function [nsv, beta, bias] = svr(X,Y,ker,C,loss,e)%SVR Support Vector Regression%%  Usage: [nsv beta bias] = svr(X,Y,ker,C,loss,e)%%  Parameters: X      - Training inputs%              Y      - Training targets%              ker    - kernel function%              C      - upper bound (non-separable case)%              loss   - loss function%              e      - insensitivity%              nsv    - number of support vectors%              beta   - Difference of Lagrange Multipliers%              bias   - bias term%%  Author: Steve Gunn (srg@ecs.soton.ac.uk)  if (nargin < 3 | nargin > 6) % check correct number of arguments    help svr  else    fprintf('Support Vector Regressing ....\n')    fprintf('______________________________\n')    n = size(X,1);    if (nargin<6) e=0.0;, end    if (nargin<5) loss='eInsensitive';, end    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) = svkernel(ker,X(i,:),X(j,:));       end    end    % Set up the parameters for the Optimisation problem    switch lower(loss)      case 'einsensitive',        Hb = [H -H; -H H];        c = [(e*ones(n,1) - Y); (e*ones(n,1) + Y)];          vlb = zeros(2*n,1);    % Set the bounds: alphas >= 0        vub = C*ones(2*n,1);   %                 alphas <= C        x0 = zeros(2*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 = [ones(1,n) -ones(1,n)];, b = 0;     % Set the constraint Ax = b        else          A = [];, b = [];         end      case 'quadratic',        Hb = H + eye(n)/(2*C);        c = -Y;        vlb = -1e30*ones(n,1);           vub = 1e30*ones(n,1);            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 = ones(1,n);, b = 0;      % Set the constraint Ax = b        else          A = [];, b = [];         end      otherwise, disp('Error: Unknown Loss Function\n');    end    % Add small amount of zero order regularisation to     % avoid problems when Hessian is badly conditioned.     % Rank is always less than or equal to n.    % Note that adding to much reg will peturb solution    Hb = Hb+1e-10*eye(size(Hb));     % Solve the Optimisation Problem        fprintf('Optimising ...\n');    st = cputime;        [alpha lambda how] = qp(Hb, c, A, b, vlb, vub, x0, neqcstr);    fprintf('Execution time : %4.1f seconds\n',cputime - st);    fprintf('Status : %s\n',how);    switch lower(loss)      case 'einsensitive',        beta =  alpha(1:n) - alpha(n+1:2*n);      case 'quadratic',        beta = alpha;    end    fprintf('|w0|^2    : %f\n',beta'*H*beta);      fprintf('Sum beta : %f\n',sum(beta));        % Compute the number of Support Vectors    svi = find( abs(beta) > epsilon );    nsv = length( svi );    fprintf('Support Vectors : %d (%3.1f%%)\n',nsv,100*nsv/n);    % Implicit bias, b0    bias = 0;    % Explicit bias, b0     if nobias(ker) ~= 0      switch lower(loss)        case 'einsensitive',          % find bias from average of support vectors with interpolation error e          % SVs with interpolation error e have alphas: 0 < alpha < C          svii = find( abs(beta) > epsilon & abs(beta) < (C - epsilon));          if length(svii) > 0            bias = (1/length(svii))*sum(Y(svii) - e*sign(beta(svii)) - H(svii,svi)*beta(svi));          else             fprintf('No support vectors with interpolation error e - cannot compute bias.\n');            bias = (max(Y)+min(Y))/2;          end        case 'quadratic',            bias = mean(Y - H*beta);      end     end  end