procrustes
Procrustes analysis
Syntax
d = procrustes(X,Y)
[d,Z] = procrustes(X,Y)
[d,Z,transform] = procrustes(X,Y)
[...] = procrustes(...,'scaling',flag)
[...] = procrustes(...,'reflection',flag)
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))
That is, 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 Y(i) to X(i).
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 array with fields:
c— Translation componentT— Orthogonal rotation and reflection componentb— Scale component
That is:
c = transform.c; T = transform.T; b = transform.b; Z = b*Y*T + c;
[...] = procrustes(...,'scaling',,
when flag)flag is false,
allows you to compute the transformation without a scale component
(that is, with b equal to 1).
The default flag is true.
[...] = procrustes(...,'reflection',,
when flag)flag is false,
allows you to compute the transformation without a reflection component
(that is, with det(T) equal to 1).
The default flag is 'best',
which computes the best-fitting transformation, whether or not it
includes a reflection component. A flag of true forces
the transformation to be computed with a reflection component (that
is, with det(T) equal to -1)
Examples
Procrustes Analysis
Generate the sample data in two dimensions.
rng('default')
n = 10;
X = normrnd(0,1,[n 2]);Rotate, scale, translate, and add some noise to sample points.
S = [0.5 -sqrt(3)/2; sqrt(3)/2 0.5]; Y = normrnd(0.5*X*S+2,0.05,n,2);
Conform Y to X using procrustes analysis.
[d,Z,tr] = procrustes(X,Y);
Plot the original X and Y with the transformed Y .
plot(X(:,1),X(:,2),'rx',Y(:,1),Y(:,2),'b.',Z(:,1),Z(:,2),'bx');
References
[1] Kendall, David G. “A Survey of the Statistical Theory of Shape.” Statistical Science. Vol. 4, No. 2, 1989, pp. 87–99.
[2] Bookstein, Fred L. Morphometric Tools for Landmark Data. Cambridge, UK: Cambridge University Press, 1991.
[3] Seber, G. A. F. Multivariate Observations. Hoboken, NJ: John Wiley & Sons, Inc., 1984.