本文是一個用Matlab實現的簡單的SVM實例，僅供參考，如有不足之處，歡迎指正。
主程序如下。
% simple SMO with 2D data
clear; close all; clc;
% generate two 'circles' of diameters 20 and 40, each consisting of 100 points
[X1, y1] = generateData(100, 2, 10, 1);
[X2, y2] = generateData(100, 2, 20, -1);
X = [X1; X2];
y = [y1; y2];
m = size(X,1);
xPos = X(y == 1,:);
xNeg = X(y == -1,:);
scatter(xPos(:,1),xPos(:,2),'y+');
hold on
scatter(xNeg(:,1),xNeg(:,2),'k+');
axis equal
hold off
% initialization of lagrange multipliers
alphas = zeros(m,1);
b = 0;
% punishment factor C
C = 1;
% update of lagrange multipliers
% the loop stops when alphas remain unchange during 5 iterations
% if |alpha_new - alpha| < delta, we don't update alpha
% num_updated = number of pairs of alpha updated during one loop
delta = 1e-10;
count_not_updated = 0;
iter = 0;
while(count_not_updated < 6)
%w = zeros(1,2);
w = (y .* alphas)' * X
b
[alphas, b, num_updated] = update_alphas(alphas, b, X, y, C, delta);
if(num_updated == 0)
count_not_updated = count_not_updated + 1;
else
count_not_updated = 0;
end
iter = iter + 1;
fprintf('iteration = %d\n',iter);
fprintf('count_not_updated = %d\n', count_not_updated);
end
% calculate w
w = (y .* alphas)' * X;
% test
[res1, res2, res3, res4, res5] = test_svm(X, y, alphas, b);
function [alphas_new, b_new, updated] = update_alphas(alphas, b, X, y, C, delta)
alphas_new = alphas;
b_new = b;
updated = 0;
m = size(X,1);
index1 = 0;
index2 = 0;
% update alphas
for index = 1:1:m
u2 = calculate_u(alphas_new, b_new, X, y, X(index,1:2));
bool_2 = is_kkt(index, y, u2, alphas_new, C);
if bool_2 == 0
index2 = index;
E2 = u2 - y(index2);
diff_E = 0;
for indexx = (index2 + 1):1:m
u1 = calculate_u(alphas_new, b_new, X, y, X(indexx,1:2));
bool_1 = is_kkt(indexx, y, u1, alphas_new, C);
if bool_1 == 0
E1 = u1 - y(indexx);
dif = abs(E2 - E1);
if dif > diff_E
diff_E = dif;
index1 = indexx;
end
end
end
if index1 == 0
index1 = index2 + 1;
end
[alpha1, alpha2, b_new, bool] = update_alpha_pair(index1,index2, alphas_new, b_new, X, y, C, delta);
if bool == 1
updated = updated + 1;
alphas_new(index1) = alpha1;
alphas_new(index2) = alpha2;
end
if bool == 0
continue;
end
end
end
endfunction [alpha1, alpha2, b, bool] = update_alpha_pair(index_a1,index_a2, alphas, b_, X, y, C, delta)
alpha1 = alphas(index_a1);
alpha2 = alphas(index_a2);
b = b_;
if index_a1 == index_a2
bool = 0;
return;
end
[L, H] = calculateLH(index_a1, index_a2, alphas, y ,C);
eta = calculateLs(index_a1, index_a2, X); %% relation with the kernel ?
x1 = X(index_a1,1:2);
x2 = X(index_a2,1:2);
y1 = y(index_a1);
y2 = y(index_a2);
u1 = calculate_u(alphas, b_, X, y, x1);
u2 = calculate_u(alphas, b_, X, y, x2);
E1 = u1 - y1;
E2 = u2 - y2;
% update alpha2
% eta > 0 ?
if eta > 0
alpha2 = alphas(index_a2) + y2 * (E1 - E2) / eta;
if alpha2 > H
alpha2 = H;
else if alpha2 < L
alpha2 = L;
end
end
else
bool = 0;
return;
end
if abs(alphas(index_a2)-alpha2) < delta
bool = 0;
return;
end
alpha1 = alphas(index_a1) + y1 * y2 * (alphas(index_a2) - alpha2);
% update b
b1 = b_ - E1 - y1 * (alpha1 - alphas(index_a1)) * ker(x1,x1) - y2 * (alpha2 - alphas(index_a2)) * ker(x1,x2);
b2 = b_ - E2 - y1 * (alpha1 - alphas(index_a1)) * ker(x1,x2) - y2 * (alpha2 - alphas(index_a2)) * ker(x2,x2);
b = update_b(alpha1, alpha2, b1, b2, C);
bool = 1;
end下面這個函數用於計算輸出函數u。
function u = calculate_u(alphas, b, X, y, x)
u = 0;
m = size(X,1);
Xk = zeros(m,1);
for index = 1:1:m
Xk(index) = ker(X(index,1:2),x);
end
% prediction of x using alphas and b that we found
u = (alphas .* y)' * Xk + b;
end下面這個函數用於計算拉格朗日乘子的上下邊界。
function [L, H] = calculateLH(index1, index2, alphas, y ,C)
% calculate the borders of alpha
if y(index1) == y(index2)
L = max(0, alphas(index2) + alphas(index1) - C);
H = min(C, alphas(index2) + alphas(index1));
else
L = max(0, alphas(index2) - alphas(index1));
H = min(C, C + alphas(index2) - alphas(index1));
end
end
下面這個函數利用核函數計算二階導。
function Ls = calculateLs(index1, index2, X)
x1 = X(index1,:);
x2 = X(index2,:);
Ls = ker(x1, x1) + ker(x2, x2) - 2 * ker(x1, x2);
end
下面這個函數更新參數b。
function b = update_b(alpha1, alpha2, b1, b2, C)
if 0 < alpha1 && alpha1 < C
b = b1;
else if 0 < alpha2 && alpha2 < C
b = b2;
else
b = (b1 + b2)/2;
end
end
end下面這個函數用於判斷拉格朗日乘子是否滿足KKT條件。
function bool = is_kkt(index, y, u, alphas, C)
bool = 1;
% three cases where the lagrange multiplier does not satisfy the kkt
% condition
if y(index)*u <= 1 && alphas(index) < C
bool = 0;
end
if y(index)*u >= 1 && alphas(index) > 0
bool = 0;
end
if (y(index)*u == 1 && alphas(index) == 0) || (y(index)*u == 1 && alphas(index) == C)
bool = 0;
end
end
下面這個函數用於定義核函數。
function kernel = ker(x1, x2)
% kernel function
% kernel = x1 * x2' + (x1.^2) * (x2.^2)' + x1(1) * x1(2) * x2(1) * x2(2);
kernel = (x1 * x2' + 1)^2;
end下面這個函數用於測試。
function [res1, res2, res3, res4, res5] = test_svm(X, y, alphas, b)
m = size(X,1);
[Xt1, ~] = generateData(100, 2, 5, 0);
[Xt2, ~] = generateData(100, 2, 15, 0);
[Xt3, ~] = generateData(100, 2, 40, 0);
% Wrong test example
%res1 = Xt1 * w' + b;
%res2 = X(1:fix(m/2),:) * w' + b;
%res3 = Xt2 * w' + b;
%res4 = X(fix(m/2)+1:m,:) * w' + b;
%res5 = Xt3 * w' + b;
res1 = zeros(100,1);
res2 = zeros(100,1);
res3 = zeros(100,1);
res4 = zeros(100,1);
res5 = zeros(100,1);
for i = 1:100
res1(i) = calculate_u(alphas, b, X, y, Xt1(i,:));
res2(i) = calculate_u(alphas, b, X, y, X(i,:));
res3(i) = calculate_u(alphas, b, X, y, Xt2(i,:));
res4(i) = calculate_u(alphas, b, X, y, X(100+i,:));
res5(i) = calculate_u(alphas, b, X, y, Xt3(i,:));
end
end
下面這個函數用於生成訓練和測試所用的數據。
function [data, labels] = generateData(m, n, r, label)
% we add some perturbations to the circle
perturbation = 1;
data = zeros(m, n);
labels = ones(m, 1) .* label;
for i = 1:m;
data(i,1) = 2*r*rand - r;
data(i,2) = (sqrt(r^2 - data(i,1)^2)) * (round(rand)*2-1) + rand * perturbation;
%labels(i) = label;
end
end生成的數據大致如下圖所示。
參考資料：
Sequential Minimal Optimization: A Fast Algorithm for Training Support Vector Machines, John C. Platt.
支持向量機通俗導論——理解SVM 的三層境界，July · pluskid (http://blog.csdn.net/v_july_v/article/details/7624837)
