Misplaced Pages

Part of a series on Statistics
Correlation and covariance
For random vectors Autocorrelation matrix Cross-correlation matrix Auto-covariance matrix Cross-covariance matrix
For stochastic processes Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
For deterministic signals Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
v t e

In probability theory and statistics, a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random vector and j-th element of another random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has either a finite number of observed empirical values or a finite or infinite number of potential values. The potential values are specified by a theoretical joint probability distribution. Intuitively, the cross-covariance matrix generalizes the notion of covariance to multiple dimensions.

The cross-covariance matrix of two random vectors ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ is typically denoted by ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}$ or ${\displaystyle \Sigma _{\mathbf {X} \mathbf {Y} }}$.

Definition

For random vectors ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$, each containing random elements whose expected value and variance exist, the cross-covariance matrix of ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ is defined by

${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} ){\stackrel {\mathrm {def} }{=}}\ \operatorname {E} }$

(Eq.1)

where ${\displaystyle \mathbf {\mu _{X}} =\operatorname {E} }$ and ${\displaystyle \mathbf {\mu _{Y}} =\operatorname {E} }$ are vectors containing the expected values of ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$. The vectors ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ need not have the same dimension, and either might be a scalar value.

The cross-covariance matrix is the matrix whose ${\displaystyle (i,j)}$ entry is the covariance

${\displaystyle \operatorname {K} _{X_{i}Y_{j}}=\operatorname {cov} =\operatorname {E} )(Y_{j}-\operatorname {E} )]}$

between the i-th element of ${\displaystyle \mathbf {X} }$ and the j-th element of ${\displaystyle \mathbf {Y} }$. This gives the following component-wise definition of the cross-covariance matrix.

${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }={\begin{bmatrix}\mathrm {E} )(Y_{1}-\operatorname {E} )]&\mathrm {E} )(Y_{2}-\operatorname {E} )]&\cdots &\mathrm {E} )(Y_{n}-\operatorname {E} )]\\\\\mathrm {E} )(Y_{1}-\operatorname {E} )]&\mathrm {E} )(Y_{2}-\operatorname {E} )]&\cdots &\mathrm {E} )(Y_{n}-\operatorname {E} )]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} )(Y_{1}-\operatorname {E} )]&\mathrm {E} )(Y_{2}-\operatorname {E} )]&\cdots &\mathrm {E} )(Y_{n}-\operatorname {E} )]\end{bmatrix}}}$

Example

For example, if ${\displaystyle \mathbf {X} =\left(X_{1},X_{2},X_{3}\right)^{\rm {T}}}$ and ${\displaystyle \mathbf {Y} =\left(Y_{1},Y_{2}\right)^{\rm {T}}}$ are random vectors, then ${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}$ is a ${\displaystyle 3\times 2}$ matrix whose ${\displaystyle (i,j)}$-th entry is ${\displaystyle \operatorname {cov} (X_{i},Y_{j})}$.

Properties

For the cross-covariance matrix, the following basic properties apply:

1. ${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=\operatorname {E} -\mathbf {\mu _{X}} \mathbf {\mu _{Y}} ^{\rm {T}}}$
2. ${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=\operatorname {cov} (\mathbf {Y} ,\mathbf {X} )^{\rm {T}}}$
3. ${\displaystyle \operatorname {cov} (\mathbf {X_{1}} +\mathbf {X_{2}} ,\mathbf {Y} )=\operatorname {cov} (\mathbf {X_{1}} ,\mathbf {Y} )+\operatorname {cov} (\mathbf {X_{2}} ,\mathbf {Y} )}$
4. ${\displaystyle \operatorname {cov} (A\mathbf {X} +\mathbf {a} ,B^{\rm {T}}\mathbf {Y} +\mathbf {b} )=A\,\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )\,B}$
5. If ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ are independent (or somewhat less restrictedly, if every random variable in ${\displaystyle \mathbf {X} }$ is uncorrelated with every random variable in ${\displaystyle \mathbf {Y} }$), then ${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=0_{p\times q}}$

where ${\displaystyle \mathbf {X} }$, ${\displaystyle \mathbf {X_{1}} }$ and ${\displaystyle \mathbf {X_{2}} }$ are random ${\displaystyle p\times 1}$ vectors, ${\displaystyle \mathbf {Y} }$ is a random ${\displaystyle q\times 1}$ vector, ${\displaystyle \mathbf {a} }$ is a ${\displaystyle q\times 1}$ vector, ${\displaystyle \mathbf {b} }$ is a ${\displaystyle p\times 1}$ vector, ${\displaystyle A}$ and ${\displaystyle B}$ are ${\displaystyle q\times p}$ matrices of constants, and ${\displaystyle 0_{p\times q}}$ is a ${\displaystyle p\times q}$ matrix of zeroes.

Definition for complex random vectors

If ${\displaystyle \mathbf {Z} }$ and ${\displaystyle \mathbf {W} }$ are complex random vectors, the definition of the cross-covariance matrix is slightly changed. Transposition is replaced by Hermitian transposition:

${\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} (\mathbf {Z} ,\mathbf {W} ){\stackrel {\mathrm {def} }{=}}\ \operatorname {E} }$

For complex random vectors, another matrix called the pseudo-cross-covariance matrix is defined as follows:

${\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} (\mathbf {Z} ,{\overline {\mathbf {W} }}){\stackrel {\mathrm {def} }{=}}\ \operatorname {E} }$

Uncorrelatedness

Two random vectors ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ are called uncorrelated if their cross-covariance matrix ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}$ matrix is a zero matrix.

Complex random vectors ${\displaystyle \mathbf {Z} }$ and ${\displaystyle \mathbf {W} }$ are called uncorrelated if their covariance matrix and pseudo-covariance matrix is zero, i.e. if ${\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {J} _{\mathbf {Z} \mathbf {W} }=0}$.

References

1. ^ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1.
2. Taboga, Marco (2010). "Lectures on probability theory and mathematical statistics".

• Covariance and correlation
• Matrices