function [nsv, alpha, b0] = svc(X,Y,ker,C)%SVC Support Vector Classification%%  Usage: [nsv alpha bias] = svc(X,Y,ker,C)%%  Parameters: X      - Training inputs%              Y      - Training targets%              ker    - kernel function%              C      - upper bound (non-separable case)%              nsv    - number of support vectors%              alpha  - Lagrange Multipliers%              b0     - bias term%%  Author: Steve Gunn (srg@ecs.soton.ac.uk)  if (nargin <2 | nargin>4) % check correct number of arguments    help svc  else    fprintf('Support Vector Classification\n')    fprintf('_____________________________\n')    n = size(X,1);    if (nargin<4) C=Inf;, end    if (nargin<3) ker='linear';, end    % tolerance for Support Vector Detection    epsilon = svtol(C);        % Construct the Kernel matrix    fprintf('Constructing ...\n');    H = zeros(n,n);      for i=1:n       for j=1:n          H(i,j) = Y(i)*Y(j)*svkernel(ker,X(i,:),X(j,:));       end    end    c = -ones(n,1);      % Add small amount of zero order regularisation to     % avoid problems when Hessian is badly conditioned.     H = H+1e-10*eye(size(H));        % Set up the parameters for the Optimisation problem    vlb = zeros(n,1);      % Set the bounds: alphas >= 0    vub = C*ones(n,1);     %                 alphas <= C    x0 = zeros(n,1);       % The starting point is [0 0 0   0]    neqcstr = nobias(ker); % Set the number of equality constraints (1 or 0)      if neqcstr       A = Y';, b = 0;     % Set the constraint Ax = b    else       A = [];, b = [];    end    % Solve the Optimisation Problem        fprintf('Optimising ...\n');    st = cputime;        [alpha lambda how] = qp(H, c, A, b, vlb, vub, x0, neqcstr);        fprintf('Execution time: %4.1f seconds\n',cputime - st);    fprintf('Status : %s\n',how);    w2 = alpha'*H*alpha;    fprintf('|w0|^2    : %f\n',w2);    fprintf('Margin    : %f\n',2/sqrt(w2));    fprintf('Sum alpha : %f\n',sum(alpha));                % Compute the number of Support Vectors    svi = find( alpha > epsilon);    nsv = length(svi);    fprintf('Support Vectors : %d (%3.1f%%)\n',nsv,100*nsv/n);    % Implicit bias, b0    b0 = 0;    % Explicit bias, b0     if nobias(ker) ~= 0      % find b0 from average of support vectors on margin      % SVs on margin have alphas: 0 < alpha < C      svii = find( alpha > epsilon & alpha < (C - epsilon));      if length(svii) > 0        b0 =  (1/length(svii))*sum(Y(svii) - H(svii,svi)*alpha(svi).*Y(svii));      else         fprintf('No support vectors on margin - cannot compute bias.\n');      end    end      end