Statistics Toolbox

procrustes

Procrustes Analysis

Syntax

• d = procrustes(X,Y)
  [d,Z] = procrustes(X,Y)
  [d,Z,transform] = procrustes(X,Y)
  

Description

d = procrustes(X,Y) determines a linear transformation (translation, reflection, orthogonal rotation, and scaling) of the points in matrix Y to best conform them to the points in matrix X. The "goodness-of-fit" criterion is the sum of squared errors. procrustes returns the minimized value of this dissimilarity measure in d. d is standardized by a measure of the scale of X, given by

• sum(sum((X-repmat(mean(X,1),size(X,1),1)).^2,1))
  

i.e., the sum of squared elements of a centered version of X. However, if X comprises repetitions of the same point, the sum of squared errors is not standardized.

X and Y must have the same number of points (rows), and procrustes matches the ith point in Y to the ith point in X. Points in Y can have smaller dimension (number of columns) than those in X. In this case, procrustes adds columns of zeros to Y as necessary.

[d,Z] = procrustes(X,Y) also returns the transformed Y values.

[d,Z,transform] = procrustes(X,Y) also returns the transformation that maps Y to Z. transform is a structure with fields:

c	Translation component
T	Orthogonal rotation and reflection component
b	Scale component

That is, Z = transform.b * Y * transform.T + transform.c.

Examples

This example creates some random points in two dimensions, then rotates, scales, translates, and adds some noise to those points. It then uses procrustes to conform Y to X, and plots the original X and Y, and the transformed Y.

•    X = normrnd(0,1,[10 2]);
     S = [0.5 -sqrt(3)/2; sqrt(3)/2 0.5];
     Y = normrnd(0.5*X*S + 2,0.05,size(X));
     [d,Z,tr] = procrustes(X,Y);
     plot(X(:,1),X(:,2),'rx',...
          Y(:,1),Y(:,2),'b.',...
          Z(:,1),Z(:,2),'bx');
  

See Also

cmdscale, factoran

References

[1]  Seber, G.A.F., Multivariate Observations, Wiley, 1984

[2]  Bulfinch, T., The Age of Fable; or, Stories of Gods and Heroes, Sanborn, Carter, and Bazin, Boston, 1855.

princomp

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