ImaGIN_nonlinear_interdependence.m
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function [PredictionX,PredictionY] = ImaGIN_nonlinear_interdependence(x,y,TimeDelay,EmbeddingDimension,Method,TimeHorizon)
%
% REFERENCES:
% - Schiff Phys Rev E 54(6) 6708 1997, Breakspear PhD Thesis, Quian Quiroga PhysRevE 65 041903 2002
% -=============================================================================
% This function is part of the ImaGIN software:
% https://f-tract.eu/
%
% This software is distributed under the terms of the GNU General Public License
% as published by the Free Software Foundation. Further details on the GPLv3
% license can be found at http://www.gnu.org/copyleft/gpl.html.
%
% FOR RESEARCH PURPOSES ONLY. THE SOFTWARE IS PROVIDED "AS IS," AND THE AUTHORS
% DO NOT ASSUME ANY LIABILITY OR RESPONSIBILITY FOR ITS USE IN ANY CONTEXT.
%
% Copyright (c) 2000-2018 Inserm U1216
% =============================================================================-
%
% Authors: Olivier David
if size(x,1)~=1
x=reshape(x,1,length(x));
end
if size(y,1)~=1
y=reshape(y,1,length(y));
end
%Normalisation
x=(x-mean(x))./std(x);
y=(y-mean(y))./std(y);
%time-delay embedding
NSample = size(x,2)-TimeDelay*(EmbeddingDimension-1)-1;
EmbeddingMatrix=[1:NSample]'*ones(1,EmbeddingDimension)+ones(NSample,1)*[0:TimeDelay:((EmbeddingDimension-1)*TimeDelay)];
X=x(EmbeddingMatrix);
Y=y(EmbeddingMatrix);
% Note: seems to be incompatible with the Theiler correction
% if exist('NSampleMax')
% if NSample>NSampleMax
% tmp=randperm(NSample);
% NSample=NSampleMax;
% X=X(tmp(1:NSample),:);
% Y=Y(tmp(1:NSample),:);
% end
% end
switch Method
case 'Schiff'
MeanX = mean(X);
MeanY = mean(Y);
%neighbourhood
% Distance = zeros(NSample,2*EmbeddingDimension);
NeighbourX = zeros(NSample,2*EmbeddingDimension);
NeighbourY = zeros(NSample,2*EmbeddingDimension);
for i1=1:NSample
DistX=sum((ones(NSample-TimeHorizon,1)*X(i1,:)-X(1:end-TimeHorizon,:)).^2,2);
DistY=sum((ones(NSample-TimeHorizon,1)*Y(i1,:)-Y(1:end-TimeHorizon,:)).^2,2);
%Theiler correction
tmp=max([1 i1-TimeDelay]):min([NSample-TimeHorizon i1+TimeDelay]);
DistX(tmp)=0;
DistY(tmp)=0;
tmp=length(tmp);
[tmp1,tmp2]=sort(DistX);
[tmp3,tmp4]=sort(DistY);
NeighbourX(i1,:) = tmp2(tmp+[1:2*EmbeddingDimension])';
NeighbourY(i1,:) = tmp4(tmp+[1:2*EmbeddingDimension])';
% [tmp1,tmp2]=sort(sum((ones(NSample,1)*X(i1,:)-X).^2,2));
% % Distance(i1,:) = sqrt(tmp1(2:(2*EmbeddingDimension+1)));
% NeighbourX(i1,:) = tmp2(2:(2*EmbeddingDimension+1))';
% [tmp1,tmp2]=sort(sum((ones(NSample,1)*Y(i1,:)-Y).^2,2));
% % Distance(i1,:) = sqrt(tmp1(2:(2*EmbeddingDimension+1)));
% NeighbourY(i1,:) = tmp2(2:(2*EmbeddingDimension+1))';
end
%Prediction error
PredictionXNum = zeros(NSample-TimeHorizon,1);
PredictionXDen = zeros(NSample-TimeHorizon,1);
PredictionYNum = zeros(NSample-TimeHorizon,1);
PredictionYDen = zeros(NSample-TimeHorizon,1);
for i1=1:NSample-TimeHorizon
tmp=TimeHorizon+NeighbourX(i1,:);
% tmp=tmp(find(tmp<=NSample));
PredictionYNum(i1)=sqrt(sum((Y(i1+TimeHorizon,:)-mean(Y(tmp,:))).^2));
PredictionYDen(i1)=sqrt(sum((Y(i1+TimeHorizon,:)-MeanY).^2));
tmp=TimeHorizon+NeighbourY(i1,:);
% tmp=tmp(find(tmp<=NSample));
PredictionXNum(i1)=sqrt(sum((X(i1+TimeHorizon,:)-mean(X(tmp,:))).^2));
PredictionXDen(i1)=sqrt(sum((X(i1+TimeHorizon,:)-MeanX).^2));
end
PredictionX=1-mean(PredictionXNum)/mean(PredictionXDen);
PredictionY=1-mean(PredictionYNum)/mean(PredictionYDen);
% case 'Quian' %with no time horizon
% RXTot=zeros(1,NSample);
% RYTot=zeros(1,NSample);
% RXY=zeros(1,NSample);
% RYX=zeros(1,NSample);
% for i1=1:NSample
% DistX=sum((ones(NSample,1)*X(i1,:)-X).^2,2);
% DistY=sum((ones(NSample,1)*Y(i1,:)-Y).^2,2);
% %Theiler correction
% tmp=max([1 i1-TimeDelay]):min([NSample i1+TimeDelay]);
% DistX(tmp)=0;
% DistY(tmp)=0;
% tmp=length(tmp);
% [tmp1,tmp2]=sort(DistX);
% [tmp3,tmp4]=sort(DistY);
% RXTot(i1)=mean(tmp1(tmp+1:end));
% RYTot(i1)=mean(tmp3(tmp+1:end));
% RXY(i1)=mean(DistX(tmp4(tmp+1:(2*EmbeddingDimension+tmp))));
% RYX(i1)=mean(DistY(tmp2(tmp+1:(2*EmbeddingDimension+tmp))));
% end
% %Prediction error
% PredictionX = mean((RXTot-RXY)./RXTot);
% PredictionY = mean((RYTot-RYX)./RYTot);
case 'Quian' %with time horizon
Criterion=-Inf;
Coarse=max([floor((TimeHorizon+1)/3) 1]);
for i0=0:Coarse:TimeHorizon
RXTot=zeros(1,NSample-i0);
RYTot=zeros(1,NSample-i0);
RXY=zeros(1,NSample-i0);
RYX=zeros(1,NSample-i0);
for i1=1:NSample-i0
DistX=sum((ones(NSample-i0,1)*X(i1,:)-X(1:end-i0,:)).^2,2);
DistY=sum((ones(NSample-i0,1)*Y(i1,:)-Y(1:end-i0,:)).^2,2);
DistXd=sum((ones(NSample-i0,1)*X(i1+i0,:)-X(1:end-i0,:)).^2,2);
DistYd=sum((ones(NSample-i0,1)*Y(i1+i0,:)-Y(1:end-i0,:)).^2,2);
%Theiler correction
tmp=max([1 i1-TimeDelay]):min([NSample-i0 i1+TimeDelay]);
DistX(tmp)=0;
DistY(tmp)=0;
tmp=length(tmp);
[tmp1,tmp2]=sort(DistX);
[tmp3,tmp4]=sort(DistY);
RXTot(i1)=mean(tmp1(tmp+1:end));
RYTot(i1)=mean(tmp3(tmp+1:end));
tmptmp=i0+tmp4(tmp+1:(2*EmbeddingDimension+tmp));
tmptmp=tmptmp(find(tmptmp<=length(DistXd)));
RXY(i1)=mean(DistXd(tmptmp));
tmptmp=i0+tmp2(tmp+1:(2*EmbeddingDimension+tmp));
tmptmp=tmptmp(find(tmptmp<=length(DistXd)));
RYX(i1)=mean(DistYd(tmptmp));
end
%Prediction error
PX=mean((RXTot-RXY)./RXTot);
PY=mean((RYTot-RYX)./RYTot);
if Criterion<abs(PX-PY)
Criterion=abs(PX-PY);
PredictionX = PX;
PredictionY = PY;
end
end
% NeighbourX = zeros(NSample,2*EmbeddingDimension);
% NeighbourY = zeros(NSample,2*EmbeddingDimension);
% RYTot=zeros(1,NSample);
% for i1=1:NSample
% [tmp1,tmp2]=sort(sum((ones(NSample,1)*X(i1,:)-X).^2,2));
% NeighbourX(i1,:) = tmp2(2:(2*EmbeddingDimension+1))';
% [tmp3,tmp4]=sort(sum((ones(NSample,1)*Y(i1,:)-Y).^2,2));
% NeighbourY(i1,:) = tmp4(2:(2*EmbeddingDimension+1))';
% RYTot(i1)=mean(tmp3(2:end));
% end
% %Prediction error
% % RY=zeros(1,NSample);
% RYX=zeros(1,NSample);
% for i1=1:NSample
% % RY(i1)=mean(sum((ones(2*EmbeddingDimension,1)*Y(i1,:)-Y(NeighbourY(i1,:),:)).^2,2));
% RYX(i1)=mean(sum((ones(2*EmbeddingDimension,1)*Y(i1,:)-Y(NeighbourX(i1,:),:)).^2,2));
% end
% Prediction = mean((RYTot-RYX)./RYTot);
case 'Chavez'
EmbeddingMatrix2=[EmbeddingMatrix EmbeddingMatrix(:,end)+TimeHorizon];
[C1,H1]=CorrelationIntegral([x(EmbeddingMatrix2) Y],TimeDelay,2);
[C2,H2]=CorrelationIntegral([X Y],TimeDelay,2);
[C3,H3]=CorrelationIntegral(x(EmbeddingMatrix2),TimeDelay,2);
[C4,H4]=CorrelationIntegral(X,TimeDelay,2);
PredictionX=log(C1)-log(C2)-log(C3)+log(C4);
% PredictionX=-H1+H2+H3-H4;
[C1,H1]=CorrelationIntegral([y(EmbeddingMatrix2) X],TimeDelay,2);
[C2,H2]=CorrelationIntegral([X Y],TimeDelay,2);
[C3,H3]=CorrelationIntegral(y(EmbeddingMatrix2),TimeDelay,2);
[C4,H4]=CorrelationIntegral(Y,TimeDelay,2);
PredictionY=log(C1)-log(C2)-log(C3)+log(C4);
% PredictionY=-H1+H2+H3-H4;
end