Editing Covariance matrix (section)

== Partial covariance matrix ==
A covariance matrix with all non-zero elements tells us that all the individual random variables are interrelated. This means that the variables are not only directly correlated, but also correlated via other variables indirectly. Often such indirect, [[common-mode interference|common-mode]] correlations are trivial and uninteresting. They can be suppressed by calculating the partial covariance matrix, that is the part of covariance matrix that shows only the interesting part of correlations.

If two vectors of random variables <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> are correlated via another vector <math>\mathbf{I}</math>, the latter correlations are suppressed in a matrix<ref name="KrzMarAnd">W J Krzanowski "Principles of Multivariate Analysis" (Oxford University Press, New York, 1988), Chap. 14.4; K V Mardia, J T Kent and J M Bibby "Multivariate Analysis (Academic Press, London, 1997), Chap. 6.5.3; T W Anderson "An Introduction to Multivariate Statistical Analysis" (Wiley, New York, 2003), 3rd ed., Chaps. 2.5.1 and 4.3.1.</ref>
<math display="block">
\operatorname{K}_\mathbf{XY \mid I}
 = \operatorname{pcov}(\mathbf{X},\mathbf{Y} \mid \mathbf{I})
 = \operatorname{cov}(\mathbf{X},\mathbf{Y})
 - \operatorname{cov}(\mathbf{X},\mathbf{I})
   \operatorname{cov}(\mathbf{I},\mathbf{I})^{-1}
   \operatorname{cov}(\mathbf{I},\mathbf{Y}).
</math>
The partial covariance matrix <math>\operatorname{K}_\mathbf{XY \mid I}</math> is effectively the simple covariance matrix <math>\operatorname{K}_\mathbf{XY}</math> as if the uninteresting random variables <math>\mathbf{I}</math> were held constant.