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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