%--- function net = train_1svm(x,KTYPE,KPAR,nu)
%--- % NET = TRAIN_1SVM(X,KTYPE,KPAR,NU)
% 1-SVM for quantile estimation % ---
% INPUTS % ---
% x - training data patterns
% KTYPE (integer) - kernel type
% 1 - linear: {default}
% 2 - linear with bias:
% x*x' + b
% 3 - polynomial of degree KPAR (X*X' + 1)^KPAR
% ((x * x') + 1)^KPAR
% 4 - gaussian kernel
% exp(-1/(2*KPAR^2)* norm(x - y)^2)
%
% KPAR - kernel hyperparamater corresponding to
% KTYPE. Thus, for a the linear, polynomial and gaussian
% kernels KPAR refers to the bias, degree, and kernel
% width. {1}
% NU - specifies size of support region \equiv (1-NU)
% --- % OUTPUTS % ---
% NET - a structure retaining the training parameters (X,KTYPE,KPAR,NU)
% as well as the hyperplane weight vector coefficients (alphas) % and offset (rho).
% author: gt jemwa 2006 m = size(x,1); %H = sv_dot(rbf_dot(gamma),x',x'); H = compute_kernel(KTYPE,KPAR,x'); H = (H+H')./2; H = H+1e-9*eye(size(H));
%setup qp optimization problem c = zeros(1,m);
Aeq = ones(1,m); eq = 1;
LB = zeros(m,1);
UB = (1/m/nu)*ones(m,1);
x0 = []; %initial starting point
options = optimset('Display','off','LargeScale','off','MaxIter',10000); [alphas,obj,exitflag,output] = quadprog(H,c,[],[],Aeq,eq,LB,UB,x0,options); % sanity check if any(alphas>((1/m/nu)+1e-3)), error('something wrong...exiting'); end
B.3 ONE-CLASS SUPPORT VECTOR CLASSIFICATION 163 %rho = H(nbsvi,nbsvi)*alphas(nbsvi,1); % Kt = sv_dot(rbf_dot(gamma),x',x'); Kt = compute_kernel(KTYPE,KPAR,x'); yp = Kt*alphas; rho = mean(yp(nbsvi)); net = struct('trainx',x,'KTYPE',KTYPE,'KPAR',KPAR,'nu',nu,'b0',rho,... 'alphas',alphas); %--- function yt = test_1svm(net,xt) %--- % YT = TEST_1SVM(NET,xt) % --- % INPUTS % ---
% NET - struct containing 1-SVM model (see TRAIN_1SVM.M)
% xt - testing data % --- % OUTPUTS % --- % YT - (non)-linear features %
% Decision function: yt = (<w,x>-rho)
% yt > 0 (point falling in estimated support region)
% yt < 0 (points falling outside estimated support region,
% "outliers") % gt jemwa 2006 Kt = compute_kernel(net.KTYPE,net.KPAR,net.trainx',xt')'; yt = Kt*net.alphas - net.b0; %--- function K=compute_kernel(KTYPE,KPAR,x1,x2) %--- if nargin < 4, x2 = x1; end switch KTYPE case 1 K = x1'*x2;
case 2 K = x1'*x2 + KPAR; case 3 K=(x1'*x2+1).^KPAR; case 4 dot_x1 = sum(x1.*x1,1); dot_x2 = sum(x2.*x2,1); unitvec = ones(size(x1,2),1); K = x1'*x2; for i=1:size(x2,2)
K(:,i) = exp(1./(2*KPAR^2)* (2 * K(:,i) - dot_x1' - dot_x2(i)... * unitvec));
end otherwise
error('Unknown kernel function'); end
165
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