% [lambda,V] = varimax(lambda[,sign,tol]) Varimax rotation

%

% Performs varimax rotation (Kaiser 1958) on the column vectors contained

% in lambda. We follow Kaiser's notation.

%

% In:

% lambda: DxL matrix (L<D) of column vectors (possibly the loadings from

% a factor analysis).

% sign: preferred majoritary sign for the vector components (-1:

% negative, 1: positive, 0: don't care). Default 0.

% tol: minimum relative increase of the objective function to keep on

% iterating (default 1e-5).

% Out:

% lambda: DxL matrix with the rotated loadings.

% V: training curve for the objective function.

% Copyright (c) 1997 by Miguel A. Carreira-Perpinan

function [lambda,V] = varimax(lambda,sign,tol)

% Argument defaults

if nargin==1 sign=0; end;

if nargin<=2 tol=1e-5; end;

[D,L] = size(lambda);

h = sqrt(sum(lambda'.^2))'+exp(-700); % Communalities

temp=lambda./(h*ones(1,L));

V = [sum(sum(temp.^4))-sum(sum(temp.^2).^2)/D]; % Objective function

V_old = V*(1-2*tol);

while abs(V(length(V))-V_old) > tol*V(length(V))

V_old = V(length(V));

for i=1:L-1

for j=i+1:L

% Optimal angle to rotate columns i, j

x = lambda(:,i)./h;

y = lambda(:,j)./h;

u = x.*x - y.*y;

v = 2*x.*y;

t = atan2( 2*(D*u'*v-sum(u)*sum(v)), D*(u'*u-v'*v)-sum(u)^2+sum(v)^2 )/4;

% Anticlockwise rotation of angle t (t+pi is valid, too)

temp = [lambda(:,i) lambda(:,j)]*[cos(t) -sin(t); sin(t) cos(t)];

lambda(:,i) = temp(:,1);

lambda(:,j) = temp(:,2);

end

end

% New value of the objective function

h = sqrt(sum(lambda'.^2))'+exp(-700); % Communalities

temp=lambda./(h*ones(1,L));

V = [V sum(sum(temp.^4))-sum(sum(temp.^2).^2)/D]; % Objective function

end

% Sign inversion so that each column vector of lambda has mainly

% components of the same sign

if sign>0

for i=1:L

if sum(lambda(:,i)) < 0

lambda(:,i) = -lambda(:,i);

end

end

elseif sign<0

for i=1:L

if sum(lambda(:,i)) > 0

lambda(:,i) = -lambda(:,i);

end

end

end