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

Famous quotes containing the word definition:

    “The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
    —William James (1842–1910)

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)

    “Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.”
    —Willard Van Orman Quine (b. 1908)