Covariance Matrix - Definition

Definition

Throughout this article, boldfaced unsubscripted X and Y are used to refer to random vectors, and unboldfaced subscripted Xi and Yi are used to refer to random scalars.

If the entries in the column vector

are random variables, each with finite variance, then the covariance matrix Σ is the matrix whose (i, j) entry is the covariance

\Sigma_{ij} = \mathrm{cov}(X_i, X_j) = \mathrm{E}\begin{bmatrix} (X_i - \mu_i)(X_j - \mu_j) \end{bmatrix}

where

\mu_i = \mathrm{E}(X_i)\,

is the expected value of the ith entry in the vector X. In other words, we have

\Sigma = \begin{bmatrix} \mathrm{E} & \mathrm{E} & \cdots & \mathrm{E} \\ \\ \mathrm{E} & \mathrm{E} & \cdots & \mathrm{E} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \mathrm{E} & \mathrm{E} & \cdots & \mathrm{E} \end{bmatrix}.

The inverse of this matrix, is the inverse covariance matrix, also known as the concentration matrix or precision matrix; see precision (statistics). The elements of the precision matrix have an interpretation in terms of partial correlations and partial variances.

Read more about this topic:  Covariance Matrix

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