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Vectors  ::  Covariance  ::  Functions  ::  Application  ::  Markov Model

Covariance

Suppose $\Xbf:(\Omega,\Fc) \rightarrow (\Rbb^n, \Bc^n)$ and $\Ybf:(\Omega,\Fc) \rightarrow (\Rbb^m, \Bc^m)$ (that is, $X$ and $Y$ are random vectors of dimension $n$ and $m$ , respectively).
\begin{definition}
\begin{displaymath}\cov(\Xbf,\Ybf) = E[(\Xbf - E[\Xbf]) (\Yb...
...1]  E[X_2]  \vdots  E[X_n]
\end{bmatrix}\end{displaymath}\end{definition}
Note: $\Sigma$ is frequently used as a symbol to denote covariance. It should not be confused with a summation sign, and is usually clear from context.

Property: $\cov(\Xbf,\Ybf) = [\cov(\Ybf,\Xbf)]^T.$

If $A$ is $\matsize{k}{n}$ and $B$ is $\matsize{l}{m}$ and $\abf \in
\Rbb^k$ and $b \in \Rbb^l$ then

\begin{displaymath}\cov(A\Xbf + \abf, B\Ybf + \bbf)= A \Sigma_{XY} B^T.
\end{displaymath}

$\cov(\Xbf,\Xbf) = \Sigma_X$ is called the ``covariance of $\Xbf$ .'' It is a symmetric matrix, non-negative definite (or positive semidefinite), and thus has all non-negative eigenvalues.

If $X_1, X_2, \ldots, X_n$ are mutually uncorrelated then

\begin{displaymath}\Sigma_X = \diag(\sigma_1^2, \sigma_2^2, \ldots, \sigma_n^2),
\end{displaymath}

where $\sigma_k^2 = \var(X_k).$

Suppose we partition $\Xbf$ of $n$ dimensions as

\begin{displaymath}\Xbf = \begin{bmatrix}\Xbf^{(1)}  \Xbf^{(2)}
\end{bmatrix}\end{displaymath}

of $k$ and $n-k$ elements, respectively. let

\begin{displaymath}\mubf = E[X] = \begin{bmatrix}\mubf^{(1)}  \mubf^{(2)}
\end{bmatrix}\end{displaymath}

where $\mubf^{(1)} = E[\Xbf^{(1)}]$ and $\mubf^{(2)} =
E[\Xbf^{(2)}]$ . Similarly,

\begin{displaymath}\Sigma_X = \begin{bmatrix}
\Sigma_{11} & \Sigma_{12}  \Sigma_{21} & \Sigma_{22}
\end{bmatrix}\end{displaymath}

where

\begin{displaymath}\Sigma_{11} = \cov(\Xbf^{(1)}, \Xbf^{(1)}) \qquad \qquad
\Sig...
...2)}) \qquad \qquad
\Sigma_{22} = \cov(\Xbf^{(2)}, \Xbf^{(2)}),
\end{displaymath}

or, in general,

\begin{displaymath}\Sigma_{ij} = \cov(\Xbf^{(i)},\Xbf^{(j)})
\end{displaymath}

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Random Vectors. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec3_2.html. This work is licensed under a Creative Commons License Creative Commons License