PCA generated initial matrix in gaussian and ellipse? - matlab

Question

I have to do PCA in Matlab for object recognition.
For now, I generate matrix randomly
[a,InputMatrix] = sort(rand(100,20)); %Rows=100 Columns=20
Average=mean(InputMatrix);
CovarianceMatrix= cov(InputMatrix);
%% Compute the Eigen Value and Eigen the Vector
[EigVector,EigValue] = eigs(Matlab_Covariance);
NewMatrix=(EigVector)*(EigValue)*(EigVector)';
e1=EigVector(:,1); % Get the all the row at the first column
e2=EigVector(:,2); % Get the all the row at the second column
%% Plotting The Matrix with Eigen Value and Eigen Vector
%creating all combinations of x and y coordinates
[x,y]=meshgrid(1:size(InputMatrix,2),1:size(InputMatrix,1)); % 2= Columns 1= Rows
x=x(:);
y=y(:);
%plotting values of A such that X-Y axis represent the column and row coordinates of A
%respectively. Z-axis represents the value at that coordinate.
scatter3(x,y,InputMatrix(:),30,'rx');
%plotting the mean at the center of the coordinate system
hold on;
scatter3(mean([1:size(InputMatrix,2)]),mean([1:size(InputMatrix,1)]),
mean2(InputMatrix),60,'go','filled');
plot(e1,'k--');
plot(e2,'k--');
But if I perform PCA in that Random Matrix (InputMatrix), the shape of eigen vector e1 and e2 that I get for the PCA result will be wrong (when I plot them with InputMatrix in the same figure).
Someone told me that for the input matrix / data, it should be fulfill the condition (to be distributed in Normal Gaussian) and in elipse shape (when I plot it).
I think, I have to do rotation, scalling and other things to do it..
But I dont understand..
Could smeone please help me to generated random matrix with Normal Gaussian and in ellipse shape??
Please.. help me T_T

Answer 1

This can be achieved by multiplying hidden components matrix to noise vectors, i. e., using underlying ICA model. To get higher dimensionality just change cnum.
close all; clear all;
cnum = 2;
nnum = 500;
C = rand(cnum, cnum); % hidden components
N1 = sort(rand(cnum, nnum)); % sorted uniform noise
D1 = C * N1; % data
N2 = rand(cnum, nnum); % uniform noise
D2 = C * N2; % data
N3 = randn(cnum, nnum); % Gaussian noise
D3 = C * N3; % data
[V1, R1] = eig(cov(D3'));
[V2, R2] = eig(cov(D3'));
[V3, R3] = eig(cov(D3'));
subplot(1, 3, 1);
axis equal
hold on
plot(D1(1,:), D1(2,:), '.');
line([C(1, 1) 0 C(1, 2)], [C(2, 1) 0 C(2, 2)], 'Color', [1 .0 .0])
line([V1(1, 1) 0 V1(1, 2)], [V1(2, 1) 0 V1(2, 2)], 'Color', [.0 .0 .0])
title('Sorted uniform')
subplot(1, 3, 2);
axis equal
hold on
plot(D2(1,:), D2(2,:), '.');
line([C(1, 1) 0 C(1, 2)], [C(2, 1) 0 C(2, 2)], 'Color', [1 .0 .0])
line([V2(1, 1) 0 V2(1, 2)], [V2(2, 1) 0 V2(2, 2)], 'Color', [.0 .0 .0])
title('Uniform')
subplot(1, 3, 3);
axis equal
hold on
plot(D3(1,:), D3(2,:), '.');
line([C(1, 1) 0 C(1, 2)], [C(2, 1) 0 C(2, 2)], 'Color', [1 .0 .0])
line([V3(1, 1) 0 V3(1, 2)], [V3(2, 1) 0 V3(2, 2)], 'Color', [.0 .0 .0])
title('Gaussian')
print('-dpng', 'pca.png')
Red lines represent hidden components and black lines represent PCA components.