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Your issue demonstrated in R with interesting solution Equity Risk Model Using PCA. Another useful link in Matlab by Nick Higham himself NCM implementation by Nick Higham written for Matlab. Another good discussion on shrinkage and other aspects of Correlation Adjustment

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Hi: I don't follow your question totally but I can comment on one aspect of it. ( So this is not an answer ). The ideas that A) stock returns are geometric brownian motion processes and B) that PCA captures some kind of similarity in stocks from two different sectors are pretty much two different things. A) comes from efficient markets theory where it is ...

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Here, I try to help a bit with the matrix norm question. Assume an $M$-dimensional multivariate normally distributed return vector $\widetilde{R}$. The covariance of these returns is $\Sigma$, and the expected returns are $\check{R}$. The probability density function of $R$ is  f(R;\check{R},\Sigma)=\left(2\pi\right)^{-\frac{M}{2}}\left|\Sigma\right|^{-\...

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Usually, when one talks about exponential smoothing, they talk about it's halflife. So, for example, suppose we exponentially smooth some quantity ( argument carries over to covariance matrix but I'd rather just rather consider the scalar quantity case ) and call the exponentially smoothed estimate $\hat{smth_t}.$ So, this means that we have: \$\hat{...

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In a physical system the eigenvectors represent "modes" of random "vibration" (random movement) of a system and the eigenvalues represent the variance (i.e. amplitude) of each of these modes. The eigenvalues added together equal the variance of the overall (combined) movements (the Decomposition Property). In the stock market the main mode (biggest ...

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From LEP's answer: there will be p−t zero eigenvalues whenever p>t and one zero eigenvalue whenever p=t. This is the main reason, your true covariance matrix will have p-t eigenvalues exactly 0. With computer arithmetic you'll have lots of eigenvalues around machine precision, usually about 10^-15. So there should not only be around 5K-6K zero ...

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I think that your problem can be solves by using another estimator for your covariance matrix. A so called shrinkage estimator leads to covariance matrix that is non-singular. Then a Cholesky decomposition should work (maybe there is even a short-cut in the shrinkage world, I will check alter on). The R package corpcor contains functions to perform ...

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