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 covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random 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.

Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself).

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

Definition

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 the column vector

${\displaystyle \mathbf {X} =(X_{1},X_{2},...,X_{n})^{\mathrm {T} }}$

are random variables, each with finite variance and expected value, then the covariance matrix ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}$ is the matrix whose ${\displaystyle (i,j)}$ entry is the covariance^{[1]: p. 177}

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

where the operator ${\displaystyle \operatorname {E} }$ denotes the expected value (mean) of its argument.

Conflicting nomenclatures and notationsedit

Nomenclatures differ. Some statisticians, following the probabilist William Feller in his two-volume book An Introduction to Probability Theory and Its Applications,^[2] call the matrix ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}$ the variance of the random vector ${\displaystyle \mathbf {X} }$, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the covariance matrix, because it is the matrix of covariances between the scalar components of the vector ${\displaystyle \mathbf {X} }$.

${\displaystyle \operatorname {var} (\mathbf {X} )=\operatorname {cov} (\mathbf {X} ,\mathbf {X} )=\operatorname {E} \left(\mathbf {X} -\operatorname {E} \mathbf {X} )(\mathbf {X} -\operatorname {E} \mathbf {X} )^{\rm {T}}\right.}$

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

By comparison, the notation for the cross-covariance matrix between two vectors is

${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=\operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} \left(\mathbf {X} -\operatorname {E} \mathbf {X} )(\mathbf {Y} -\operatorname {E} \mathbf {Y} )^{\rm {T}}\right.}$

Propertiesedit

Relation to the autocorrelation matrixedit

The auto-covariance matrix ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}$ is related to the autocorrelation matrix ${\displaystyle {\mathrm{R}}_{\mathbf{X}\mathbf{X}}}$Zdroj:https://en.wikipedia.org?pojem=Variance-covariance_matrix
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