Editing Covariance matrix (section)

=\operatorname{K}_{\mathbf{X}\mathbf{Y}}=
\operatorname{E}
\left[
 (\mathbf{X} - \operatorname{E}[\mathbf{X}])
 (\mathbf{Y} - \operatorname{E}[\mathbf{Y}])^\mathsf{T}
\right].
</math>

==Properties==
===Relation to the autocorrelation matrix===
The auto-covariance matrix <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}</math> is related to the [[autocorrelation matrix]] <math>\operatorname{R}_{\mathbf{X}\mathbf{X}}</math> by
<math display="block">\operatorname{K}_{\mathbf{X}\mathbf{X}} = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^\mathsf{T}] =  \operatorname{R}_{\mathbf{X}\mathbf{X}} - \operatorname{E}[\mathbf{X}] \operatorname{E}[\mathbf{X}]^\mathsf{T}</math>
where the autocorrelation matrix is defined as <math>\operatorname{R}_{\mathbf{X}\mathbf{X}} = \operatorname{E}[\mathbf{X} \mathbf{X}^\mathsf{T}]</math>.

===Relation to the correlation matrix===
{{further|Correlation matrix}}
An entity closely related to the covariance matrix is the matrix of [[Pearson product-moment correlation coefficient]]s between each of the random variables in the random vector <math>\mathbf{X}</math>, which can be written as
<math display="block">\operatorname{corr}(\mathbf{X}) = \big(\operatorname{diag}(\operatorname{K}_{\mathbf{X}\mathbf{X}})\big)^{-\frac{1}{2}} \, \operatorname{K}_{\mathbf{X}\mathbf{X}} \, \big(\operatorname{diag}(\operatorname{K}_{\mathbf{X}\mathbf{X}})\big)^{-\frac{1}{2}},</math>
where  <math>\operatorname{diag}(\operatorname{K}_{\mathbf{X}\mathbf{X}})</math> is the matrix of the diagonal elements of <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}</math> (i.e., a [[diagonal matrix]] of the variances of <math>X_i</math>  for <math>i = 1, \dots, n</math>).

Equivalently, the correlation matrix can be seen as the covariance matrix of the [[standardized variable|standardized random variables]] <math>X_i/\sigma(X_i)</math> for <math>i = 1, \dots, n</math>.
<math display="block">
\operatorname{corr}(\mathbf{X})
= \begin{bmatrix}
 1 & \frac{\operatorname{E}[(X_1 - \mu_1)(X_2 - \mu_2)]}{\sigma(X_1)\sigma(X_2)} & \cdots & \frac{\operatorname{E}[(X_1 - \mu_1)(X_n - \mu_n)]}{\sigma(X_1)\sigma(X_n)} \\ \\
 \frac{\operatorname{E}[(X_2 - \mu_2)(X_1 - \mu_1)]}{\sigma(X_2)\sigma(X_1)} & 1 & \cdots & \frac{\operatorname{E}[(X_2 - \mu_2)(X_n - \mu_n)]}{\sigma(X_2)\sigma(X_n)} \\ \\
 \vdots & \vdots & \ddots & \vdots \\ \\
 \frac{\operatorname{E}[(X_n - \mu_n)(X_1 - \mu_1)]}{\sigma(X_n)\sigma(X_1)} & \frac{\operatorname{E}[(X_n - \mu_n)(X_2 - \mu_2)]}{\sigma(X_n)\sigma(X_2)} & \cdots & 1
\end{bmatrix}.
</math>

Each element on the principal diagonal of a correlation matrix is the correlation of a random variable with itself, which always equals 1. Each [[off-diagonal element]] is between −1 and +1 inclusive.

===Inverse of the covariance matrix===
The [[invertible matrix|inverse of this matrix]], <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}^{-1}</math>, if it exists, is the inverse covariance matrix (or inverse concentration matrix{{dubious|reason=An inverse concentration gets smaller the more concentrated something is, not larger as it reasonably should be. Besides, we already call it "concentration matrix" later on in this sentence, which is kind of contradictory.|date=September 2024}}), also known as the ''[[precision matrix]]'' (or ''concentration matrix'').<ref>{{cite book |title=All of Statistics: A Concise Course in Statistical Inference |url=https://archive.org/details/springer_10.1007-978-0-387-21736-9 |first=Larry |last=Wasserman |year=2004 |publisher=Springer |isbn=0-387-40272-1}}</ref>

Just as the covariance matrix can be written as the rescaling of a correlation matrix by the marginal variances: 
<math display="block">\operatorname{cov}(\mathbf{X})
= \begin{bmatrix}
 \sigma_{x_1} & & & 0\\
 & \sigma_{x_2}\\
 & &  \ddots\\
 0 & & &  \sigma_{x_n}
\end{bmatrix}
\begin{bmatrix}
 1 & \rho_{x_1, x_2} & \cdots & \rho_{x_1, x_n}\\
 \rho_{x_2, x_1} & 1 & \cdots & \rho_{x_2, x_n}\\
 \vdots & \vdots & \ddots & \vdots\\
 \rho_{x_n, x_1} & \rho_{x_n, x_2} & \cdots & 1\\
\end{bmatrix}
\begin{bmatrix}
 \sigma_{x_1} & & & 0\\
 & \sigma_{x_2}\\
 & &  \ddots\\
 0 & & &  \sigma_{x_n}
\end{bmatrix}</math>

So, using the idea of [[partial correlation]], and partial variance, the inverse covariance matrix can be expressed analogously:
<math display="block">\operatorname{cov}(\mathbf{X})^{-1}
= \begin{bmatrix}
 \frac{1}{\sigma_{x_1|x_2...}} & & & 0\\
 & \frac{1}{\sigma_{x_2|x_1,x_3...}}\\
 & &  \ddots\\
 0 & & &  \frac{1}{\sigma_{x_n|x_1...x_{n-1}}}
\end{bmatrix}
\begin{bmatrix}
 1 & -\rho_{x_1, x_2\mid x_3...} & \cdots & -\rho_{x_1, x_n\mid x_2...x_{n-1}}\\
 -\rho_{x_2, x_1\mid x_3...} & 1 & \cdots & -\rho_{x_2, x_n\mid x_1,x_3...x_{n-1}}\\
 \vdots & \vdots & \ddots & \vdots\\
 -\rho_{x_n, x_1\mid x_2...x_{n-1}} & -\rho_{x_n, x_2\mid x_1,x_3...x_{n-1}} & \cdots & 1\\
\end{bmatrix}
\begin{bmatrix}
 \frac{1}{\sigma_{x_1|x_2...}} & & & 0\\
 & \frac{1}{\sigma_{x_2|x_1,x_3...}}\\
 & &  \ddots\\
 0 & & &  \frac{1}{\sigma_{x_n|x_1...x_{n-1}}}
\end{bmatrix}</math>
This duality motivates a number of other dualities between marginalizing and conditioning for Gaussian random variables.

===Basic properties===
For <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}=\operatorname{var}(\mathbf{X}) = \operatorname{E} \left[ \left( \mathbf{X} - \operatorname{E}[\mathbf{X}] \right) \left( \mathbf{X} - \operatorname{E}[\mathbf{X}] \right)^\mathsf{T} \right]</math> and <math> \boldsymbol{\mu}_\mathbf{X} = \operatorname{E}[\textbf{X}]</math>, where <math>\mathbf{X} = (X_1,\ldots,X_n)^\mathsf{T}</math> is an <math>n</math>-dimensional random variable, the following basic properties apply:<ref name=taboga>{{cite web |last1=Taboga |first1=Marco |url=http://www.statlect.com/varian2.htm |title=Lectures on probability theory and mathematical statistics |year=2010}}</ref>
# <math> \operatorname{K}_{\mathbf{X}\mathbf{X}} = \operatorname{E}(\mathbf{X X^\mathsf{T}}) - \boldsymbol{\mu}_\mathbf{X}\boldsymbol{\mu}_\mathbf{X}^\mathsf{T} </math>
# <math> \operatorname{K}_{\mathbf{X}\mathbf{X}} \,</math> is [[Positive-semidefinite matrix|positive-semidefinite]], i.e. <math>\mathbf{a}^T \operatorname{K}_{\mathbf{X}\mathbf{X}} \mathbf{a} \ge 0 \quad \text{for all } \mathbf{a} \in \mathbb{R}^n</math>
{{Hidden|
  title = ''Proof'' |
  content = 
<!------------------------------------------------------------------------------------->
Indeed, from the property 4 it follows that under linear transformation of random variable <math>\mathbf{X}</math> with covariation matrix <math>\mathbf{\Sigma_{X}} = \mathrm{cov}(\mathbf{X})</math> by linear operator <math>\mathbf{A}</math> s.a. <math>\mathbf{Y} = \mathbf{A}\mathbf{X}</math>, the covariation matrix is tranformed as
: <math>\mathbf{\Sigma_{Y}} = \mathrm{cov}\left(\mathbf{Y}\right) = \mathbf{A\, \Sigma_{X}\,A}^{\top}</math>.
As according to the property 3 matrix <math>\mathbf{\Sigma_{X}}</math> is symmetric, it can be diagonalized by a linear orthogonal transformation, i.e. there exists such orthogonal matrix <math>\mathbf{A}</math> (meanwhile <math>\mathbf{A}^{\top} = \mathbf{A}^{-1}</math>), that
: <math>\mathbf{A\, \Sigma_{X}\,A}^{\top} = \mathbf{A\, \Sigma_{X}\,A}^{-1} = \mbox{diag}(\sigma_1,\ldots,\sigma_n),</math>
and <math>\sigma_1,\ldots,\sigma_n</math> are eigenvalues of <math>\mathbf{\Sigma_{X}}</math>. But this means that this matrix is a covariation matrix for a random variable <math>\mathbf{Y} = \mathbf{A}\mathbf{X}</math>, and the main diagonal of <math>\mathbf{\Sigma_{Y}} = \mathrm{cov}\left(\mathbf{Y}\right)</math> consists of variances of elements of <math>\mathbf{Y}</math> vector. As variance is always non-negative, we conclude that <math>\sigma_i \geq 0</math> for any <math>i</math>. But this means that matrix <math>\mathbf{\Sigma_{X}}</math> is positive-semidefinite.
<!------------------------------------------------------------------------------------->
}}
# <math> \operatorname{K}_{\mathbf{X}\mathbf{X}} \,</math> is [[symmetric matrix|symmetric]], i.e. <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}^\mathsf{T} = \operatorname{K}_{\mathbf{X}\mathbf{X}}</math>
# For any constant (i.e. non-random) <math>m \times n</math> matrix <math>\mathbf{A}</math> and constant <math>m \times 1</math> vector <math>\mathbf{a}</math>, one has <math> \operatorname{var}(\mathbf{A X} + \mathbf{a}) = \mathbf{A}\, \operatorname{var}(\mathbf{X})\, \mathbf{A}^\mathsf{T} </math>
# If <math>\mathbf{Y}</math> is another random vector with the same dimension as <math>\mathbf{X}</math>, then <math>\operatorname{var}(\mathbf{X} + \mathbf{Y}) = \operatorname{var}(\mathbf{X}) + \operatorname{cov}(\mathbf{X},\mathbf{Y}) + \operatorname{cov}(\mathbf{Y}, \mathbf{X}) + \operatorname{var}(\mathbf{Y}) </math> where <math>\operatorname{cov}(\mathbf{X}, \mathbf{Y})</math> is the [[cross-covariance matrix]] of <math>\mathbf{X}</math> and <math>\mathbf{Y}</math>.

=== Block matrices ===

The joint mean <math>\boldsymbol\mu</math> and [[cross-covariance matrix|joint covariance matrix]] <math>\boldsymbol\Sigma</math> of <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> can be written in block form
<math display="block">
\boldsymbol\mu
=
\begin{bmatrix}
 \boldsymbol{\mu}_X \\
 \boldsymbol{\mu}_Y
\end{bmatrix}, \qquad
\boldsymbol\Sigma
=
\begin{bmatrix}
 \operatorname{K}_\mathbf{XX} & \operatorname{K}_\mathbf{XY} \\
 \operatorname{K}_\mathbf{YX} & \operatorname{K}_\mathbf{YY}
\end{bmatrix}
</math>
where <math> \operatorname{K}_\mathbf{XX} = \operatorname{var}(\mathbf{X}) </math>, <math> \operatorname{K}_\mathbf{YY} = \operatorname{var}(\mathbf{Y}) </math> and <math> \operatorname{K}_\mathbf{XY} = \operatorname{K}^\mathsf{T}_\mathbf{YX} = \operatorname{cov}(\mathbf{X}, \mathbf{Y}) </math>.

<math> \operatorname{K}_\mathbf{XX} </math> and <math> \operatorname{K}_\mathbf{YY} </math> can be identified as the variance matrices of the [[marginal distribution]]s for <math> \mathbf{X} </math> and <math> \mathbf{Y} </math> respectively.

If <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> are [[Multivariate normal distribution|jointly normally distributed]],
<math display="block">
\mathbf{X}, \mathbf{Y} \sim\ \mathcal{N}(\boldsymbol\mu, \operatorname{\boldsymbol\Sigma}),
</math>
then the [[conditional distribution]] for <math>\mathbf{Y}</math> given <math>\mathbf{X}</math> is given by<ref name=eaton>{{cite book|last=Eaton|first=Morris L.|title=Multivariate Statistics: a Vector Space Approach|year=1983|publisher=John Wiley and Sons|isbn=0-471-02776-6|pages=116–117}}</ref>
<math display="block">
\mathbf{Y} \mid \mathbf{X} \sim\ \mathcal{N}(\boldsymbol{\mu}_\mathbf{Y|X}, \operatorname{K}_\mathbf{Y|X}),
</math>
defined by [[conditional mean]]
<math display="block">
\boldsymbol{\mu}_{\mathbf{Y}|\mathbf{X}}
=
\boldsymbol{\mu}_\mathbf{Y} + \operatorname{K}_\mathbf{YX} \operatorname{K}_\mathbf{XX}^{-1}
\left(
 \mathbf{X} - \boldsymbol{\mu}_\mathbf{X}
\right)
</math>
and [[conditional variance]]
<math display="block">
  \operatorname{K}_\mathbf{Y|X}
  = \operatorname{K}_\mathbf{YY} - \operatorname{K}_\mathbf{YX} \operatorname{K}_\mathbf{XX}^{-1} \operatorname{K}_\mathbf{XY}.
</math>

The matrix <math> \operatorname{K}_\mathbf{YX} \operatorname{K}_\mathbf{XX}^{-1} </math> is known as the matrix of [[regression analysis|regression]] coefficients, while in linear algebra <math> \operatorname{K}_\mathbf{Y|X} </math> is the [[Schur complement]] of <math> \operatorname{K}_\mathbf{XX} </math> in <math> \boldsymbol\Sigma </math>.

The matrix of regression coefficients may often be given in transpose form, <math> \operatorname{K}_\mathbf{XX}^{-1} \operatorname{K}_\mathbf{XY} </math>, suitable for post-multiplying a row vector of explanatory variables <math> \mathbf{X}^\mathsf{T} </math> rather than pre-multiplying a column vector <math> \mathbf{X} </math>. In this form they correspond to the coefficients obtained by inverting the matrix of the [[normal equations]] of [[ordinary least squares]] (OLS).

== 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.

== Standard deviation matrix ==
{{Main|Standard deviation#Standard deviation matrix}}

The standard deviation matrix <math>\mathbf{S}</math> is the extension of the standard deviation to multiple dimensions.  It is the symmetric [[Square root of a matrix|square root]] of the covariance matrix <math>\mathbf{\Sigma}</math>. <ref name="Das">{{cite arXiv |eprint=2012.14331 |last1=Das |first1=Abhranil |author2=Wilson S Geisler |title=Methods to integrate multinormals and compute classification measures |date=2020 |class=stat.ML }}</ref>

==Covariance matrix as a parameter of a distribution==
If a column vector <math> \mathbf{X} </math> of <math> n </math> possibly correlated random variables is [[Multivariate normal distribution|jointly normally distributed]], or more generally [[Elliptical distribution|elliptically distributed]], then its [[probability density function]] <math> \operatorname{f}(\mathbf{X}) </math> can be expressed in terms of the covariance matrix <math> \boldsymbol{\Sigma} </math> as follows<ref name="KrzMarAnd"/>
<math display="block">
\operatorname{f}(\mathbf{X})
 = (2 \pi)^{-n/2} |\boldsymbol{\Sigma}|^{-1/2}
   \exp \left ( - \tfrac{1}{2} \mathbf{(X - \mu)^\mathsf{T} \Sigma^{-1} (X - \mu)} \right ),
</math>
where <math> \boldsymbol{\mu} = \operatorname{E}[\mathbf{X}] </math> and <math> |\boldsymbol{\Sigma}| </math> is the [[determinant]] of <math> \boldsymbol{\Sigma} </math>.

==Covariance matrix as a linear operator==
{{main|Covariance operator}}
Applied to one vector, the covariance matrix maps a linear combination '''c''' of the random variables '''X''' onto a vector of covariances with those variables: <math>\mathbf c^\mathsf{T} \Sigma = \operatorname{cov}(\mathbf c^\mathsf{T} \mathbf X, \mathbf X)</math>. Treated as a [[bilinear form]], it yields the covariance between the two linear combinations: <math>\mathbf d^\mathsf{T} \boldsymbol\Sigma \mathbf c = \operatorname{cov}(\mathbf d^\mathsf{T} \mathbf X, \mathbf c^\mathsf{T} \mathbf X)</math>. The variance of a linear combination is then <math>\mathbf c^\mathsf{T} \boldsymbol\Sigma \mathbf c</math>, its covariance with itself.

Similarly, the (pseudo-)inverse covariance matrix provides an inner product <math>\langle c - \mu| \Sigma^+ |c - \mu\rangle</math>, which induces the [[Mahalanobis distance]], a measure of the "unlikelihood" of ''c''.{{Citation needed|date=February 2012}}

==Which matrices are covariance matrices?==
From basic property 4. above, let <math>\mathbf{b}</math> be a <math>(p \times 1)</math> real-valued vector, then
<math display="block">\operatorname{var}(\mathbf{b}^\mathsf{T}\mathbf{X}) = \mathbf{b}^\mathsf{T} \operatorname{var}(\mathbf{X}) \mathbf{b},\,</math>
which must always be nonnegative, since it is the [[variance#Properties|variance]] of a real-valued random variable, so a covariance matrix is always a [[positive-semidefinite matrix]].

The above argument can be expanded as follows:<math display="block">
\begin{align}
& w^\mathsf{T} \operatorname{E} \left[(\mathbf{X} - \operatorname{E}[\mathbf{X}]) (\mathbf{X} - \operatorname{E}[\mathbf{X}])^\mathsf{T}\right] w
= \operatorname{E} \left[w^\mathsf{T}(\mathbf{X} - \operatorname{E}[\mathbf{X}]) (\mathbf{X} - \operatorname{E}[\mathbf{X}])^\mathsf{T}w\right] \\
&= \operatorname{E} \big[\big( w^\mathsf{T}(\mathbf{X} - \operatorname{E}[\mathbf{X}]) \big)^2 \big] \geq 0,
\end{align}
</math>where the last inequality follows from the observation that <math>w^\mathsf{T}(\mathbf{X} - \operatorname{E}[\mathbf{X}])</math> is a scalar.

Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose <math>M</math> is a <math>p \times p</math> symmetric positive-semidefinite matrix. From the finite-dimensional case of the [[spectral theorem]], it follows that <math>M</math> has a nonnegative symmetric [[Square root of a matrix|square root]], which can be denoted by '''M'''<sup>1/2</sup>. Let <math>\mathbf{X}</math> be any <math>p \times 1</math> column vector-valued random variable whose covariance matrix is the <math>p \times p</math> identity matrix. Then
<math display="block">\operatorname{var}(\mathbf{M}^{1/2} \mathbf{X}) = \mathbf{M}^{1/2} \, \operatorname{var}(\mathbf{X}) \, \mathbf{M}^{1/2} = \mathbf{M}.</math>

==Complex random vectors==
{{further|Complex random vector#Covariance matrix and pseudo-covariance matrix}}

The [[Variance#Generalizations|variance]] of a [[complex number|complex]] ''scalar-valued'' random variable with expected value <math>\mu</math> is conventionally defined using [[complex conjugation]]:
<math display="block">
\operatorname{var}(Z)
= \operatorname{E}\left[ (Z - \mu_Z)\overline{(Z - \mu_Z)} \right],
</math>
where the complex conjugate of a complex number <math>z</math> is denoted <math>\overline{z}</math>; thus the variance of a complex random variable is a real number.

If <math>\mathbf{Z} = (Z_1,\ldots,Z_n) ^\mathsf{T}</math> is a column vector of complex-valued random variables, then the [[conjugate transpose]] <math>\mathbf{Z}^\mathsf{H}</math> is formed by ''both'' transposing and conjugating. In the following expression, the product of a vector with its conjugate transpose results in a square matrix called the '''covariance matrix''', as its expectation:<ref name=Lapidoth>{{cite book |first=Amos |last=Lapidoth |year=2009 |title=A Foundation in Digital Communication |publisher=Cambridge University Press |isbn=978-0-521-19395-5}}</ref>{{rp|p=293}}
<math display="block">
\operatorname{K}_{\mathbf{Z}\mathbf{Z}} = \operatorname{cov}[\mathbf{Z},\mathbf{Z}] =
\operatorname{E}
\left[
 (\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})(\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})^\mathsf{H}
\right],
</math>
The matrix so obtained will be [[Hermitian matrix|Hermitian]] [[Positive-semidefinite matrix|positive-semidefinite]],<ref>{{Cite web |url=http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/expect.html |first=Mike |last=Brookes |title=The Matrix Reference Manual}}</ref> with real numbers in the main diagonal and complex numbers off-diagonal.

;Properties
* The covariance matrix is a [[Hermitian matrix]], i.e. <math>\operatorname{K}_{\mathbf{Z}\mathbf{Z}}^\mathsf{H} = \operatorname{K}_{\mathbf{Z}\mathbf{Z}}</math>.<ref name=KunIlPark/>{{rp||page=179}}
* The diagonal elements of the covariance matrix are real.<ref name=KunIlPark/>{{rp||page=179}}

===Pseudo-covariance matrix{{anchor|Pseudo}}===
For complex random vectors, another kind of second central moment, the '''pseudo-covariance matrix''' (also called '''relation matrix''') is defined as follows:
<math display="block">
\operatorname{J}_{\mathbf{Z}\mathbf{Z}} = \operatorname{cov}[\mathbf{Z},\overline{\mathbf{Z}}] =
\operatorname{E}
\left[
 (\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})(\mathbf{Z} - \boldsymbol{\mu}_\mathbf{Z})^\mathsf{T}
\right]
</math>

In contrast to the covariance matrix defined above, Hermitian transposition gets replaced by transposition in the definition.
Its diagonal elements may be complex valued; it is a [[complex symmetric matrix]].

==Estimation==
{{Main|Estimation of covariance matrices}}
If <math>\mathbf{M}_{\mathbf{X}}</math> and <math>\mathbf{M}_{\mathbf{Y}}</math> are centered [[Data matrix (multivariate statistics)|data matrices]] of dimension <math>p \times n</math> and <math>q \times n</math> respectively, i.e. with ''n'' columns of observations of ''p'' and ''q'' rows of variables, from which the row means have been subtracted, then, if the row means were estimated from the data, sample covariance matrices <math>\mathbf{Q}_{\mathbf{XX}}</math> and <math>\mathbf{Q}_{\mathbf{XY}}</math> can be defined to be
<math display="block"> \mathbf{Q}_{\mathbf{XX}} = \frac{1}{n-1} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{X}}^\mathsf{T}, \qquad \mathbf{Q}_{\mathbf{XY}} = \frac{1}{n-1} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{Y}}^\mathsf{T} </math>
or, if the row means were known a priori,
<math display="block"> \mathbf{Q}_{\mathbf{XX}} = \frac{1}{n} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{X}}^\mathsf{T}, \qquad \mathbf{Q}_{\mathbf{XY}} = \frac{1}{n} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{Y}}^\mathsf{T}. </math>

These empirical sample covariance matrices are the most straightforward and most often used estimators for the covariance matrices, but other estimators also exist, including regularised or shrinkage estimators, which may have better properties.

==Applications==
The covariance matrix is a useful tool in many different areas. From it a [[transformation matrix]] can be derived, called a [[whitening transformation]], that allows one to completely decorrelate the data <ref>{{Cite journal |last1=Kessy |first1=Agnan |last2=Strimmer |first2=Korbinian |last3=Lewin |first3=Alex |year = 2018 |title=Optimal Whitening and Decorrelation |url=https://www.tandfonline.com/doi/full/10.1080/00031305.2016.1277159 |journal=The American Statistician | publisher = Taylor & Francis |volume=72 |issue=4 |pages=309–314 |doi=10.1080/00031305.2016.1277159 | arxiv = 1512.00809}} </ref> or, from a different point of view, to find an optimal basis for representing the data in a compact way{{citation needed|date=February 2012}} (see [[Rayleigh quotient]] for a formal proof and additional properties of covariance matrices).
This is called [[principal component analysis]] (PCA) and the [[Karhunen–Loève transform]] (KL-transform).

The covariance matrix plays a key role in [[financial economics]], especially in [[Modern portfolio theory|portfolio theory]] and its [[mutual fund separation theorem]] and in the [[capital asset pricing model]]. The matrix of covariances among various assets' returns is used to determine, under certain assumptions, the relative amounts of different assets that investors should (in a [[Normative economics|normative analysis]]) or are predicted to (in a [[Positive economics|positive analysis]]) choose to hold in a context of [[Diversification (finance)|diversification]].

===Use in optimization===
The [[evolution strategy]], a particular family of Randomized Search Heuristics, fundamentally relies on a covariance matrix in its mechanism. The characteristic mutation operator draws the update step from a multivariate normal distribution using an evolving covariance matrix. There is a formal proof that the [[evolution strategy]]'s covariance matrix adapts to the inverse of the [[Hessian matrix]] of the search landscape, [[up to]] a scalar factor and small random fluctuations (proven for a single-parent strategy and a static model, as the population size increases, relying on the quadratic approximation).<ref>{{cite journal | doi = 10.1016/j.tcs.2019.09.002 | first = O.M. | last = Shir | author2 = A. Yehudayoff | title = On the covariance-Hessian relation in evolution strategies | journal = Theoretical Computer Science | volume = 801 | pages = 157–174 | publisher = Elsevier | year = 2020 | doi-access = free | arxiv = 1806.03674 }}</ref>
Intuitively, this result is supported by the rationale that the optimal covariance distribution can offer mutation steps whose equidensity probability contours match the level sets of the landscape, and so they maximize the progress rate.

===Covariance mapping===
In '''covariance mapping''' the values of the <math> \operatorname{cov}(\mathbf{X}, \mathbf{Y}) </math> or <math> \operatorname{pcov}(\mathbf{X}, \mathbf{Y} \mid \mathbf{I}) </math> matrix are plotted as a 2-dimensional map. When vectors <math> \mathbf{X} </math> and <math> \mathbf{Y} </math> are discrete [[random function]]s, the map shows statistical relations between different regions of the random functions. Statistically independent regions of the functions show up on the map as zero-level flatland, while positive or negative correlations show up, respectively, as hills or valleys.

In practice the column vectors <math> \mathbf{X}, \mathbf{Y} </math>, and <math> \mathbf{I} </math> are acquired experimentally as rows of <math> n </math> samples, e.g.
<math display="block"> \left[\mathbf{X}_1, \mathbf{X}_2, \dots, \mathbf{X}_n\right] =
\begin{bmatrix}
  X_1(t_1) & X_2(t_1) & \cdots & X_n(t_1) \\ \\
  X_1(t_2) & X_2(t_2) & \cdots & X_n(t_2) \\ \\
  \vdots   &   \vdots & \ddots & \vdots   \\ \\
  X_1(t_m) & X_2(t_m) & \cdots & X_n(t_m)
\end{bmatrix} ,
</math>
where <math> X_j(t_i) </math> is the ''i''-th discrete value in sample ''j'' of the random function <math> X(t) </math>. The expected values needed in the covariance formula are estimated using the [[sample mean]], e.g.
<math display="block"> \langle \mathbf{X} \rangle = \frac{1}{n} \sum_{j=1}^{n} \mathbf{X}_j </math>
and the covariance matrix is estimated by the [[sample covariance]] matrix
<math display="block">
  \operatorname{cov}(\mathbf{X},\mathbf{Y})
  \approx \langle \mathbf{XY^\mathsf{T}} \rangle
  - \langle \mathbf{X} \rangle \langle \mathbf{Y}^\mathsf{T} \rangle ,
</math>
where the angular brackets denote sample averaging as before except that the [[Bessel's correction]] should be made to avoid [[bias of an estimator|bias]]. Using this estimation the partial covariance matrix can be calculated as
<math display="block">
  \operatorname{pcov}(\mathbf{X},\mathbf{Y} \mid \mathbf{I})
  = \operatorname{cov}(\mathbf{X},\mathbf{Y})
  - \operatorname{cov}(\mathbf{X},\mathbf{I})
  \left (    \operatorname{cov}(\mathbf{I},\mathbf{I})
  \backslash \operatorname{cov}(\mathbf{I},\mathbf{Y}) \right ),
</math>
where the backslash denotes the [[Division (mathematics)#Left and right division|left matrix division]] operator, which bypasses the requirement to invert a matrix and is available in some computational packages such as [[Matlab]].<ref name="LJF16">L J Frasinski "Covariance mapping techniques" ''J. Phys. B: At. Mol. Opt. Phys.'' '''49''' 152004 (2016), {{doi|10.1088/0953-4075/49/15/152004}}</ref>

[[Image:Stages of partial covariance mapping.png|thumb|600px|'''Figure 1: Construction of a partial covariance map of N<sub>2</sub> molecules undergoing Coulomb explosion induced by a free-electron laser.'''<ref name="OK13"/> Panels '''a''' and '''b''' map the two terms of the covariance matrix, which is shown in panel '''c'''. Panel '''d''' maps common-mode correlations via intensity fluctuations of the laser. Panel '''e''' maps the partial covariance matrix that is corrected for the intensity fluctuations. Panel '''f''' shows that 10% overcorrection improves the map and makes ion-ion correlations clearly visible. Owing to momentum conservation these correlations appear as lines approximately perpendicular to the autocorrelation line (and to the periodic modulations which are caused by detector ringing).]]
Fig. 1 illustrates how a partial covariance map is constructed on an example of an experiment performed at the [[DESY#FLASH|FLASH]] [[free-electron laser]] in Hamburg.<ref name="OK13">O Kornilov, M Eckstein, M Rosenblatt, C P Schulz, K Motomura, A Rouzée, J Klei, L Foucar, M Siano, A Lübcke, F. Schapper, P Johnsson, D M P Holland, T Schlatholter, T Marchenko, S Düsterer, K Ueda, M J J Vrakking and L J Frasinski "Coulomb explosion of diatomic molecules in intense XUV fields mapped by partial covariance" ''J. Phys. B: At. Mol. Opt. Phys.'' '''46''' 164028 (2013), {{doi|10.1088/0953-4075/46/16/164028}}</ref> The random function <math> X(t) </math> is the [[Time-of-flight_mass_spectrometry|time-of-flight]] spectrum of ions from a [[Coulomb explosion]] of nitrogen molecules multiply ionised by a laser pulse. Since only a few hundreds of molecules are ionised at each laser pulse, the single-shot spectra are highly fluctuating. However, collecting typically <math> m=10^4 </math> such spectra, <math> \mathbf{X}_j(t) </math>, and averaging them over <math> j </math> produces a smooth spectrum <math> \langle \mathbf{X}(t) \rangle </math>, which is shown in red at the bottom of Fig. 1. The average spectrum <math> \langle \mathbf{X} \rangle </math> reveals several nitrogen ions in a form of peaks broadened by their kinetic energy, but to find the correlations between the ionisation stages and the ion momenta requires calculating a covariance map.

In the example of Fig. 1 spectra <math> \mathbf{X}_j(t) </math> and <math> \mathbf{Y}_j(t) </math> are the same, except that the range of the time-of-flight <math> t </math> differs. Panel '''a''' shows <math> \langle \mathbf{XY^\mathsf{T}} \rangle </math>, panel '''b''' shows <math> \langle \mathbf{X} \rangle \langle \mathbf{Y}^\mathsf{T} \rangle </math> and panel '''c''' shows their difference, which is <math> \operatorname{cov}(\mathbf{X},\mathbf{Y}) </math> (note a change in the colour scale). Unfortunately, this map is overwhelmed by uninteresting, common-mode correlations induced by laser intensity fluctuating from shot to shot. To suppress such correlations the laser intensity <math> I_j </math> is recorded at every shot, put into <math> \mathbf{I} </math> and <math> \operatorname{pcov}(\mathbf{X},\mathbf{Y} \mid \mathbf{I}) </math> is calculated as panels '''d''' and '''e''' show. The suppression of the uninteresting correlations is, however, imperfect because there are other sources of common-mode fluctuations than the laser intensity and in principle all these sources should be monitored in vector <math> \mathbf{I} </math>. Yet in practice it is often sufficient to overcompensate the partial covariance correction as panel '''f''' shows, where interesting correlations of ion momenta are now clearly visible as straight lines centred on ionisation stages of atomic nitrogen.

===Two-dimensional infrared spectroscopy===
Two-dimensional infrared spectroscopy employs [[two-dimensional correlation analysis|correlation analysis]] to obtain 2D spectra of the [[condensed matter physics|condensed phase]]. There are two versions of this analysis: [[Two-dimensional correlation analysis#Calculation of the synchronous spectrum|synchronous]] and [[Two-dimensional correlation analysis#Calculation of the asynchronous spectrum|asynchronous]]. Mathematically, the former is expressed in terms of the sample covariance matrix and the technique is equivalent to covariance mapping.<ref>{{cite journal |first=I. |last=Noda |title=Generalized two-dimensional correlation method applicable to infrared, Raman, and other types of spectroscopy |journal=Appl. Spectrosc. |volume=47 |issue= 9|pages=1329–36 |year=1993 |doi=10.1366/0003702934067694 |bibcode=1993ApSpe..47.1329N }}</ref>

==See also==
* [[Covariance function]]
* [[Eigenvalue decomposition]]
* [[Gramian matrix]]
* [[Lewandowski-Kurowicka-Joe distribution]]
* [[Multivariate statistics]]
* [[Principal components]]
* [[Quadratic form (statistics)]]

==References==
{{Reflist}}

==Further reading==
* {{springer|title=Covariance matrix|id=p/c026820}}
* "[https://thekalmanfilter.com/covariance-matrix-explained/ Covariance Matrix Explained With Pictures]", an easy way to visualize covariance matrices!
* {{mathworld|urlname=CovarianceMatrix|title= Covariance Matrix}}
* {{cite book|first=N. G. |last=van Kampen|title=Stochastic processes in physics and chemistry|url=https://archive.org/details/stochasticproces0000kamp |url-access=registration |location= New York|publisher=North-Holland|year= 1981|isbn=0-444-86200-5}}

{{Statistics}}
{{Matrix classes}}

{{DEFAULTSORT:Covariance Matrix}}
[[Category:Covariance and correlation]]
[[Category:Matrices (mathematics)]]
[[Category:Summary statistics]]