www.gusucode.com > 支持向量机工具箱 - LIBSVM OSU_SVM LS_SVM源码程序 > 支持向量机工具箱 - LIBSVM OSU_SVM LS_SVM\stprtool\svm\kerskfmat.m
function [Alpha,bias,sol,t,flps,margin,up,lo]=...
kerskfmat(X,I,epsilon,ker,arg,tmax,C)
% KERSKFMAT fast version of KERNELSK.
% [Alpha,bias,sol,t,flps,margin,up,lo]=kerskfmat(X,I,epsilon,ker,arg,tmax,C)
%
% Faster version (with respect to number of floating point operations)
% of KERNELSK algorithm. It does not compute the whole kernel matrix.
%
% Inputs:
% X [NxL] training patterns, N is dimension and L number of patterns.
% I [1xL] labels, 1 for 1st class and 2 for 2nd class.
% epsilon [1x1] precision of found solution. The margin of found
% hyperplane is less than the optimal margin at most by epsilon.
% ker [string] kernel, see 'help kernel'.
% arg [...] argument of given kernel, see 'help kernel'.
% tmax [int] maximal number of iterations.
% C [real] trade-off between margin and training error.
%
% Outputs:
% Alpha [1xL] Weights (Lagrangians) of patterns.
% bias [real] bias (threshold) of found decision rule.
% sol [int] 1 solution is found
% 0 algorithm stoped (t == tmax) before converged.
% -1 hyperplane with margin greater then epsilon
% does not exist.
% t [int] number of iterations.
% margin [real] margin between classes.
% flps [int] number of used floating point operations.
% up [1,t] evolution of the upper bound on the optimal margin.
% lo [1,t] evolution of the lower bound on the optimal margin.
% CE [real] classification error on training patterns.
%
% See also KERNELSK, SVM.
%
% Statistical Pattern Recognition Toolbox, Vojtech Franc, Vaclav Hlavac
% (c) Czech Technical University Prague, http://cmp.felk.cvut.cz
% Written Vojtech Franc (diploma thesis) 02.11.1999, 13.4.2000
% Modifications
% 19-September-2001, V.Franc, comments changed.
% 18-August-2001, V.Franc, up and lo bounds added
% 9-August-2001, V.Franc, version without computing kernel matrices
% 13-July-2001, V.Franc, comments
% 12-July-2001, V.Franc, C, bias and normal vect. normalized.
% 11-July-2001, V.Franc, Rosta Horcik proved that the computation
% of threshold is OK.
% 10-July-2001, V.Franc, derived from kekozinec2
flops(0);
% set default values of the input argiments
if nargin < 7,
C = inf;
end
% maximal number of iteraions
if nargin < 6,
tmax = inf;
end
% indexes of pattens in the 1st and 2nd class
xinx1 = find(I == 1);
xinx2 = find(I == 2);
X1=X(:,xinx1); % patters from 1st class
X2=X(:,xinx2); % patters from 1st class
l1 = size(X1,2); % number os patterns
l2 = size(X2,2);
% make problem lin-separable in high dimensional space
if C ~= 0,
kadd = 1/(2*C);
else
kadd = 0;
end
% convex coeficients defining normal of the decision hyperplane
% (they correspond to the Lagrangian multiplyers).
s1 = zeros(l1, 1);
s2 = zeros(l2, 1);
% initial sol
s1(1)=1; % take the 1st pattern from the 1st class
s2(1)=1; % take the 2nd pattern from the 2st class
sol=0;
t = 0;
minXDA1=-inf;
minXDA2=-inf;
% -- Inicalization of temp. variables ---------------------------------
Di1 = zeros(l1,1);
Fi1 = zeros(l1,1);
for i=1:l1,
Di1(i) = kernel(X1(:,1),X1(:,i), ker,arg);
Fi1(i) = kernel(X2(:,1),X1(:,i), ker,arg);
end
Di1(1) = Di1(1) + kadd;
Di2 = zeros(l2,1);
Fi2 = zeros(l2,1);
for i=1:l2,
Di2(i) = kernel(X1(:,1),X2(:,i), ker,arg);
Fi2(i) = kernel(X2(:,1),X2(:,i), ker,arg);
end
Fi2(1) = Fi2(1)+kadd;
a = kernel(X1(:,1),X1(:,1),ker,arg) + kadd;
b = kernel(X2(:,1),X2(:,1),ker,arg) + kadd;
c = kernel(X1(:,1),X2(:,1),ker,arg );
% upper and lower bounds on the optimal margin
up=[];
lo=[];
% main cycle
while sol == 0 & tmax > t,
t = t + 1;
sol = 1;
% -- compute auxciliary variables --
if sqrt( a -2*c +b) <= 0,
% algorithm has converged to the zero vector --> classes overlap
sol = -1;
break;
end
[emin,inx1] = min(Di1-Fi1);
[gmin,inx2] = min(Fi2-Di2);
% projection x \in X_1 on (w_1 - w_2)
proj1 = (emin + b -c )/sqrt(a-2*c+b);
% projection x \in X_2 on (w_2 - w_1)
proj2 = (gmin + a - c)/sqrt(a-2*c+b);
% --- compute stop condition for the alpha1 (1st class) ------
% (proj1 < proj2) ~ the worst point will be used for update
if (proj1 < proj2) & (proj1 <= (sqrt(a-2*c+b) - epsilon)),
% -- Adaptation phase of vector alpha1 ----------------------------
k = (a - emin - c)/...
(a+kernel(X1(:,inx1),X1(:,inx1),ker,arg)+kadd-2*(Di1(inx1)-Fi1(inx1)) );
k = min( 1, k );
s1 = s1 * (1-k);
s1(inx1) = s1(inx1) + k;
sol = 0;
% -------------------------------------------------------------
a = a*(1-k)^2 + 2*(1-k)*k*Di1(inx1) + ...
k^2 * (kernel(X1(:,inx1),X1(:,inx1),ker,arg)+kadd );
c = c*(1-k) + k*Fi1(inx1);
for i=1:l1,
Di1(i) = Di1(i)*(1-k) + k*kernel(X1(:,i),X1(:,inx1),ker,arg);
end
Di1(inx1) = Di1(inx1) + k*kadd;
for i=1:l2,
Di2(i) = Di2(i)*(1-k) + k*kernel(X2(:,i),X1(:,inx1),ker,arg);
end
else
% --- compute stop condition for the alpha2 (2st class) ------
if proj2 <= (sqrt(a-2*c+b) - epsilon ),
% -- Adaptation phase ----------------------------------
k = (b - gmin -c)/...
(b+kernel(X2(:,inx2),X2(:,inx2),ker,arg)+kadd-2*(Fi2(inx2)-Di2(inx2)));
k = min( 1, k );
s2 = s2 * (1-k);
s2(inx2) = s2(inx2) + k;
sol = 0;
% ------------------------------------------------------
b = b*(1-k)^2 + 2*(1-k)*k*Fi2(inx2) + ...
k^2 * (kernel(X2(:,inx2),X2(:,inx2),ker,arg)+kadd );
c = c*(1-k) + k*Di2(inx2);
for i=1:l1,
Fi1(i) = (1-k)*Fi1(i) + k*kernel(X2(:,inx2),X1(:,i),ker,arg);
end
for i=1:l2,
Fi2(i) = (1-k)*Fi2(i) + k*kernel(X2(:,inx2),X2(:,i),ker,arg);
end
Fi2(inx2) = Fi2(inx2) + k*kadd;
end
end
% store up=||w||/2 and current margin m(w1-w2,theta) = min( m1, m2)
m = min([proj1,proj2]) - 0.5*sqrt(a-2*c+b);
up = [up,sqrt(a-2*c+b)/2];
lo = [lo,m ];
%% disp(sprintf('step = %d', t ));
end
if sol == 1 & (proj1 < 0 | proj2 < 0),
sol = -2;
end
% -- compute threshold -----------------------
% sqared margin in transfromed space
margin2 = a - 2*c + b;
% threshold after normalization
theta = (a-b)/margin2;
% solution (normal vect. in the transformed space) after normalization
s1=2*s1/margin2;
s2=2*s2/margin2;
% -- make SVM-like output --------------------
Alpha=zeros(1,l1+l2);
Alpha(xinx1)=s1;
Alpha(xinx2)=s2;
bias = -theta;
% -- compute margin -----------------------------------
if nargout >= 6,
margin = 0;
for i=find(Alpha ~= 0),
for j=find(Alpha ~= 0 ),
margin = margin + Alpha(i)*Alpha(j)*itosgn(I(i))*...
itosgn(I(j))* kernel(X(:,i),X(:,j),ker,arg);
end
end
margin = 2/sqrt(margin);
end
% overall number of used float point operations
flps=flops;
return;