Plot the 1st Principal Component of the 2D sample set. The script uses a standard 1PC procedure and do not canter the matrix as it recommnded.

Contents

## Get the data

The matrix A has objects in its rows and features (the function princomp uses this feature )

A =[ 2.66 30.00 1.00 44.63 3.00 40.33 1.98 50.50 1.23 60.00 2.49 34.55 1.89 55.50 ]; [m,n]=size(A);

## Normalize the matrix

In real projects it is needed to descale the data. Be careful, this precedure is not robust to outlier objects.

for j = 1:n A(:,j)=( A(:,j)-min(A(:,j)) ) / ( max(A(:,j))-min(A(:,j)) ); end % center the matrix, since the embedded princomp centers the matrix any way %for j = 1:n, A(:,j)=A(:,j)-mean(A(:,j)); end

## Caclulate the Singular Values Decomposition

[U,L,V] = svd(A); % put in 1/sqrt(n-1) later pc1 = V(:,1);% pc1 is the first right singular vector of A % so the principal components are the rows of the matrix A % the Matlab procedure 'princomp' center the matrix and then makes svd % coeff of the 'princomps' are the matrix V. norm_pc1 = pc1'*pc1; % just check it if (norm_pc1-1.0>1e-10), error('Norm of the 1PC is not 1!'); end

## Calculate projections

pr = [ A * pc1 .* pc1(1), A * pc1 .* pc1(2)]; % calculate projections q = sqrt(sum(pr.*pr,2)); % caclulate norms of projections, so-called integral indicators

## Check the residuals

Z = A*V; [q - abs(Z(:,1)) < 1e-10]' % q % the distance to the projections are equal to Z (rotated coordinates) % abs(Z(:,1))

ans = 1 1 1 1 1 1 1

## Plot the raw vectors projection to the 1-st PC

h = figure; % create a new figure xlabel Feature1 ylabel Feature2 set(gca,'PlotBoxAspectRatio',[1 1 1]'); % set 1:1 axis aspect ratio grid on % show the axis grid %plot([0,1,1,0,0],[0,0,1,1,0],'-g'); % green square [0,1]x[0,1]; hold on % keep all the painting on plot(A(:,1),A(:,2),'*k'); % plot the objects with asterisks % plot objects names (order numbers) for i = 1:m, text(A(i,1),A(i,2),[' Vec',num2str(i)]); end plot(-pc1(1),-pc1(2),'+r'); % the 1-st PC, vector plot([0,-pc1(1)],[0,-pc1(2)],'-r'); % the same, plot a line from (0.0) to th 1PC text(-pc1(1),-pc1(2),' 1-st PC'); % show the text about it for i=1:m % m objects to plot % TO DO plot([A(i,1),pr(i,1)],[A(i,2),pr(i,2)],'b-'); text(pr(i,1),pr(i,2),sprintf(' %0.2f',q(i))); plot(pr(i,1),pr(i,2),'.k'); end title('1-st Principal Component'); %saveas(h, 'demo_principal_component_1','png') close(h);

```
% this file: demo_principal_component
```