一、簡介

支持向量機(Support Vector Machine)是Cortes和Vapnik于1995年首先提出的，它在解決小樣本、非線性及高維模式識別中表現出許多特有的優勢，并能夠推廣應用到函數擬合等其他機器學習問題中。
1 數學部分
1.1 二維空間









2 算法部分

二、源代碼

function [model,H] = lssvmMATLAB(model) % Only for intern LS-SVMlab use; % % MATLAB implementation of the LS-SVM algorithm. This is slower % than the C-mex implementation, but it is more reliable and flexible; % % % This implementation is quite straightforward, based on MATLAB's % backslash matrix division (or PCG if available) and total kernel % matrix construction. It has some extensions towards advanced % techniques, especially applicable on small datasets (weighed % LS-SVM, gamma-per-datapoint) % Copyright (c) 2002, KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlab %fprintf('~'); % % is it weighted LS-SVM ? % weighted = (length(model.gam)>model.y_dim); if and(weighted,length(model.gam)~=model.nb_data),warning('not enough gamma''s for Weighted LS-SVMs, simple LS-SVM applied');weighted=0; end % computation omega and H omega = kernel_matrix(model.xtrain(model.selector, 1:model.x_dim), ...model.kernel_type, model.kernel_pars); % initiate alpha and b model.b = zeros(1,model.y_dim); model.alpha = zeros(model.nb_data,model.y_dim); for i=1:model.y_dim,H = omega;model.selector=~isnan(model.ytrain(:,i));nb_data=sum(model.selector);if size(model.gam,2)==model.nb_data, try invgam = model.gam(i,:).^-1; catch, invgam = model.gam(1,:).^-1;endfor t=1:model.nb_data, H(t,t) = H(t,t)+invgam(t); endelsetry invgam = model.gam(i,1).^-1; catch, invgam = model.gam(1,1).^-1;endfor t=1:model.nb_data, H(t,t) = H(t,t)+invgam; endend v = H(model.selector,model.selector)\model.ytrain(model.selector,i);%eval('v = pcg(H,model.ytrain(model.selector,i), 100*eps,model.nb_data);','v = H\model.ytrain(model.selector, i);');nu = H(model.selector,model.selector)\ones(nb_data,1);%eval('nu = pcg(H,ones(model.nb_data,i), 100*eps,model.nb_data);','nu = H\ones(model.nb_data,i);');s = ones(1,nb_data)*nu(:,1);model.b(i) = (nu(:,1)'*model.ytrain(model.selector,i))./s;model.alpha(model.selector,i) = v(:,1)-(nu(:,1)*model.b(i)); end % Copyright (c) 2010, KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.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 data nb =200; sig = .20;% construct data leng = 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 data figure; hold on; plot(samplesyin(:,1),samplesyin(:,2),'+','Color',[0.6 0.6 0.6]); plot(samplesyang(:,1),samplesyang(:,2),'+','Color',[0.6 0.6 0.6]); xlabel('X_1'); ylabel('X_2'); title('Structured dataset'); disp(' (press any key)'); pause% % kernel based Principal Component Analysis % disp(' '); disp(' extract the principal eigenvectors in feature space'); disp(' >> nb_pcs=4;'); nb_pcs = 4; disp(' >> sig2 = .8;'); sig2 = .8; 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(' '); disp(' make a grid over the inputspace:'); disp('>> Xax = -3:0.1:3; Yax = -2.0:0.1:2.5;'); Xax = -3:0.1:3; Yax = -2.0:0.1:2.5; 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)))'']; ');

三、運行結果

總結

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