Principal Component Analysis

1. 範例：指定五個點做 PCA。可以看清楚哪個點打到哪個點。

A = [1 2 -2 0 1; 1 -1.3 0 2 -1];

H = eye(5)-0.2*ones(5);

C = A*H;

B = cov(C(1,:),C(2,:),1); % this cov uses n-1 as denominator

[V,L] = eig(B);

L = real(diag(L));

[L,I] = sort(L,'descend');

X = V'*C;

Y = V(:,I(1))*X(I(1),:);

% fix ratio to see orthogonal projection

a = max([abs(C(1,:)) abs(C(2,:)) abs(Y(1,:)) abs(Y(2,:))]);

axis([-a a -a a]);

pbaspect([1 1 1]);

hold on

for i = 1:5

plot(C(1,i),C(2,i), '.', 'MarkerSize', 18, 'Color',[0 0 1]);

plot(Y(1,i),Y(2,i), '.', 'MarkerSize', 18, 'Color',[1 0 0]);

line([C(1,i);Y(1,i)],[C(2,i);Y(2,i)]);

if i < 5

line([Y(1,i);Y(1,i+1)],[Y(2,i);Y(2,i+1)]);

end

end

2. PCA函數(d = 2 or 3)

function Y = pca(X,d)

m = size(X,1);

n = size(X,2);

H = eye(n)-(1/n)*ones(n);

C = X*H;

B = C*C'; % this cov uses n-1 as denominator

[V,L] = eig(B);

L = real(diag(L));

[~,I] = sort(L,'descend');

D = V'*C;

Y = zeros(m,n);

Y(1:d,:) = D(I(1:d),:);

hold on;

if d == 2

for i = 1:n

plot(Y(1,i),Y(2,i),'.', 'MarkerSize', 12, 'Color',[0 i/n i/n]);

end

else

for i = 1:n

plot3(Y(1,i),Y(2,i),Y(3,i),'.', 'MarkerSize', 12, 'Color',[0 i/n i/n]);

end

end

3.如何使用 PCA 函數？

打開 matlab，輸入 swiss roll 或其他現成資料矩陣 X，然後執行 pca(X,3); 即可。

ps. 可用 scatter 指令取代 plot。
ps. 當然可以用 svd 來寫，此處是配合教學所以用 eigen-decomposition.