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consider:

$Y_1,\dots,Y_n \sim N(0,\sigma^2)$ iid.

If I have a high variance I will see a very heterogenous sample and a large bell around the mean. It is sufficient to look at $\sigma^2$ to have an idea about it.

Now consider:

$\mathbf{Y_1},\dots,\mathbf{Y_n} \sim N_p(\mathbf{0},\Sigma)$.

My question is simple:

How do I infer the 'heterogeneity' of samples by looking at the covariance matrix?

Is it possible to say something in general by looking for example at the determinant of the matrix (not considering the $p=2$ case)? Or should I just look at the single entries of the matrix?

Thank you in advance

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Heterogeneity is large variance? If so, then for $Y$ that is MVN with $\Sigma$ as a covariance matrix, its main diagonal is the variance of each component of $Y$, where the non-diagonal terms are covariance of $y_i$ and $y_j$, as such they can be very "large" due to high variance or/and strong correlation. Besides, any judgment as large or small depends on your application. Looking at its determinant can be useful for some purposes as $|\Sigma| = \prod_{i=1}^n\lambda_i$ when the eigenvalues can give you some indication of the dominant elements.

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  • $\begingroup$ I realized that my question is not well posed, because even I don't know what I mean with heterogeneity. My question would be more like: ok, in a univariate case, by looking at the variance I know how data are dispersed around mean. Is there something in the covariance matrix which gives me the same information? Nice insight about the fact that strong covariance components may be such because of high variance of each one of them even if not correlated. Could you expand about when the eigenvalues can give you some indication of the dominant elements? Thank you :) $\endgroup$ Commented Feb 18, 2017 at 16:53
  • $\begingroup$ You can read this explanation to understand the intuition behind it. $\endgroup$
    – V. Vancak
    Commented Feb 18, 2017 at 17:30

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