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Let's assume we have 4 independent random vectors that have values $\in [0.0, 1.0]$ and a lot of elements. Label them: $\{r_1,r_2,r_3,r_4\}$. Now, let's consider a simple case of 3 variables and try understanding the relation between them. Consider two distinct sets:

$S_1 = \{ r_1 + r_2 , r_1 + r_3, r_1 + r_4\}$

and

$S_1 = \{ r_1 + r_2 , r_2 + r_3, r_3 + r_1\}$

Both sets have identical covariance matrices because each pair has only one common random vector:

$ Cov(S_1) = Cov(S_2) = \left( \begin{array}{ccc} 1.0 & 0.5 & 0.5 \\ 0.5 & 1.0 & 0.5 \\ 0.5 & 0.5 & 1.0 \\ \end{array} \right) $

but obviously the relationship between the vectors in a set are very different. How to quantify and estimate the relationship between all the vectors in a set?

PS: this question if a follow up from my previous question on covariance matrices.

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Karolis
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  • What does it mean for a vector to have values in $[0.0,1.0]$? What does it mean for a vector (or something else?) to have "a lot of elements"? – Dilip Sarwate Feb 15 '16 at 23:34
  • Hey @DilipSarwate, i meant that random vectors $r_i$ have some arbitrary large number of elements (dimensions) such that the errors on our $Cov$ estimates are small. By range $[0,1]$, I mean that all the elements within the vector are within that range. I don't think it changes anything, but it is easier for me to understand if I think about specific example. thanks! – Karolis Feb 15 '16 at 23:53

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