how to find percentage of total variation from population covariance matrix and population mean matrix

Question

I have a matrix with dimension nxp. No:of observations are not known. But no: of variables is 3. So p=3. Population mean is defined as µ1=1,µ2=-1,µ3=2.

$\sum$=

\begin{bmatrix} 1 & k & 0\\ k & 1 & k\\ 0 & k & 1\end{bmatrix} How to find the value of k such that the principal components PC1 and PC2 account for more than some percentage (say 75%) of total variation of X?

It looks like Sigma is a correlation matrix not a covariance matrix because of the ones on the diagonal. Could you precise what is X ? — manu190466, Apr 10 '20 at 07:19
X = (X1, X2, X3) distributed as N3(µ, Σ). I did determinant( Σ - lamda * I ) = 0. To get value of k in terms of lamda. — StatsMonkey, Apr 10 '20 at 11:52
When you say "they account ..." what would you mean exactly ? The two first eigenvectors ? Please edit your question to give more details. — manu190466, Apr 10 '20 at 12:28

manu190466 · Accepted Answer · 2020-04-10T15:16:24.423

0

Given that the cumulative ratio of variance explained is equal to the sum of the reduced eigenvalues (see 1 for details), you have to

calculate the three eigenvalues of your matrix in function of k
reduce and sum the two first eigenvalues
find the value of k that gives you the needed percentage (ie : 0.75)

edited Apr 10 '20 at 15:16

answered Apr 10 '20 at 13:35

manu190466

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I think we have to find all the eigenvalues of the matrix to find sum of the reduced eigenvalues. (λ1+λ2) / (λ1+λ2+λ3) – StatsMonkey Apr 10 '20 at 15:07
I didn't mentionned it but you are right ! Answer edited. – manu190466 Apr 10 '20 at 15:14

how to find percentage of total variation from population covariance matrix and population mean matrix

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