% Copyright (c) 2002,  KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlabdisp('  This demo illustrates facilities of LS-SVMlab');disp('  with respect to unsupervised learning.');disp(' a demo dataset is generated...');clear yin yang samplesyin samplesyang mema% initiate variables and construct the datanb = 25;sig = .4;% construct dataleng = 1;for t=1:nb,   yin(t,:) = [2.*sin(t/nb*pi*leng) 2.*cos(.61*t/nb*pi*leng) (t/nb*sig)];   yang(t,:) = [-2.*sin(t/nb*pi*leng) .45-2.*cos(.61*t/nb*pi*leng) (t/nb*sig)];   samplesyin(t,:)  = [yin(t,1)+yin(t,3).*randn   yin(t,2)+yin(t,3).*randn];  samplesyang(t,:) = [yang(t,1)+yang(t,3).*randn   yang(t,2)+yang(t,3).*randn];end% plot the datafigure; hold onplot(samplesyin(:,1),samplesyin(:,2),'b+');plot(samplesyang(:,1),samplesyang(:,2),'r*');xlabel('X_1');ylabel('X_2');title('Structured dataset');disp('  (press any key)');pause%% kernel based Principal Component Analysis%disp('  extract the principal eigenvectors in feature space');disp(' >> nb_pcs=5;'); nb_pcs = 5;disp(' >> sig2 = .75;'); sig2 = .75;disp(' >> [lam,U] = kpca([samplesyin;samplesyang],''RBF_kernel'',sig2,[],''eigs'',nb_pcs); ');[lam,U] = kpca([samplesyin;samplesyang],'RBF_kernel',sig2,[],'eigs',nb_pcs);disp('  (press any key)');pause%% make a grid over the inputspace%disp(' make a grid over the inputspace:');disp('>> Xax = -3:1:3; Yax = -3.2:1:3.2;'); Xax = -3:.2:3; Yax = -3.2:.2:3.2;disp('>> [A,B] = meshgrid(Xax,Yax);'); [A,B] = meshgrid(Xax,Yax);disp('>> grid = [reshape(A,prod(size(A)),1) reshape(B,1,prod(size(B)))'']; ');grid = [reshape(A,prod(size(A)),1) reshape(B,1,prod(size(B)))'];%% compute projections of each point of the inputspace on the% principal components%disp('  compute projections of each point of the inputspace on the  ');disp('  principal components');disp('>> k = kernel_matrix([samplesyin;samplesyang],''RBF_kernel'',sig2,grid)'';  ');k = kernel_matrix([samplesyin;samplesyang],'RBF_kernel',sig2,grid)';disp('>> projections = k*U;'); projections = k*U; disp('>> contour(Xax,Yax,reshape(projections(:,1),length(Yax),length(Xax)));'); contour(Xax,Yax,reshape(projections(:,1),length(Yax),length(Xax)));title('projections of the inputspace on the principal components in feature space');disp('  (press any key)');pause%% minimize the reconstruction error using the first principal components%disp(' ');disp(' Minimize the reconstruction error using the first principal components');try  disp(' Fo every point, the reconstruction point is minimzed using');  disp(' ----------------------------------------------------------');  disp(' ');  disp('>> Xd = denoise_kpca([samplesyin;samplesyang],''RBF_kernel'',sig2,[],''eigs'',5); ');  Xd = denoise_kpca([samplesyin;samplesyang],'RBF_kernel',sig2,[],'eigs',5);   disp('>> plot(Xd(:,1),Xd(:,2),''ko''); '); plot(Xd(:,1),Xd(:,2),'ko');  disp(' ');  title('Denoising (''o'') by minimizing the reconstruction error in feature space');  disp(' ');  disp(' In the last figure, one can see the original datapoints');  disp('(''*'') and the reconstructed data (''o'').  ');  disp(' ');catch  disp(' Denoising does not work when the ''fminunc'' function is unavailable ');end  disp(' ');disp('  This  concludes this demo');hold off