted as points close together. Justices with low similarity scores are plotted as points far apart.

The method starts with a similarity matrix A. The diagonal elements of A are ones, indicating a justice’s perfect similarity with themselves. (Alternatively, a distance matrix D = (d[i,j]) could be used to build A, with diagonal entries equal to zero and the a[i,j] off diagonal entries equal to -½*d[i,j]^2).

The matrix A is transformed by subtracting out of each row the mean of that row of A, then subtracting out from each column the mean of that column of A, then adding the overall mean of A to each entry. The eigenvectors and eigenvalues of this transformed version of A are then computed. The elements of the first two eigenvectors multiplied by their respective eigenvalues give the co-ordinates of the nine justices in 2-space. This provides the visual representation of the original similarity matrix shown above. A measure of goodness of fit is given by the sum of the first two eigenvalues divided by the trace of the transformed A. Here about 75%.

An arbitrary algebraic sign places Justices Stevens, Ginsburg, Souter, and Breyer on the left of our plot and Justices Thomas, Scalia, Roberts and Alito on the right. Justice Kennedy is the most centrist justice falling just a little right of center.

An anagram note: The first letter of the justice’s last names: KGB’S STARS
(May 2010: My mnemonic KGB'S STARS for the justice's last names has held out through two replacements on the Court. Rehnquist left and Roberts replaced him. Souter left and Sotamayor replaced him.
But now it may fail. Stevens left and Kagan is the current candidate to replace him. Perhaps STARK KGBS. Not very good.)