Part of a series onStatistics
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

Inprobability theory andstatistics, acovariance matrix (also known asauto-covariance matrix,dispersion matrix,variance matrix, orvariance–covariance matrix) is a squarematrix giving thecovariance between each pair of elements of a givenrandom vector.

Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the${\displaystyle x}$ and${\displaystyle y}$ directions contain all of the necessary information; a${\displaystyle 2\times 2}$ matrix would be necessary to fully characterize the two-dimensional variation.

Anycovariance matrix issymmetric andpositive semi-definite and its main diagonal containsvariances (i.e., the covariance of each element with itself).

The covariance matrix of a random vector${\displaystyle \mathbf{X}}$ is typically denoted by${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$,${\displaystyle \mathrm{\Sigma }}$ or${\displaystyle S}$.

Throughout this article, boldfaced unsubscripted${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ are used to refer to random vectors, and Roman subscripted${\displaystyle X_i}$ and${\displaystyle Y_i}$ are used to refer to scalar random variables.

If the entries in thecolumn vector$${\displaystyle \mathbf{X}=(X_1,X_2,\dots ,X_n{)}^{\mathsf{T}}}$$arerandom variables, each with finitevariance andexpected value, then the covariance matrix${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$ is the matrix whose${\displaystyle (i,j)}$ entry is thecovariance^[1]: 177$${\displaystyle {\mathrm{K}}_{X_i{X}_j}=\mathrm{cov}[X_i,X_j]=\mathrm{E}[(X_i-\mathrm{E}[X_i])(X_j-\mathrm{E}[X_j])]}$$where the operator${\displaystyle \mathrm{E}}$ denotes the expected value (mean) of its argument.

Nomenclatures differ. Some statisticians, following the probabilistWilliam Feller in his two-volume bookAn Introduction to Probability Theory and Its Applications,^[2] call the matrix${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$ thevariance of the random vector${\displaystyle \mathbf{X}}$, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it thecovariance matrix, because it is the matrix of covariances between the scalar components of the vector${\displaystyle \mathbf{X}}$.$${\displaystyle \mathrm{var}(\mathbf{X})=\mathrm{cov}(\mathbf{X},\mathbf{X})=\mathrm{E}\left[(\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{X}-\mathrm{E}[\mathbf{X}]{)}^{\mathsf{T}}\right].}$$

Both forms are quite standard, and there is no ambiguity between them. The matrix${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$ is also often called thevariance-covariance matrix, since the diagonal terms are in fact variances.

By comparison, the notation for thecross-covariance matrixbetween two vectors is$${\displaystyle \mathrm{cov}(\mathbf{X},\mathbf{Y})={\mathrm{K}}_{\mathbf{X}\mathbf{Y}}=\mathrm{E}\left[(\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{Y}-\mathrm{E}[\mathbf{Y}]{)}^{\mathsf{T}}\right].}$$

The auto-covariance matrix${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$ is related to theautocorrelation matrix${\displaystyle {\mathrm{R}}_{\mathbf{X}\mathbf{X}}}$ by$${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}=\mathrm{E}[(\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{X}-\mathrm{E}[\mathbf{X}]{)}^{\mathsf{T}}]={\mathrm{R}}_{\mathbf{X}\mathbf{X}}-\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{X}{]}^{\mathsf{T}}}$$where the autocorrelation matrix is defined as${\displaystyle {\mathrm{R}}_{\mathbf{X}\mathbf{X}}=\mathrm{E}[\mathbf{X}{\mathbf{X}}^{\mathsf{T}}]}$.

An entity closely related to the covariance matrix is the matrix ofPearson product-moment correlation coefficients between each of the random variables in the random vector${\displaystyle \mathbf{X}}$, which can be written as$${\displaystyle \mathrm{corr}(\mathbf{X})=(\mathrm{diag}({\mathrm{K}}_{\mathbf{X}\mathbf{X}}){)}^{-\frac{1}{2}}{\mathrm{K}}_{\mathbf{X}\mathbf{X}}(\mathrm{diag}({\mathrm{K}}_{\mathbf{X}\mathbf{X}}){)}^{-\frac{1}{2}},}$$where${\displaystyle \mathrm{diag}({\mathrm{K}}_{\mathbf{X}\mathbf{X}})}$ is the matrix of the diagonal elements of${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$ (i.e., adiagonal matrix of the variances of${\displaystyle X_i}$ for${\displaystyle i=1,\dots ,n}$).

Equivalently, the correlation matrix can be seen as the covariance matrix of thestandardized random variables${\displaystyle X_i/\sigma (X_i)}$ for${\displaystyle i=1,\dots ,n}$.

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

Theinverse of this matrix,${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}^{-1}}$, if it exists, is the inverse covariance matrix (or inverse concentration matrix^{[dubious –discuss]}), also known as theprecision matrix (orconcentration matrix).^[3]

Just as the covariance matrix can be written as the rescaling of a correlation matrix by the marginal variances:

So, using the idea ofpartial correlation, and partial variance, the inverse covariance matrix can be expressed analogously:

This duality motivates a number of other dualities between marginalizing and conditioning for Gaussian random variables.

For${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}=\mathrm{var}(\mathbf{X})=\mathrm{E}\left[\left(\mathbf{X}-\mathrm{E}[\mathbf{X}]\right){\left(\mathbf{X}-\mathrm{E}[\mathbf{X}]\right)}^{\mathsf{T}}\right]}$ and${\displaystyle {\mathit{\mu }}_{\mathbf{X}}=\mathrm{E}[\mathbf{\text{X}}]}$, where${\displaystyle \mathbf{X}=(X_1,\dots ,X_n{)}^{\mathsf{T}}}$ is an${\displaystyle n}$-dimensional random variable, the following basic properties apply:^[4]

1. ${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}=\mathrm{E}(\mathbf{X}{\mathbf{X}}^{\mathsf{T}})-{\mathit{\mu }}_{\mathbf{X}}{\mathit{\mu }}_{\mathbf{X}}^{\mathsf{T}}}$
2. ${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$ ispositive-semidefinite, i.e.${\displaystyle {\mathbf{a}}^T{\mathrm{K}}_{\mathbf{X}\mathbf{X}}\mathbf{a}\ge 0\text{for all }\mathbf{a}\in {\mathbb{R}}^n}$

Proof

Indeed, from the property 4 it follows that under linear transformation of random variable${\displaystyle \mathbf{X}}$ with covariation matrix${\displaystyle {\mathbf{\Sigma }}_{\mathbf{X}}=\mathrm{c}\mathrm{o}\mathrm{v}(\mathbf{X})}$ by linear operator${\displaystyle \mathbf{A}}$ s.a.${\displaystyle \mathbf{Y}=\mathbf{A}\mathbf{X}}$, the covariation matrix is transformed as

${\displaystyle {\mathbf{\Sigma }}_{\mathbf{Y}}=\mathrm{c}\mathrm{o}\mathrm{v}\left(\mathbf{Y}\right)={\mathbf{A}{\mathbf{\Sigma }}_{\mathbf{X}}\mathbf{A}}^{\mathrm{\top }}}$.

As according to the property 3 matrix${\displaystyle {\mathbf{\Sigma }}_{\mathbf{X}}}$ is symmetric, it can be diagonalized by a linear orthogonal transformation, i.e. there exists such orthogonal matrix${\displaystyle \mathbf{A}}$ (meanwhile${\displaystyle {\mathbf{A}}^{\mathrm{\top }}={\mathbf{A}}^{-1}}$), that

${\displaystyle {\mathbf{A}{\mathbf{\Sigma }}_{\mathbf{X}}\mathbf{A}}^{\mathrm{\top }}={\mathbf{A}{\mathbf{\Sigma }}_{\mathbf{X}}\mathbf{A}}^{-1}=\text{diag}({\sigma }_1,\dots ,{\sigma }_n),}$

and${\displaystyle {\sigma }_1,\dots ,{\sigma }_n}$ are eigenvalues of${\displaystyle {\mathbf{\Sigma }}_{\mathbf{X}}}$. But this means that this matrix is a covariation matrix for a random variable${\displaystyle \mathbf{Y}=\mathbf{A}\mathbf{X}}$, and the main diagonal of${\displaystyle {\mathbf{\Sigma }}_{\mathbf{Y}}=\mathrm{c}\mathrm{o}\mathrm{v}\left(\mathbf{Y}\right)}$ consists of variances of elements of${\displaystyle \mathbf{Y}}$ vector. As variance is always non-negative, we conclude that${\displaystyle {\sigma }_i\ge 0}$ for any${\displaystyle i}$. But this means that matrix${\displaystyle {\mathbf{\Sigma }}_{\mathbf{X}}}$ is positive-semidefinite.

3. ${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$ issymmetric, i.e.${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}^{\mathsf{T}}={\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$
4. For any constant (i.e. non-random)${\displaystyle m\times n}$ matrix${\displaystyle \mathbf{A}}$ and constant${\displaystyle m\times 1}$ vector${\displaystyle \mathbf{a}}$, one has${\displaystyle \mathrm{var}(\mathbf{A}\mathbf{X}+\mathbf{a})=\mathbf{A}\mathrm{var}(\mathbf{X}){\mathbf{A}}^{\mathsf{T}}}$
5. If${\displaystyle \mathbf{Y}}$ is another random vector with the same dimension as${\displaystyle \mathbf{X}}$, then${\displaystyle \mathrm{var}(\mathbf{X}+\mathbf{Y})=\mathrm{var}(\mathbf{X})+\mathrm{cov}(\mathbf{X},\mathbf{Y})+\mathrm{cov}(\mathbf{Y},\mathbf{X})+\mathrm{var}(\mathbf{Y})}$ where${\displaystyle \mathrm{cov}(\mathbf{X},\mathbf{Y})}$ is thecross-covariance matrix of${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$.

The joint mean${\displaystyle \mathit{\mu }}$ andjoint covariance matrix${\displaystyle \mathbf{\Sigma }}$ of${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ can be written in block formwhere${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}=\mathrm{var}(\mathbf{X})}$,${\displaystyle {\mathrm{K}}_{\mathbf{Y}\mathbf{Y}}=\mathrm{var}(\mathbf{Y})}$ and${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{Y}}={\mathrm{K}}_{\mathbf{Y}\mathbf{X}}^{\mathsf{T}}=\mathrm{cov}(\mathbf{X},\mathbf{Y})}$.

${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$ and${\displaystyle {\mathrm{K}}_{\mathbf{Y}\mathbf{Y}}}$ can be identified as the variance matrices of themarginal distributions for${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ respectively.

If${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ arejointly normally distributed,$${\displaystyle \mathbf{X},\mathbf{Y}\sim \mathcal{N}(\mathit{\mu },\mathbf{\Sigma }),}$$then theconditional distribution for${\displaystyle \mathbf{Y}}$ given${\displaystyle \mathbf{X}}$ is given by^[5]$${\displaystyle \mathbf{Y}\mid \mathbf{X}\sim \mathcal{N}({\mathit{\mu }}_{\mathbf{Y}\mathbf{|}\mathbf{X}},{\mathrm{K}}_{\mathbf{Y}\mathbf{|}\mathbf{X}}),}$$defined byconditional mean$${\displaystyle {\mathit{\mu }}_{\mathbf{Y}|\mathbf{X}}={\mathit{\mu }}_{\mathbf{Y}}+{\mathrm{K}}_{\mathbf{Y}\mathbf{X}}{\mathrm{K}}_{\mathbf{X}\mathbf{X}}^{-1}\left(\mathbf{X}-{\mathit{\mu }}_{\mathbf{X}}\right)}$$andconditional variance$${\displaystyle {\mathrm{K}}_{\mathbf{Y}\mathbf{|}\mathbf{X}}={\mathrm{K}}_{\mathbf{Y}\mathbf{Y}}-{\mathrm{K}}_{\mathbf{Y}\mathbf{X}}{\mathrm{K}}_{\mathbf{X}\mathbf{X}}^{-1}{\mathrm{K}}_{\mathbf{X}\mathbf{Y}}.}$$

The matrix${\displaystyle {\mathrm{K}}_{\mathbf{Y}\mathbf{X}}{\mathrm{K}}_{\mathbf{X}\mathbf{X}}^{-1}}$ is known as the matrix ofregression coefficients, while in linear algebra${\displaystyle {\mathrm{K}}_{\mathbf{Y}\mathbf{|}\mathbf{X}}}$ is theSchur complement of${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}}$ in${\displaystyle \mathbf{\Sigma }}$.

The matrix of regression coefficients may often be given in transpose form,${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{X}}^{-1}{\mathrm{K}}_{\mathbf{X}\mathbf{Y}}}$, suitable for post-multiplying a row vector of explanatory variables${\displaystyle {\mathbf{X}}^{\mathsf{T}}}$ rather than pre-multiplying a column vector${\displaystyle \mathbf{X}}$. In this form they correspond to the coefficients obtained by inverting the matrix of thenormal equations ofordinary least squares (OLS).

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 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${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ are correlated via another vector${\displaystyle \mathbf{I}}$, the latter correlations are suppressed in a matrix^[6]$${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{Y}\mid \mathbf{I}}=\mathrm{pcov}(\mathbf{X},\mathbf{Y}\mid \mathbf{I})=\mathrm{cov}(\mathbf{X},\mathbf{Y})-\mathrm{cov}(\mathbf{X},\mathbf{I})\mathrm{cov}(\mathbf{I},\mathbf{I}{)}^{-1}\mathrm{cov}(\mathbf{I},\mathbf{Y}).}$$The partial covariance matrix${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{Y}\mid \mathbf{I}}}$ is effectively the simple covariance matrix${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{Y}}}$ as if the uninteresting random variables${\displaystyle \mathbf{I}}$ were held constant.

The standard deviation matrix${\displaystyle \mathbf{S}}$ is the extension of the standard deviation to multiple dimensions. It is the symmetricsquare root of the covariance matrix${\displaystyle \mathbf{\Sigma }}$.^[7]

If a column vector${\displaystyle \mathbf{X}}$ of${\displaystyle n}$ possibly correlated random variables isjointly normally distributed, or more generallyelliptically distributed, then itsprobability density function${\displaystyle \mathrm{f}(\mathbf{X})}$ can be expressed in terms of the covariance matrix${\displaystyle \mathbf{\Sigma }}$ as follows^[6]$${\displaystyle \mathrm{f}(\mathbf{X})=(2\pi {)}^{-n/2}|\mathbf{\Sigma }{|}^{-1/2}\exp \left(-\frac{1}{2}\mathbf{(}\mathbf{X}\mathbf{-}\mu {\mathbf{)}}^{\mathsf{T}}{\mathbf{\Sigma }}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{X}\mathbf{-}\mu \mathbf{)}\right),}$$where${\displaystyle \mathit{\mu }=\mathrm{E}[\mathbf{X}]}$ and${\displaystyle |\mathbf{\Sigma }|}$ is thedeterminant of${\displaystyle \mathbf{\Sigma }}$, the so-calledgeneralized variance.

Applied to one vector, the covariance matrix maps a linear combinationc of the random variablesX onto a vector of covariances with those variables:${\displaystyle {\mathbf{c}}^{\mathsf{T}}\mathrm{\Sigma }=\mathrm{cov}({\mathbf{c}}^{\mathsf{T}}\mathbf{X},\mathbf{X})}$. Treated as abilinear form, it yields the covariance between the two linear combinations:${\displaystyle {\mathbf{d}}^{\mathsf{T}}\mathbf{\Sigma }\mathbf{c}=\mathrm{cov}({\mathbf{d}}^{\mathsf{T}}\mathbf{X},{\mathbf{c}}^{\mathsf{T}}\mathbf{X})}$. The variance of a linear combination is then${\displaystyle {\mathbf{c}}^{\mathsf{T}}\mathbf{\Sigma }\mathbf{c}}$, its covariance with itself.

Similarly, the (pseudo-)inverse covariance matrix provides an inner product${\displaystyle ⟨c-\mu |{\mathrm{\Sigma }}^{+}|c-\mu ⟩}$, which induces theMahalanobis distance, a measure of the "unlikelihood" ofc.^{[citation needed]}

From basic property 4. above, let${\displaystyle \mathbf{b}}$ be a${\displaystyle (p\times 1)}$ real-valued vector, then$${\displaystyle \mathrm{var}({\mathbf{b}}^{\mathsf{T}}\mathbf{X})={\mathbf{b}}^{\mathsf{T}}\mathrm{var}(\mathbf{X})\mathbf{b},}$$which must always be nonnegative, since it is thevariance of a real-valued random variable, so a covariance matrix is always apositive-semidefinite matrix.

The above argument can be expanded as follows:where the last inequality follows from the observation that${\displaystyle w^{\mathsf{T}}(\mathbf{X}-\mathrm{E}[\mathbf{X}])}$ is a scalar.

Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose${\displaystyle M}$ is a${\displaystyle p\times p}$ symmetric positive-semidefinite matrix. From the finite-dimensional case of thespectral theorem, it follows that${\displaystyle M}$ has a nonnegative symmetricsquare root, which can be denoted byM^1/2. Let${\displaystyle \mathbf{X}}$ be any${\displaystyle p\times 1}$ column vector-valued random variable whose covariance matrix is the${\displaystyle p\times p}$ identity matrix. Then$${\displaystyle \mathrm{var}({\mathbf{M}}^{1/2}\mathbf{X})={\mathbf{M}}^{1/2}\mathrm{var}(\mathbf{X}){\mathbf{M}}^{1/2}=\mathbf{M}.}$$

Thevariance of acomplexscalar-valued random variable with expected value${\displaystyle \mu }$ is conventionally defined usingcomplex conjugation:$${\displaystyle \mathrm{var}(Z)=\mathrm{E}\left[(Z-{\mu }_Z)\overline{(Z-{\mu }_Z)}\right],}$$where the complex conjugate of a complex number${\displaystyle z}$ is denoted${\displaystyle \overline{z}}$; thus the variance of a complex random variable is a real number.

If${\displaystyle \mathbf{Z}=(Z_1,\dots ,Z_n{)}^{\mathsf{T}}}$ is a column vector of complex-valued random variables, then theconjugate transpose${\displaystyle {\mathbf{Z}}^{\mathsf{H}}}$ is formed byboth transposing and conjugating. In the following expression, the product of a vector with its conjugate transpose results in a square matrix called thecovariance matrix, as its expectation:^[8]: 293$${\displaystyle {\mathrm{K}}_{\mathbf{Z}\mathbf{Z}}=\mathrm{cov}[\mathbf{Z},\mathbf{Z}]=\mathrm{E}\left[(\mathbf{Z}-{\mathit{\mu }}_{\mathbf{Z}})(\mathbf{Z}-{\mathit{\mu }}_{\mathbf{Z}}{)}^{\mathsf{H}}\right],}$$The matrix so obtained will beHermitianpositive-semidefinite,^[9] with real numbers in the main diagonal and complex numbers off-diagonal.

Properties

• The covariance matrix is aHermitian matrix, i.e.${\displaystyle {\mathrm{K}}_{\mathbf{Z}\mathbf{Z}}^{\mathsf{H}}={\mathrm{K}}_{\mathbf{Z}\mathbf{Z}}}$.^[1]: 179
• The diagonal elements of the covariance matrix are real.^[1]: 179

For complex random vectors, another kind of second central moment, thepseudo-covariance matrix (also calledrelation matrix) is defined as follows:$${\displaystyle {\mathrm{J}}_{\mathbf{Z}\mathbf{Z}}=\mathrm{cov}[\mathbf{Z},\overline{\mathbf{Z}}]=\mathrm{E}\left[(\mathbf{Z}-{\mathit{\mu }}_{\mathbf{Z}})(\mathbf{Z}-{\mathit{\mu }}_{\mathbf{Z}}{)}^{\mathsf{T}}\right]}$$

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 acomplex symmetric matrix.

If${\displaystyle {\mathbf{M}}_{\mathbf{X}}}$ and${\displaystyle {\mathbf{M}}_{\mathbf{Y}}}$ are centereddata matrices of dimension${\displaystyle p\times n}$ and${\displaystyle q\times n}$ respectively, i.e. withn columns of observations ofp andq rows of variables, from which the row means have been subtracted, then, if the row means were estimated from the data, sample covariance matrices${\displaystyle {\mathbf{Q}}_{\mathbf{X}\mathbf{X}}}$ and${\displaystyle {\mathbf{Q}}_{\mathbf{X}\mathbf{Y}}}$ can be defined to be$${\displaystyle {\mathbf{Q}}_{\mathbf{X}\mathbf{X}}=\frac{1}{n-1}{\mathbf{M}}_{\mathbf{X}}{\mathbf{M}}_{\mathbf{X}}^{\mathsf{T}},{\mathbf{Q}}_{\mathbf{X}\mathbf{Y}}=\frac{1}{n-1}{\mathbf{M}}_{\mathbf{X}}{\mathbf{M}}_{\mathbf{Y}}^{\mathsf{T}}}$$or, if the row means were known a priori,$${\displaystyle {\mathbf{Q}}_{\mathbf{X}\mathbf{X}}=\frac{1}{n}{\mathbf{M}}_{\mathbf{X}}{\mathbf{M}}_{\mathbf{X}}^{\mathsf{T}},{\mathbf{Q}}_{\mathbf{X}\mathbf{Y}}=\frac{1}{n}{\mathbf{M}}_{\mathbf{X}}{\mathbf{M}}_{\mathbf{Y}}^{\mathsf{T}}.}$$

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.

The covariance matrix is a useful tool in many different areas. From it atransformation matrix can be derived, called awhitening transformation, that allows one to completely decorrelate the data^[10] or, from a different point of view, to find an optimal basis for representing the data in a compact way^{[citation needed]} (seeRayleigh quotient for a formal proof and additional properties of covariance matrices).This is calledprincipal component analysis (PCA) and theKarhunen–Loève transform (KL-transform).

The covariance matrix plays a key role infinancial economics, especially inportfolio theory and itsmutual fund separation theorem and in thecapital 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 anormative analysis) or are predicted to (in apositive analysis) choose to hold in a context ofdiversification.

Theevolution 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 theevolution strategy's covariance matrix adapts to the inverse of theHessian 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).^[11]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.

Incovariance mapping the values of the${\displaystyle \mathrm{cov}(\mathbf{X},\mathbf{Y})}$ or${\displaystyle \mathrm{pcov}(\mathbf{X},\mathbf{Y}\mid \mathbf{I})}$ matrix are plotted as a 2-dimensional map. When vectors${\displaystyle \mathbf{X}}$ and${\displaystyle \mathbf{Y}}$ are discreterandom functions, 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${\displaystyle \mathbf{X},\mathbf{Y}}$, and${\displaystyle \mathbf{I}}$ are acquired experimentally as rows of${\displaystyle n}$ samples, e.g.where${\displaystyle X_j(t_i)}$ is thei-th discrete value in samplej of the random function${\displaystyle X(t)}$. The expected values needed in the covariance formula are estimated using thesample mean, e.g.$${\displaystyle ⟨\mathbf{X}⟩=\frac{1}{n}\sum _{j=1}^n{\mathbf{X}}_j}$$and the covariance matrix is estimated by thesample covariance matrix$${\displaystyle \mathrm{cov}(\mathbf{X},\mathbf{Y})\approx ⟨\mathbf{X}{\mathbf{Y}}^{\mathsf{T}}⟩-⟨\mathbf{X}⟩⟨{\mathbf{Y}}^{\mathsf{T}}⟩,}$$where the angular brackets denote sample averaging as before except that theBessel's correction should be made to avoidbias. Using this estimation the partial covariance matrix can be calculated as$${\displaystyle \mathrm{pcov}(\mathbf{X},\mathbf{Y}\mid \mathbf{I})=\mathrm{cov}(\mathbf{X},\mathbf{Y})-\mathrm{cov}(\mathbf{X},\mathbf{I})\left(\mathrm{cov}(\mathbf{I},\mathbf{I})\mathrm{\setminus }\mathrm{cov}(\mathbf{I},\mathbf{Y})\right),}$$where the backslash denotes theleft matrix division operator, which bypasses the requirement to invert a matrix and is available in some computational packages such asMatlab.^[12]

Fig. 1 illustrates how a partial covariance map is constructed on an example of an experiment performed at theFLASHfree-electron laser in Hamburg.^[13] The random function${\displaystyle X(t)}$ is thetime-of-flight spectrum of ions from aCoulomb 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${\displaystyle m={10}^4}$ such spectra,${\displaystyle {\mathbf{X}}_j(t)}$, and averaging them over${\displaystyle j}$ produces a smooth spectrum${\displaystyle ⟨\mathbf{X}(t)⟩}$, which is shown in red at the bottom of Fig. 1. The average spectrum${\displaystyle ⟨\mathbf{X}⟩}$ 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${\displaystyle {\mathbf{X}}_j(t)}$ and${\displaystyle {\mathbf{Y}}_j(t)}$ are the same, except that the range of the time-of-flight${\displaystyle t}$ differs. Panela shows${\displaystyle ⟨\mathbf{X}{\mathbf{Y}}^{\mathsf{T}}⟩}$, panelb shows${\displaystyle ⟨\mathbf{X}⟩⟨{\mathbf{Y}}^{\mathsf{T}}⟩}$ and panelc shows their difference, which is${\displaystyle \mathrm{cov}(\mathbf{X},\mathbf{Y})}$ (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${\displaystyle I_j}$ is recorded at every shot, put into${\displaystyle \mathbf{I}}$ and${\displaystyle \mathrm{pcov}(\mathbf{X},\mathbf{Y}\mid \mathbf{I})}$ is calculated as panelsd ande 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${\displaystyle \mathbf{I}}$. Yet in practice it is often sufficient to overcompensate the partial covariance correction as panelf 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 employscorrelation analysis to obtain 2D spectra of thecondensed phase. There are two versions of this analysis:synchronous andasynchronous. Mathematically, the former is expressed in terms of the sample covariance matrix and the technique is equivalent to covariance mapping.^[14]

• Covariance function
• Eigenvalue decomposition
• Gramian matrix
• Lewandowski-Kurowicka-Joe distribution
• Multivariate statistics
• Principal components
• Quadratic form (statistics)

• "Covariance matrix",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• "Covariance Matrix Explained With Pictures", an easy way to visualize covariance matrices!
• Weisstein, Eric W."Covariance Matrix".MathWorld.
• van Kampen, N. G. (1981).Stochastic processes in physics and chemistry. New York: North-Holland.ISBN 0-444-86200-5.

• Covariance and correlation
• Matrices (mathematics)
• Summary statistics

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