I'm studying multivariate analysis for self-taught. In particular I have doubts and I realize when I do the exercises. I read and reread the theory ...
On a sample of 48 units, variables were measured.
The data correlation matrix has eigenvalues equal to
$ (2.609,0.923,0.396.0.072) $
and eigenvectors equal to:
$ (- 0.474 - 0.605 +0.376 + 0.517 $
$ - 0.589 -0237 -0.063 -0.770$
$ + 0.393 -0.705 -0.589 -0.036 $
$ + 0.523 -0.284 + 0.713 - 0.371) $
1) The standardized values observed on the first unit are $ (- 0.819,0.077, -1.530, -0.935) $ to determine all the main components related to this observation; 2) Which starting variable is most correlated with the first main component? 3) Use all methods to determine the number of main components to keep.
My solutions: 1) The analysis of the main components was done by spectral decomposition of the correlation matrix. To find the values of the main components: from the theory I read that the main components are the linear combinations of the starting variables $ y_i = a_1 x1 + ... + a_n x_n $. Having said this, having the eigenvalues and the eigenvectors, my $ a $ what are they? 2) This just can not understand it 3) Having the variances placed in descending order on the main diagonal of the eigenvalue matrix, to find the standard deviations take the second then: $ (- 0.474) ^ 2 $ $ (- 0.237) ^ 2 $ etc. Having standard deviations, I can trace the scree plot and see the elbow. The second way I can find the standard deviation proportions:
$ (- 0.474) ^ 2 + (-0.237) ^ 2 + (-0.589) ^ 2 + (- 0.371) ^ 2 $ = $0.7654$
Standard deviation (diagonal eigenvalues at the second): 0.2248 0.0562 0.3469 0.1376 Proportion (eigenvalues diagonal to the second divided 0.7654): 0.2937 0.0734 0.4532 0.1798 Cumulative: 0.2937 0.3671 0.8203 1
I find a threshold c = 0.7654 / n ° main components then I do 0.2248> c 0.0562> c ... I hold only the major ones
I would like clarification, I have also read that the main components can also derive from the decomposition into singular values. Thank you.