function varargout=ace(x,y,w,ii,oi)
%ACE Finds optimal transformation and maximal correlation.
% MCOR=ACE(X,Y [,W,II,OI]) estimates the maximal correlation
% of the system theta(X)=phi(Y), where X is one-dimensional
% and Y can be multi-dimensional.
%
% [THETA, PHI]=ACE(X,Y [,W,II,OI]) estimates the optimal
% transformations THETA and PHI of the system theta(X)=phi(Y).
%
% [THETA, PHI, MCOR]=ACE(X,Y [,W,II,OI]) estimates the optimal
% transformations THETA and PHI and the maximal correlation
% MCOR of the system theta(X)=phi(Y).
%
% [THETA, PHI, MCOR, I, O, Imax, Omax]=ACE(X,Y [,W,II,OI])
% estimates the THETA, PHI and MCOR and outputs the number of
% inner iterations I, break-up number of inner inner iterations,
% number of outer iterations O and break-up number of outer
% inner iterations. If the algorithm doesn't converge, the
% number of iterations will be negative signed.
%
% ACE(...) without parameters plots the optimal transformations
% THETA and PHI.
%
% Optional parameters:
% W is the half-length of the boxcar window, II is the maximal
% number of inner iterations, OI is the minimal number of outer
% iterations. If W=[], the default boxcar window size is 11.
%
% Examples: x = (-1:.002:1) + .3 * rand(1,1001);
% y = (-1:.002:1) .^ 2 + .3* rand(1,1001);
% corrcoef(x,y)
% ace(y,x)
%
% References:
% Breiman, L., Friedman, J. H.:
% Estimating Optimal Transformations for Multiple regression
% and Correlation, J. Am. Stat. Assoc., Vol. 80, No. 391, 1985.
% Voss, H., Kurths, J.:
% Reconstruction of nonlinear time delay models from data by the
% use of optimal transformations, Phys. Lett. A, 234, 1997.
%
% See also: MCF
% Copyright (c) 2001-2003 by AMRON
% Norbert Marwan, Potsdam University, Germany
% http://www.agnld.uni-potsdam.de
%
% $Date$
% $Revision$
%
% $Log$
% Revision 1.5 2004/11/10 07:05:02 marwan
% initial import
%
%
% This program is free software; you can redistribute it and/or
% modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or any later version.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% check the in/out
error(nargchk(2,5,nargin))
error(nargoutchk(0,7,nargout))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% splash the GPL
filename='ace';
which_res=which([filename,'.m']);
gplrc_path=[strrep(which_res,[filename,'.m'],''), 'private'];
gplrc_file=[gplrc_path, filesep, '.gpl.',filename];
if ~exist(gplrc_path)
mkdir(strrep(which_res,[filename,'.m'],''),'private')
end
if ~exist(gplrc_file)
fid=fopen(gplrc_file,'w');
fprintf(fid,'%s\n','If you delete this file, the GNU Public License will');
fprintf(fid,'%s','splash up at the next time the programme starts.');
fclose(fid);
if exist('gpl')
txt=gpl;
else
txt={'The GNU General Public License was not found.'};
end
h=figure('NumberTitle','off',...,
'ButtonDownFcn','close',...
'Name','GNU General Public License');
ha=get(h,'Position');
h=uicontrol('Style','Listbox',...
'ButtonDownFcn','close',...
'CallBack','close',...
'Position',[0 0 ha(3) ha(4)],...
'FontName','Courier',...
'BackgroundColor',[.8 .8 .8],...
'String',txt);
waitfor(h)
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% initialization
flag=1;
if size(y,1)<size(y,2)
y=y';
end
if size(x,1)<size(x,2)
x=x';
end
if size(x,2)>1, error('The dimension of x must be one.'); end
if nargin<5 | isempty(oi)
oi=1000;
end
if nargin<4 | isempty(ii)
ii=100;
end
if nargin<3 | isempty(w)
w=round(.1*size(y,1));w=5;
end
N=size(y,1);
dim=size(y,2);
if size(x,1)~=N
error('The lengths of x and y must match.')
end
ocrit=1*eps; icrit=1*eps; % relative accuracy of the CPU
try
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sorting
[xs ix]=sort(x); [ys iy]=sort(y);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IID distributed data
xr(ix,1)=(1:N)';
for d=1:dim, yr(iy(:,d),d)=(1:N)'; end
theta=(xr-(N+1)/2)/sqrt(N*(N+1)/12);
phi=(yr-(N+1)/2)/sqrt(N*(N+1)/12);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iteration process
ieps=1.; oeps=1.;
o=1; ocrit1=1;
maxi=0;
h0=waitbar(0,'','Name','Iteration`s Progress');
pos=get(h0,'Position'); set(h0,'Position',[pos(1) pos(2)+100 pos(3) pos(4)]);
h1=get(get(h0,'Children'),'Title');
while o<=oi & ocrit1>ocrit
% waitbar(log(eps/ocrit1)/36+1), set(h1,'String',['convergence: ', num2str(ocrit1)])
waitbar(o/oi), set(h1,'String',['convergence: ', num2str(ocrit1)])
o=o+1;
i=1; icrit1=1;
while i<=ii & icrit1>icrit
i=i+1;
for d=1:dim; sum0=0;
for dd=1:dim
if dd~=d sum0=sum0+phi(:,dd); end;
end;
A=theta-sum0;
A=A(iy(:,d)); % A=phi(xk) = sortierung
ww=-w:w;
% r=conv(A,exp((-ww.^2)/(.5*(ww(end)-ww(1)))^2)); % faltung = gleitender mittelwert
r=conv(A,ones(2*w+1,1)); % faltung = gleitender mittelwert
r=r(w+1:N+w)/(2*w+1); % bringt r wieder auf die laenge von y
switch flag
case 1
r(1:w)=interp1([w+1:2*w+1],r([w+1:2*w+1]),[1:w],'linear','extrap');
r(N-w+1:N)=interp1([N-2*w:N-w],r([N-2*w:N-w]),[N-w+1:N],'linear','extrap');
case 2
r(1:w)=mean(r(w+1:w+2)); r(N-w+1:N)=mean(r(N-w-1:N-w));
end
% phi(:,d)=r(:,d);
phi(:,d)=r(yr(:,d)); % sortiert r wieder auf ausgangssortierung
end;
icrit1=ieps;
if dim==1 sum0=phi;
else sum0=sum(phi')';
end;
ieps=sum((sum0-theta).^2)/N;
icrit1=abs(icrit1-ieps); % konvergiert die varianz noch?
end;
A=sum0(ix);
% A=sum0;
% r=conv(A,exp((-ww.^2)/(.5*(ww(end)-ww(1)))^2)); % faltung = gleitender mittelwert
r=conv(A,ones(2*w+1,1));
r=r(w+1:N+w)/(2*w+1);
switch flag
case 1
r(1:w)=interp1([w+1:2*w+1],r([w+1:2*w+1]),[1:w],'linear','extrap');%
r(N-w+1:N)=interp1([N-2*w:N-w],r([N-2*w:N-w]),[N-w+1:N],'linear','extrap');
case 2
r(1:w)=mean(r(w+1:w+2)); r(N-w+1:N)=mean(r(N-w-1:N-w));
end
theta=r(xr);
% theta=r;
theta=(theta-mean(theta))/std(theta);
ocrit1=oeps;
oeps=sum((sum0-theta).^2)/N;
ocrit1=abs(ocrit1-oeps);
if maxi<i, maxi=i; end
end
waitbar(1), if ocrit1>ocrit, set(h1,'String','doesn`t converge!'), ocrit1=-ocrit1; pause(.8), end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% output
temp=corrcoef(theta,sum0);
if nargout==7
varargout(7)={ocrit1};
end
if nargout>=6
varargout(6)={icrit1};
end
if nargout>=5
varargout(5)={o-1};
end
if nargout>=4
varargout(4)={maxi-1};
end
if nargout>=3
varargout(3)={temp(1,2)};
end
if nargout>=2;
varargout(1)={theta};
varargout(2)={phi};
end
if nargout==1;
varargout(1)={temp(1,2)};
elseif nargout==0
a=theta;
b=phi;
c=temp(1,2);
d=maxi-1;
e=o-1;
f=icrit1;
g=ocrit1;
N=size(b,1);
dim=size(b,2);
figure
subplot(ceil((dim+2)/2),2,1)
plot(x,a,'k.','MarkerSize',0.5)
if isempty(inputname(1));
tx=['\Theta(x)'];
xlabel('x')
else
tx=['\Theta(',inputname(1) ,')'];
xlabel(inputname(1))
end
ylabel(tx)
for i=2:dim+1,
[s1 s2]=sort(y(:,i-1));
thetainv=interp1(a+.0000001.*randn(length(a),1),x,b(s2,i-1),'nearest');
subplot(ceil((dim+2)/2),2,i)
[hAX h1 h2]=plotyy(y(:,i-1),b(:,i-1), s1,thetainv);
set(h1,'Color','k','LineStyle','none','Marker','.','MarkerSize',0.5)
set(h2,'Color',[.6 .6 .6],'LineStyle','none','Marker','.','MarkerSize',0.5)
set(hAX(1),'ycolor','k'), set(hAX(2),'ycolor',.5.*[.6 .6 .6])
if isempty(inputname(2));
tx=['\Phi_{' num2str(i-1) '}(y)'];
tx2=['\Theta^{-1}(\Phi_{' num2str(i-1) '}(y))'];
xlabel('y_*')
else
tx=['\Phi_{' num2str(i-1) '}(',inputname(2) ,')'];
tx2=['\Theta^{-1}(\Phi_{' num2str(i-1) '}(',inputname(2) ,'))'];
xlabel(inputname(2))
end
ylabel(tx)
s1(find(isnan(thetainv)))=[];
thetainv(find(isnan(thetainv)))=[];
[smax sind]=max(s1);
h1=text(smax+.051*abs(max(s1)-min(s1)),thetainv(sind),tx2,'FontSize',8);
set(h1,'Parent',hAX(2))
end
sig=0;
h=subplot(ceil((dim+2)/2),2,2*ceil((dim+2)/2));
txt=char;
if f<0 | g<0
txt='doesn''t converge!';
end
tf=f/eps;tg=g/eps;
if tf>9999, tf=num2str(tf,'%1.0e'); else, tf=num2str(tf); end
if tg>9999, tg=num2str(tg,'%1.0e'); else, tg=num2str(tg); end
set(h,'Visible','off')
text(.1,-.18,date,'FontSize',8,'FontAngle','Italic')
text(.1,.8,['A C E'],'FontSize',14,'FontWeight','Bold','FontName','new century schoolbook')
text(.1,.6,['Max. Correlation \Psi: ' num2str(c)])
text(.1,.5,['5% Significance level: ...' ]) %num2str(sig)])
text(.1,.35,['Data length: ' num2str(N)])
text(.1,.25,['Window length: ' num2str(2*w+1)])
text(.1,.15,['Number of Iterations: ', num2str(d), '/ ', num2str(e)])
text(.1,.05,['divergence criteria/eps: ', tf, '/ ', tg])
if ~isempty(txt)
text(.1,-.05,txt,'Color',[.8 0 0],'FontWeight','Bold')
end
end
close(h0)
catch
delete(h0)
if ~strcmpi(lasterr,'Interrupt')
disp('Could not compute the optimal transformations.')
disp('Try other input arguments.')
end
for i=1:nargout,varargout(i)={NaN};end
end