How to find the weight of PC_1 in my measurements, after doing PCA?

Hi,

I'm trying to use the following code to understand PCA , SVD and it's releation:

% PCA_vs_SVD (my sand_box)

%% generate fake data points
    my_fake_dataPoints = [-4 0 ; -2 1 ; -1 -1 ; 1 1 ; 3 2 ; 4 2];

    % remove mean
        my_fake_dataPoints_noMean = my_fake_dataPoints - mean(my_fake_dataPoints , 2);

% do PCA
    [coeff        , score        , latent        , tsquared        , explained        , mu]        = pca(my_fake_dataPoints);
    [coeff_noMean , score_noMean , latent_noMean , tsquared_noMean , explained_noMean , mu_noMean] = pca(my_fake_dataPoints_noMean);

% do SVD
    [U        , S        , V]        = svd(my_fake_dataPoints);    
    [U_noMean , S_noMean , V_noMean] = svd(my_fake_dataPoints_noMean);

%% plots
figure(1)
    biplot(coeff , 'scores' , score , 'MarkerSize' , 30 , 'varlabels' ,{'var_1' , 'var_2'});

figure(2)
    scatter(score(:,1) , score(:,2))
    axis equal  
    xlabel('1st Principal Component')
    ylabel('2nd Principal Component')
    grid on
    

Have some questions:

1) How can I know the weight of PC_1 in my measurments? is it simply first column fo "score", or something else?

2) What's exactly the connection between the output of PCA and SVD? Which case should I compare, standard? with mean subtraction?

3) Am I missing something in the following:

PC1 = alpha_1 * v1 + alpha_2 * v2, right? my alphas are the first column of "coeff" variable, right?

So, the 1st data point projected on PC1 should be: 0.95 * (-4) + 0.28 * (0) = -3.8, right? But it doesn't match score(1,1), which is -4.23... what am I missing here?

Answer 1

[Githin George]

Hello Mark,

I understand you have a few doubts related to PCA. To answer your queries:

1. The "explained"/ "explained_noMean" variable contains the percentage weight of data, captured by each of the Principal Component (PC_1 and PC_2 in this case).
2. PCA with standardized data yields the same result as doing SVD. I suggest you refer to the following answer know more: https://www.mathworks.com/matlabcentral/answers/774902-pca-vs-svd-or-eig-functions?s_tid=srchtitle_site_search_1_pca%20vs%20svd
3. The equation "PC1 = alpha_1 * v1 + alpha_2 * v2" gives the projected value of original data point (v1,v2), on the principal axis. But note that "score" is a measure of correlation of data points to the corresponding PC. It does not equal to the projected value in the Principal Component.

I hope this addresses your queries.