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建立smo.m
% function [alpha,bias] = smo(X, y, C, tol) function model = smo(X, y, C, tol) % SMO: SMO algorithm for SVM % %Implementation of the Sequential Minimal Optimization (SMO) %training algorithm for Vapnik‘s Support Vector Machine (SVM) % % This is a modified code from Gavin Cawley‘s MATLAB Support % Vector Machine Toolbox % (c) September 2000. % % Diego Andres Alvarez. % % USAGE: [alpha,bias] = smo(K, y, C, tol) % % INPUT: % % K: n x n kernel matrix % y: 1 x n vector of labels, -1 or 1 % C: a regularization parameter such that 0 <= alpha_i <= C/n % tol: tolerance for terminating criterion % % OUTPUT: % % alpha: 1 x n lagrange multiplier coefficient % bias: scalar bias (offset) term % Input/output arguments modified by JooSeuk Kim and Clayton Scott, 2007 global SMO; y = y‘; ntp = size(X,1); %recompute C % C = C/ntp; %initialize ii0 = find(y == -1); ii1 = find(y == 1); i0 = ii0(1); i1 = ii1(1); alpha_init = zeros(ntp, 1); alpha_init(i0) = C; alpha_init(i1) = C; bias_init = C*(X(i0,:)*X(i1,:)‘ -X(i0,:)*X(i1,:)‘) + 1; %Inicializando las variables SMO.epsilon = 10^(-6); SMO.tolerance = tol; SMO.y = y‘; SMO.C = C; SMO.alpha = alpha_init; SMO.bias = bias_init; SMO.ntp = ntp; %number of training points %CACHES: SMO.Kcache = X*X‘; %kernel evaluations SMO.error = zeros(SMO.ntp,1); %error numChanged = 0; examineAll = 1; %When all data were examined and no changes done the loop reachs its %end. Otherwise, loops with all data and likely support vector are %alternated until all support vector be found. while ((numChanged > 0) || examineAll) numChanged = 0; if examineAll %Loop sobre todos los puntos for i = 1:ntp numChanged = numChanged + examineExample(i); end; else %Loop sobre KKT points for i = 1:ntp %Solo los puntos que violan las condiciones KKT if (SMO.alpha(i)>SMO.epsilon) && (SMO.alpha(i)<(SMO.C-SMO.epsilon)) numChanged = numChanged + examineExample(i); end; end; end; if (examineAll == 1) examineAll = 0; elseif (numChanged == 0) examineAll = 1; end; end; alpha = SMO.alpha‘; alpha(alpha < SMO.epsilon) = 0; alpha(alpha > C-SMO.epsilon) = C; bias = -SMO.bias; model.w = (y.*alpha)* X; %%%%%%%%%%%%%%%%%%%%%% model.b = bias; return; function RESULT = fwd(n) global SMO; LN = length(n); RESULT = -SMO.bias + sum(repmat(SMO.y,1,LN) .* repmat(SMO.alpha,1,LN) .* SMO.Kcache(:,n))‘; return; function RESULT = examineExample(i2) %First heuristic selects i2 and asks to examineExample to find a %second point (i1) in order to do an optimization step with two %Lagrange multipliers global SMO; alpha2 = SMO.alpha(i2); y2 = SMO.y(i2); if ((alpha2 > SMO.epsilon) && (alpha2 < (SMO.C-SMO.epsilon))) e2 = SMO.error(i2); else e2 = fwd(i2) - y2; end; % r2 < 0 if point i2 is placed between margin (-1)-(+1) % Otherwise r2 is > 0. r2 = f2*y2-1 r2 = e2*y2; %KKT conditions: % r2>0 and alpha2==0 (well classified) % r2==0 and 0% r2<0 and alpha2==C (support vectors between margins) % % Test the KKT conditions for the current i2 point. % % If a point is well classified its alpha must be 0 or if % it is out of its margin its alpha must be C. If it is at margin % its alpha must be between 0%take action only if i2 violates Karush-Kuhn-Tucker conditions if ((r2 < -SMO.tolerance) && (alpha2 < (SMO.C-SMO.epsilon))) || ... ((r2 > SMO.tolerance) && (alpha2 > SMO.epsilon)) % If it doens‘t violate KKT conditions then exit, otherwise continue. %Try i2 by three ways; if successful, then immediately return 1; RESULT = 1; % First the routine tries to find an i1 lagrange multiplier that % maximizes the measure |E1-E2|. As large this value is as bigger % the dual objective function becames. % In this first test, only support vectors will be tested. POS = find((SMO.alpha > SMO.epsilon) & (SMO.alpha < (SMO.C-SMO.epsilon))); [MAX,i1] = max(abs(e2 - SMO.error(POS))); if ~isempty(i1) if takeStep(i1, i2, e2), return; end; end; %The second heuristic choose any Lagrange Multiplier that is a SV and tries to optimize for i1 = randperm(SMO.ntp) if (SMO.alpha(i1) > SMO.epsilon) & (SMO.alpha(i1) < (SMO.C-SMO.epsilon)) %if a good i1 is found, optimise if takeStep(i1, i2, e2), return; end; end end %if both heuristc above fail, iterate over all data set for i1 = randperm(SMO.ntp) if ~((SMO.alpha(i1) > SMO.epsilon) & (SMO.alpha(i1) < (SMO.C-SMO.epsilon))) if takeStep(i1, i2, e2), return; end; end end; end; %no progress possible RESULT = 0; return; function RESULT = takeStep(i1, i2, e2) % for a pair of alpha indexes, verify if it is possible to execute % the optimisation described by Platt. global SMO; RESULT = 0; if (i1 == i2), return; end; % compute upper and lower constraints, L and H, on multiplier a2 alpha1 = SMO.alpha(i1); alpha2 = SMO.alpha(i2); y1 = SMO.y(i1); y2 = SMO.y(i2); C = SMO.C; K = SMO.Kcache; s = y1*y2; if (y1 ~= y2) L = max(0, alpha2-alpha1); H = min(C, alpha2-alpha1+C); else L = max(0, alpha1+alpha2-C); H = min(C, alpha1+alpha2); end; if (L == H), return; end; if (alpha1 > SMO.epsilon) & (alpha1 < (C-SMO.epsilon)) e1 = SMO.error(i1); else e1 = fwd(i1) - y1; end; %if (alpha2 > SMO.epsilon) & (alpha2 < (C-SMO.epsilon)) % e2 = SMO.error(i2); %else % e2 = fwd(i2) - y2; %end; %compute eta k11 = K(i1,i1); k12 = K(i1,i2); k22 = K(i2,i2); eta = 2.0*k12-k11-k22; %recompute Lagrange multiplier for pattern i2 if (eta < 0.0) a2 = alpha2 - y2*(e1 - e2)/eta; %constrain a2 to lie between L and H if (a2 < L) a2 = L; elseif (a2 > H) a2 = H; end; else %When eta is not negative, the objective function W should be %evaluated at each end of the line segment. Only those terms in the %objective function that depend on alpha2 need be evaluated... ind = find(SMO.alpha>0); aa2 = L; aa1 = alpha1 + s*(alpha2-aa2); Lobj = aa1 + aa2 + sum((-y1*aa1/2).*SMO.y(ind).*K(ind,i1) + (-y2*aa2/2).*SMO.y(ind).*K(ind,i2)); aa2 = H; aa1 = alpha1 + s*(alpha2-aa2); Hobj = aa1 + aa2 + sum((-y1*aa1/2).*SMO.y(ind).*K(ind,i1) + (-y2*aa2/2).*SMO.y(ind).*K(ind,i2)); if (Lobj>Hobj+SMO.epsilon) a2 = H; elseif (Lobj<Hobj-SMO.epsilon) a2 = L; else a2 = alpha2; end; end; if (abs(a2-alpha2) < SMO.epsilon*(a2+alpha2+SMO.epsilon)) return; end; % recompute Lagrange multiplier for pattern i1 a1 = alpha1 + s*(alpha2-a2); w1 = y1*(a1 - alpha1); w2 = y2*(a2 - alpha2); %update threshold to reflect change in Lagrange multipliers b1 = SMO.bias + e1 + w1*k11 + w2*k12; bold = SMO.bias; if (a1>SMO.epsilon) & (a1<(C-SMO.epsilon)) SMO.bias = b1; else b2 = SMO.bias + e2 + w1*k12 + w2*k22; if (a2>SMO.epsilon) & (a2<(C-SMO.epsilon)) SMO.bias = b2; else SMO.bias = (b1 + b2)/2; end; end; % update error cache using new Lagrange multipliers SMO.error = SMO.error + w1*K(:,i1) + w2*K(:,i2) + bold - SMO.bias; SMO.error(i1) = 0.0; SMO.error(i2) = 0.0; % update vector of Lagrange multipliers SMO.alpha(i1) = a1; SMO.alpha(i2) = a2; %report progress made RESULT = 1; return;
画图文件:start_SMOforSVM.m(点击自动生成二维两类数据,画图,这里只是线性的,非线性的可以对应修改)
clear X = []; Y=[]; figure; % Initialize training data to empty; will get points from user % Obtain points froom the user: trainPoints=X; trainLabels=Y; clf; axis([-5 5 -5 5]); if isempty(trainPoints) % Define the symbols and colors we‘ll use in the plots later symbols = {‘o‘,‘x‘}; classvals = [-1 1]; trainLabels=[]; hold on; % Allow for overwriting existing plots xlim([-5 5]); ylim([-5 5]); for c = 1:2 title(sprintf(‘Click to create points from class %d. Press enter when finished.‘, c)); [x, y] = getpts; plot(x,y,symbols{c},‘LineWidth‘, 2, ‘Color‘, ‘black‘); % Grow the data and label matrices trainPoints = vertcat(trainPoints, [x y]); trainLabels = vertcat(trainLabels, repmat(classvals(c), numel(x), 1)); end end % C = 10;tol = 0.001; % par = SMOforSVM(trainPoints, trainLabels , C, tol ); % p=length(par.b); m=size(trainPoints,2); % if m==2 % % for i=1:p % % plot(X(lc(i)-l(i)+1:lc(i),1),X(lc(i)-l(i)+1:lc(i),2),‘bo‘) % % hold on % % end % k = -par.w(1)/par.w(2); % b0 = - par.b/par.w(2); % bdown=(-par.b-1)/par.w(2); % bup=(-par.b+1)/par.w(2); % for i=1:p % hold on % h = refline(k,b0(i)); % set(h, ‘Color‘, ‘r‘) % hdown=refline(k,bdown(i)); % set(hdown, ‘Color‘, ‘b‘) % hup=refline(k,bup(i)); % set(hup, ‘Color‘, ‘b‘) % end % end % xlim([-5 5]); ylim([-5 5]); % % pause C = 10;tol = 0.001; par = smo(trainPoints, trainLabels, C, tol); p=length(par.b); m=size(trainPoints,2); if m==2 % for i=1:p % plot(X(lc(i)-l(i)+1:lc(i),1),X(lc(i)-l(i)+1:lc(i),2),‘bo‘) % hold on % end k = -par.w(1)/par.w(2); b0 = - par.b/par.w(2); bdown=(-par.b-1)/par.w(2); bup=(-par.b+1)/par.w(2); for i=1:p hold on h = refline(k,b0(i)); set(h, ‘Color‘, ‘r‘) hdown=refline(k,bdown(i)); set(hdown, ‘Color‘, ‘b‘) hup=refline(k,bup(i)); set(hup, ‘Color‘, ‘b‘) end end xlim([-5 5]); ylim([-5 5]);
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原文地址:http://www.cnblogs.com/huadongw/p/4994657.html