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Covariance

From Wikipedia, the free encyclopedia

Measure of the joint variability

This article is about the degree to which random variables vary similarly. For other uses, seeCovariance (disambiguation).

The sign of the covariance of two random variablesX andY

Inprobability theory andstatistics,covariance is a measure of the joint variability of tworandom variables.^[1]

The sign of the covariance, therefore, shows the tendency in thelinear relationship between the variables. If greater values of one variable mainly correspond with greater values of the other variable, and the same holds for lesser values (that is, the variables tend to show similar behavior), the covariance is positive.^[2] In the opposite case, when greater values of one variable mainly correspond to lesser values of the other (that is, the variables tend to show opposite behavior), the covariance is negative. One feature of covariance is that it has units of measurement and the magnitude of the covariance is affected by said units. This means changing the units (e.g., from meters to millimeters) changes the covariance value proportionally, making it difficult to assess the strength of the relationship from the covariance alone; In some situations, it is desirable to compare the strength of the joint association between different pairs of random variables that do not necessarily have the same units.^[3] In those situations, we use thecorrelation coefficient, which normalizes the covariance by dividing by thegeometric mean of the totalvariances (i.e., the product of thestandard deviations) for the two random variables to get a result between -1 and 1 and makes the units irrelevant.^[4]

A distinction must be made between (1) the covariance of two random variables, which is apopulation parameter that can be seen as a property of thejoint probability distribution, and (2) thesample covariance, which in addition to serving as a descriptor of the sample, also serves as anestimated value of the population parameter.

Definition

For twojointly distributed real-valued random variables $X {\displaystyle X}$ and $Y {\displaystyle Y}$ with finitesecond moments, the covariance is defined as theexpected value (or mean) of the product of their deviations from their individual expected values:^[5]^[6]^: 119

$\operatorname {cov} (X,Y)=\operatorname {E} {{\big [}(X-\operatorname {E} [X])(Y-\operatorname {E} [Y]){\big ]}}$

where $\operatorname {E} [X]$ is the expected value of $X {\displaystyle X}$ , also known as the mean of $X {\displaystyle X}$ . The covariance is also sometimes denoted $\sigma _{XY}$ or $\sigma (X,Y)$ , in analogy tovariance. By using the linearity property of expectations, this can be simplified to the expected value of their product minus the product of their expected values: ${\begin{aligned}\operatorname {cov} (X,Y)&=\operatorname {E} \left[\left(X-\operatorname {E} \left[X\right]\right)\left(Y-\operatorname {E} \left[Y\right]\right)\right]\\&=\operatorname {E} \left[XY-X\operatorname {E} \left[Y\right]-\operatorname {E} \left[X\right]Y+\operatorname {E} \left[X\right]\operatorname {E} \left[Y\right]\right]\\&=\operatorname {E} \left[XY\right]-\operatorname {E} \left[X\right]\operatorname {E} \left[Y\right]-\operatorname {E} \left[X\right]\operatorname {E} \left[Y\right]+\operatorname {E} \left[X\right]\operatorname {E} \left[Y\right]\\&=\operatorname {E} \left[XY\right]-\operatorname {E} \left[X\right]\operatorname {E} \left[Y\right].\end{aligned}}$ This identity is useful for mathematical derivations. From the viewpoint of numerical computation, however, it is susceptible tocatastrophic cancellation (see the section onnumerical computation below).

Theunits of measurement of the covariance $\operatorname {cov} (X,Y)$ are those of $X {\displaystyle X}$ times those of $Y {\displaystyle Y}$ . By contrast,correlation coefficients, which depend on the covariance, are adimensionless measure of linear dependence. (In fact, correlation coefficients can simply be understood as a normalized version of covariance.)

Complex random variables

Main article:Complex random variable § Covariance

The covariance between twocomplex random variables $Z, W {\displaystyle Z,W}$ is defined as^[6]^: 119 $\operatorname {cov} (Z,W)=\operatorname {E} \left[(Z-\operatorname {E} [Z]){\overline {(W-\operatorname {E} [W])}}\right]=\operatorname {E} \left[Z{\overline {W}}\right]-\operatorname {E} [Z]\operatorname {E} \left[{\overline {W}}\right]$

Notice the complex conjugation of the second factor in the definition.

A relatedpseudo-covariance can also be defined.

Discrete random variables

If the (real) random variable pair $(X,Y)$ can take on the values $(x_{i},y_{i})$ for $i=1,\ldots ,n$ , with equal probabilities $p_{i}=1/n$ , then the covariance can be equivalently written in terms of the means $\operatorname {E} [X]$ and $\operatorname {E} [Y]$ as $\operatorname {cov} (X,Y)={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-E(X))(y_{i}-E(Y)).$

It can also be equivalently expressed, without directly referring to the means, as^[7] $\operatorname {cov} (X,Y)={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {1}{2}}(x_{i}-x_{j})(y_{i}-y_{j})={\frac {1}{n^{2}}}\sum _{i}\sum _{j>i}(x_{i}-x_{j})(y_{i}-y_{j}).$

More generally, if there are $n {\displaystyle n}$ possible realizations of $(X,Y)$ , namely $(x_{i},y_{i})$ but with possibly unequal probabilities $p_{i}$ for $i=1,\ldots ,n$ , then the covariance is $\operatorname {cov} (X,Y)=\sum _{i=1}^{n}p_{i}(x_{i}-E(X))(y_{i}-E(Y)).$

In the case where two discrete random variables $X {\displaystyle X}$ and $Y {\displaystyle Y}$ have a joint probability distribution, represented by elements $p_{i,j}$ corresponding to the joint probabilities of $P(X=x_{i},Y=y_{j})$ , the covariance is calculated using a double summation over the indices of the matrix:

$\operatorname {cov} (X,Y)=\sum _{i=1}^{n}\sum _{j=1}^{n}p_{i,j}(x_{i}-E[X])(y_{j}-E[Y]).$

Examples

Consider three independent random variables $A, B, C {\displaystyle A,B,C}$ and two constants $q, r {\displaystyle q,r}$ . ${\begin{aligned}X&=qA+B\\Y&=rA+C\\\operatorname {cov} (X,Y)&=qr\operatorname {var} (A)\end{aligned}}$ In the special case, $q=1$ and $r=1$ , the covariance between $X {\displaystyle X}$ and $Y {\displaystyle Y}$ is just the variance of $A {\displaystyle A}$ and the name covariance is entirely appropriate.

Geometric interpretation of the covariance example.Each cuboid is theaxis-aligned bounding box of its point(x,y,f (x,y)), and theX andY means (magenta point).The covariance is the sum of the volumes of the cuboids in the 1st and 3rd quadrants (red) and in the 2nd and 4th (blue).

Suppose that $X {\displaystyle X}$ and $Y {\displaystyle Y}$ have the followingjoint probability mass function,^[8] in which the six central cells give the discrete joint probabilities $f(x,y)$ of the six hypothetical realizations $(x,y)\in S=\left\{(5,8),(6,8),(7,8),(5,9),(6,9),(7,9)\right\}$ :

$f(x,y)$		5	6	7		$f_{Y}(y)$
$f(x,y)$		x				$f_{Y}(y)$
y	8	0	0.4	0.1		0.5
y	9	0.3	0	0.2		0.5

$f_{X}(x)$		0.3	0.4	0.3		1

$X {\displaystyle X}$ can take on three values (5, 6 and 7) while $Y {\displaystyle Y}$ can take on two (8 and 9). Their means are $\mu _{X}=5(0.3)+6(0.4)+7(0.1+0.2)=6$ and $\mu _{Y}=8(0.4+0.1)+9(0.3+0.2)=8.5$ . Then, ${\begin{aligned}\operatorname {cov} (X,Y)={}&\sigma _{XY}=\sum _{(x,y)\in S}f(x,y)\left(x-\mu _{X}\right)\left(y-\mu _{Y}\right)\\[4pt]={}&(0)(5-6)(8-8.5)+(0.4)(6-6)(8-8.5)+(0.1)(7-6)(8-8.5)+{}\\[4pt]&(0.3)(5-6)(9-8.5)+(0)(6-6)(9-8.5)+(0.2)(7-6)(9-8.5)\\[4pt]={}&{-0.1}\;.\end{aligned}}$

Properties

Covariance with itself

Thevariance is a special case of the covariance in which the two variables are identical:^[6]^: 121 $\operatorname {cov} (X,X)=\operatorname {var} (X)\equiv \sigma ^{2}(X)\equiv \sigma _{X}^{2}.$

Covariance of linear combinations

If $X {\displaystyle X}$ , $Y {\displaystyle Y}$ , $W {\displaystyle W}$ , and $V {\displaystyle V}$ are real-valued random variables and $a, b, c, d {\displaystyle a,b,c,d}$ are real-valued constants, then the following facts are a consequence of the definition of covariance: ${\begin{aligned}\operatorname {cov} (X,a)&=0\\\operatorname {cov} (X,X)&=\operatorname {var} (X)\\\operatorname {cov} (X,Y)&=\operatorname {cov} (Y,X)\\\operatorname {cov} (aX,bY)&=ab\,\operatorname {cov} (X,Y)\\\operatorname {cov} (X+a,Y+b)&=\operatorname {cov} (X,Y)\\\operatorname {cov} (aX+bY,cW+dV)&=ac\,\operatorname {cov} (X,W)+ad\,\operatorname {cov} (X,V)+bc\,\operatorname {cov} (Y,W)+bd\,\operatorname {cov} (Y,V)\end{aligned}}$

For a sequence $X_{1},\ldots ,X_{n}$ of random variables in real-valued, and constants $a_{1},\ldots ,a_{n}$ , we have $\operatorname {var} \left(\sum _{i=1}^{n}a_{i}X_{i}\right)=\sum _{i=1}^{n}a_{i}^{2}\sigma ^{2}(X_{i})+2\sum _{i,j\,:\,i<j}a_{i}a_{j}\operatorname {cov} (X_{i},X_{j})=\sum _{i,j}{a_{i}a_{j}\operatorname {cov} (X_{i},X_{j})}$

Hoeffding's covariance identity

A useful identity to compute the covariance between two random variables $X, Y {\displaystyle X,Y}$ is the Hoeffding's covariance identity:^[9] $\operatorname {cov} (X,Y)=\int _{\mathbb {R} }\int _{\mathbb {R} }\left(F_{(X,Y)}(x,y)-F_{X}(x)F_{Y}(y)\right)\,dx\,dy$ where $F_{(X,Y)}(x,y)$ is the joint cumulative distribution function of the random vector $(X,Y)$ and $F_{X}(x),F_{Y}(y)$ are themarginals.

Uncorrelatedness and independence

Main article:Correlation and dependence

Random variables whose covariance is zero are calleduncorrelated.^[6]^: 121 Similarly, the components of random vectors whosecovariance matrix is zero in every entry outside the main diagonal are also called uncorrelated.

If $X {\displaystyle X}$ and $Y {\displaystyle Y}$ areindependent random variables, then their covariance is zero.^[6]^: 123^[10] This follows because under independence, $\operatorname {E} [XY]=\operatorname {E} [X]\cdot \operatorname {E} [Y].$

The converse, however, is not generally true. For example, let $X {\displaystyle X}$ be uniformly distributed in $[-1,1]$ and let $Y=X^{2}$ . Clearly, $X {\displaystyle X}$ and $Y {\displaystyle Y}$ are not independent, but ${\begin{aligned}\operatorname {cov} (X,Y)&=\operatorname {cov} \left(X,X^{2}\right)\\&=\operatorname {E} \left[X\cdot X^{2}\right]-\operatorname {E} [X]\cdot \operatorname {E} \left[X^{2}\right]\\&=\operatorname {E} \left[X^{3}\right]-\operatorname {E} [X]\operatorname {E} \left[X^{2}\right]\\&=0-0\cdot \operatorname {E} [X^{2}]\\&=0.\end{aligned}}$

In this case, the relationship between $Y {\displaystyle Y}$ and $X {\displaystyle X}$ is non-linear, while correlation and covariance are measures of linear dependence between two random variables. This example shows that if two random variables are uncorrelated, that does not in general imply that they are independent. However, if two variables arejointly normally distributed (but not if they are merelyindividually normally distributed), uncorrelatednessdoes imply independence.^[11]

$X {\displaystyle X}$ and $Y {\displaystyle Y}$ whose covariance is positive are called positively correlated, which implies if $X>E[X]$ then likely $Y>E[Y]$ . Conversely, $X {\displaystyle X}$ and $Y {\displaystyle Y}$ with negative covariance are negatively correlated, and if $X>E[X]$ then likely $Y<E[Y]$ .

Relationship to inner products

Many of the properties of covariance can be extracted elegantly by observing that it satisfies similar properties to those of aninner product:

bilinear: for constants $a {\displaystyle a}$ and $b {\displaystyle b}$ and random variables $X, Y, Z, {\displaystyle X,Y,Z,}$ $\operatorname {cov} (aX+bY,Z)=a\operatorname {cov} (X,Z)+b\operatorname {cov} (Y,Z)$
symmetric: $\operatorname {cov} (X,Y)=\operatorname {cov} (Y,X)$
positive semi-definite: $\sigma ^{2}(X)=\operatorname {cov} (X,X)\geq 0$ for all random variables $X {\displaystyle X}$ , and $\operatorname {cov} (X,X)=0$ implies that $X {\displaystyle X}$ is constantalmost surely.

In fact these properties imply that the covariance defines an inner product over thequotient vector space obtained by taking the subspace of random variables with finite second moment and identifying any two that differ by a constant. (This identification turns the positive semi-definiteness above into positive definiteness.) That quotient vector space is isomorphic to the subspace of random variables with finite second moment and mean zero; on that subspace, the covariance is exactly theL² inner product of real-valued functions on the sample space.

As a result, for random variables with finite variance, the inequality $\left|\operatorname {cov} (X,Y)\right|\leq {\sqrt {\sigma ^{2}(X)\sigma ^{2}(Y)}}$ holds via theCauchy–Schwarz inequality.

Proof: If $\sigma ^{2}(Y)=0$ , then it holds trivially. Otherwise, let random variable $Z=X-{\frac {\operatorname {cov} (X,Y)}{\sigma ^{2}(Y)}}Y.$

Then we have ${\begin{aligned}0\leq \sigma ^{2}(Z)&=\operatorname {cov} \left(X-{\frac {\operatorname {cov} (X,Y)}{\sigma ^{2}(Y)}}Y,\;X-{\frac {\operatorname {cov} (X,Y)}{\sigma ^{2}(Y)}}Y\right)\\[12pt]&=\sigma ^{2}(X)-{\frac {(\operatorname {cov} (X,Y))^{2}}{\sigma ^{2}(Y)}}\\\implies (\operatorname {cov} (X,Y))^{2}&\leq \sigma ^{2}(X)\sigma ^{2}(Y)\\\left|\operatorname {cov} (X,Y)\right|&\leq {\sqrt {\sigma ^{2}(X)\sigma ^{2}(Y)}}\end{aligned}}$

Calculating the sample covariance

Further information:Sample mean and sample covariance

The sample covariances among $K {\displaystyle K}$ variables based on $N {\displaystyle N}$ observations of each, drawn from an otherwise unobserved population, are given by the $K\times K$ matrix $\textstyle {\overline {\mathbf {q} }}=\left[q_{jk}\right]$ with the entries

q_{jk}={\frac {1}{N-1}}\sum _{i=1}^{N}\left(X_{ij}-{\bar {X}}_{j}\right)\left(X_{ik}-{\bar {X}}_{k}\right),

which is an estimate of the covariance between variable $j {\displaystyle j}$ and variable $k {\displaystyle k}$ .

The sample mean and the sample covariance matrix areunbiased estimates of themean and the covariance matrix of therandom vector $\textstyle \mathbf {X}$ , a vector whosejth element $(j=1,\,\ldots ,\,K)$ is one of the random variables. The reason the sample covariance matrix has $\textstyle N-1$ in the denominator rather than $\textstyle N$ is essentially that the population mean $\operatorname {E} (\mathbf {X} )$ is not known and is replaced by the sample mean $\mathbf {\bar {X}}$ . If the population mean $\operatorname {E} (\mathbf {X} )$ is known, the analogous unbiased estimate is given by

q_{jk}={\frac {1}{N}}\sum _{i=1}^{N}\left(X_{ij}-\operatorname {E} \left(X_{j}\right)\right)\left(X_{ik}-\operatorname {E} \left(X_{k}\right)\right)

.

Generalizations

Auto-covariance matrix of real random vectors

Main article:Auto-covariance matrix

For a vector $\mathbf {X} ={\begin{bmatrix}X_{1}&X_{2}&\dots &X_{m}\end{bmatrix}}^{\mathrm {T} }$ of $m {\displaystyle m}$ jointly distributed random variables with finite second moments, its auto-covariance matrix (also known as thevariance–covariance matrix or simply thecovariance matrix) $\operatorname {K} _{\mathbf {X} \mathbf {X} }$ (also denoted by $\Sigma (\mathbf {X} )$ or $\operatorname {cov} (\mathbf {X} ,\mathbf {X} )$ ) is defined as^[12]^: 335 ${\begin{aligned}\operatorname {K} _{\mathbf {XX} }=\operatorname {cov} (\mathbf {X} ,\mathbf {X} )&=\operatorname {E} \left[(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\mathrm {T} }\right]\\&=\operatorname {E} \left[\mathbf {XX} ^{\mathrm {T} }\right]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{\mathrm {T} }.\end{aligned}}$

Let $\mathbf {X}$ be arandom vector with covariance matrixΣ, and letA be a matrix that can act on $\mathbf {X}$ on the left. The covariance matrix of the matrix-vector productA X is: ${\begin{aligned}\operatorname {cov} (\mathbf {AX} ,\mathbf {AX} )&=\operatorname {E} \left[\mathbf {AX(A} \mathbf {X)} ^{\mathrm {T} }\right]-\operatorname {E} [\mathbf {AX} ]\operatorname {E} \left[(\mathbf {A} \mathbf {X} )^{\mathrm {T} }\right]\\&=\operatorname {E} \left[\mathbf {AXX} ^{\mathrm {T} }\mathbf {A} ^{\mathrm {T} }\right]-\operatorname {E} [\mathbf {AX} ]\operatorname {E} \left[\mathbf {X} ^{\mathrm {T} }\mathbf {A} ^{\mathrm {T} }\right]\\&=\mathbf {A} \operatorname {E} \left[\mathbf {XX} ^{\mathrm {T} }\right]\mathbf {A} ^{\mathrm {T} }-\mathbf {A} \operatorname {E} [\mathbf {X} ]\operatorname {E} \left[\mathbf {X} ^{\mathrm {T} }\right]\mathbf {A} ^{\mathrm {T} }\\&=\mathbf {A} \left(\operatorname {E} \left[\mathbf {XX} ^{\mathrm {T} }\right]-\operatorname {E} [\mathbf {X} ]\operatorname {E} \left[\mathbf {X} ^{\mathrm {T} }\right]\right)\mathbf {A} ^{\mathrm {T} }\\&=\mathbf {A} \Sigma \mathbf {A} ^{\mathrm {T} }.\end{aligned}}$

This is a direct result of the linearity ofexpectation and is usefulwhen applying alinear transformation, such as awhitening transformation, to a vector.

Cross-covariance matrix of real random vectors

Main article:Cross-covariance matrix

For realrandom vectors $\mathbf {X} \in \mathbb {R} ^{m}$ and $\mathbf {Y} \in \mathbb {R} ^{n}$ , the $m\times n$ cross-covariance matrix is equal to^[12]^: 336

{\begin{aligned}\operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )&=\operatorname {E} \left[(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\mathrm {T} }\right]\\&=\operatorname {E} \left[\mathbf {X} \mathbf {Y} ^{\mathrm {T} }\right]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\mathrm {T} }\end{aligned}}

Eq.2

where $\mathbf {Y} ^{\mathrm {T} }$ is thetranspose of the vector (or matrix) $\mathbf {Y}$ .

The $(i,j)$ -th element of this matrix is equal to the covariance $\operatorname {cov} (X_{i},Y_{j})$ between thei-th scalar component of $\mathbf {X}$ and thej-th scalar component of $\mathbf {Y}$ . In particular, $\operatorname {cov} (\mathbf {Y} ,\mathbf {X} )$ is thetranspose of $\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )$ .

Cross-covariance sesquilinear form of random vectors in a real or complex Hilbert space

More generally let $H_{1}=(H_{1},\langle \,,\rangle _{1})$ and $H_{2}=(H_{2},\langle \,,\rangle _{2})$ , beHilbert spaces over $\mathbb {R}$ or $\mathbb {C}$ with $\langle \,,\rangle$ anti linear in the first variable, and let $\mathbf {X} ,\mathbf {Y}$ be $H_{1}$ resp. $H_{2}$ valued random variables. Then the covariance of $\mathbf {X}$ and $\mathbf {Y}$ is thesesquilinear form on $H_{1}\times H_{2}$ (anti linear in the first variable) given by ${\begin{aligned}\operatorname {K} _{X,Y}(h_{1},h_{2})=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )(h_{1},h_{2})&=\operatorname {E} \left[\langle h_{1},(\mathbf {X} -\operatorname {E} [\mathbf {X} ])\rangle _{1}\langle (\mathbf {Y} -\operatorname {E} [\mathbf {Y} ]),h_{2}\rangle _{2}\right]\\&=\operatorname {E} [\langle h_{1},\mathbf {X} \rangle _{1}\langle \mathbf {Y} ,h_{2}\rangle _{2}]-\operatorname {E} [\langle h,\mathbf {X} \rangle _{1}]\operatorname {E} [\langle \mathbf {Y} ,h_{2}\rangle _{2}]\\&=\langle h_{1},\operatorname {E} \left[(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\dagger }\right]h_{2}\rangle _{1}\\&=\langle h_{1},\left(\operatorname {E} [\mathbf {X} \mathbf {Y} ^{\dagger }]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\dagger }\right)h_{2}\rangle _{1}\\\end{aligned}}$

Numerical computation

Main article:Algorithms for calculating variance § Covariance

When $\operatorname {E} [XY]\approx \operatorname {E} [X]\operatorname {E} [Y]$ , the equation $\operatorname {cov} (X,Y)=\operatorname {E} \left[XY\right]-\operatorname {E} \left[X\right]\operatorname {E} \left[Y\right]$ is prone tocatastrophic cancellation if $\operatorname {E} \left[XY\right]$ and $\operatorname {E} \left[X\right]\operatorname {E} \left[Y\right]$ are not computed exactly and thus should be avoided in computer programs when the data has not been centered before.^[13]Numerically stable algorithms should be preferred in this case.^[14]

Comments

The covariance is sometimes called a measure of "linear dependence" between the two random variables. That does not mean the same thing as in the context oflinear algebra (seelinear dependence). When the covariance is normalized, one obtains thePearson correlation coefficient, which gives the goodness of the fit for the best possible linear function describing the relation between the variables. In this sense covariance is a linear gauge of dependence.

Applications

In genetics and molecular biology

Covariance is an important measure inbiology. Certain sequences ofDNA are conserved more than others among species, and thus to study secondary and tertiary structures ofproteins, or ofRNA structures, sequences are compared in closely related species. If sequence changes are found or no changes at all are found innoncoding RNA (such asmicroRNA), sequences are found to be necessary for common structural motifs, such as an RNA loop. In genetics, covariance serves a basis for computation of Genetic Relationship Matrix (GRM) (aka kinship matrix), enabling inference on population structure from sample with no known close relatives as well as inference on estimation of heritability of complex traits.

In the theory ofevolution andnatural selection, theprice equation describes how agenetic trait changes in frequency over time. The equation uses a covariance between a trait andfitness, to give a mathematical description of evolution and natural selection. It provides a way to understand the effects that gene transmission and natural selection have on the proportion of genes within each new generation of a population.^[15]^[16]

In financial economics

Covariances play a key role infinancial economics, especially inmodern portfolio theory and in thecapital asset pricing model. Covariances among various assets' returns are 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.

In meteorological and oceanographic data assimilation

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The covariance matrix is important in estimating the initial conditions required for running weather forecast models, a procedure known asdata assimilation. The "forecast error covariance matrix" is typically constructed between perturbations around a mean state (either a climatological or ensemble mean). The "observation error covariance matrix" is constructed to represent the magnitude of combined observational errors (on the diagonal) and the correlated errors between measurements (off the diagonal). This is an example of its widespread application toKalman filtering and more generalstate estimation for time-varying systems.

In micrometeorology

Theeddy covariance technique is a key atmospherics measurement technique where the covariance between instantaneous deviation in vertical wind speed from the mean value and instantaneous deviation in gas concentration is the basis for calculating the vertical turbulent fluxes.

In signal processing

The covariance matrix is used to capture the spectral variability of a signal.^[17]

In statistics

Correlation

Main article:Correlation

The Pearson Correlation coefficient between two random variables $X {\displaystyle X}$ and $Y {\displaystyle Y}$ is defined as $\rho _{X,Y}={\frac {\operatorname {cov} (X,Y)}{\sigma _{X}\sigma _{Y}}}$

where

$\operatorname {cov}$ is thecovariance
$\sigma _{X}$ is thestandard deviation of $X {\displaystyle X}$
$\sigma _{Y}$ is the standard deviation of $Y {\displaystyle Y}$ .

The denominator can also be written as ${\sqrt {\operatorname {var} (X)\operatorname {var} (Y)}}$ , which is thegeometric mean of the variances.

Thus we see that the correlation coefficient is a normalized version of the covariance. It is always a number between $-1$ and $1 {\displaystyle 1}$ , and is unitless (unlike the covariance).

The correlation coefficient is often denoted with an $r {\displaystyle r}$ , and is frequently reported in scientific studies.

Principal Component Analysis

Main article:Principal Component Analysis

The covariance matrix is used inprincipal component analysis to reduce feature dimensionality indata preprocessing. The principal components are the dimensions that explain the most variance in the data. A well known application is to intelligence, producing theg factor. Another is to personality, with models like thefive factor model being derived from principal component analysis.

See also

References

^Rice, John (2007).Mathematical Statistics and Data Analysis. Brooks/Cole Cengage Learning. p. 138.ISBN 9780534399429.
^Weisstein, Eric W."Covariance".MathWorld.
^Kim, Hae-Young (February 2018)."Statistical notes for clinical researchers: covariance and correlation".Restorative Dentistry & Endodontics.43 (1) e4.doi:10.5395/rde.2018.43.e4.ISSN 2234-7658.PMC 5816993.PMID 29487835.
^"4.3. Covariance and correlation coefficient — TU Delft textbook".mude.citg.tudelft.nl. Retrieved2025-10-30.
^Oxford Dictionary of Statistics, Oxford University Press, 2002, p. 104.
^^a ^b ^c ^d ^ePark, Kun Il (2018).Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer.ISBN 9783319680743.
^Yuli Zhang; Huaiyu Wu; Lei Cheng (June 2012). "Some new deformation formulas about variance and covariance".Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012). pp. 987–992.
^"Covariance of X and Y | STAT 414/415". The Pennsylvania State University. Archived fromthe original on August 17, 2017. RetrievedAugust 4, 2019.
^Papoulis (1991).Probability, Random Variables and Stochastic Processes. McGraw-Hill.
^Siegrist, Kyle."Covariance and Correlation". University of Alabama in Huntsville. RetrievedOct 3, 2022.
^Dekking, Michel, ed. (2005).A modern introduction to probability and statistics: understanding why and how. Springer texts in statistics. London [Heidelberg]: Springer.ISBN 978-1-85233-896-1.
^^a ^bGubner, John A. (2006).Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press.ISBN 978-0-521-86470-1.
^Donald E. Knuth (1998).The Art of Computer Programming, volume 2:Seminumerical Algorithms, 3rd edn., p. 232. Boston: Addison-Wesley.
^Schubert, Erich; Gertz, Michael (2018)."Numerically stable parallel computation of (Co-)variance".Proceedings of the 30th International Conference on Scientific and Statistical Database Management. Bozen-Bolzano, Italy: ACM Press. pp. 1–12.doi:10.1145/3221269.3223036.ISBN 978-1-4503-6505-5.S2CID 49665540.
^Price, George (1970). "Selection and covariance".Nature.227 (5257):520–521.Bibcode:1970Natur.227..520P.doi:10.1038/227520a0.PMID 5428476.S2CID 4264723.
^Harman, Oren (2020)."When science mirrors life: on the origins of the Price equation".Philosophical Transactions of the Royal Society B: Biological Sciences.375 (1797). royalsocietypublishing.org:1–7.doi:10.1098/rstb.2019.0352.PMC 7133509.PMID 32146891.
^Sahidullah, Md.; Kinnunen, Tomi (March 2016)."Local spectral variability features for speaker verification".Digital Signal Processing.50:1–11.Bibcode:2016DSP....50....1S.doi:10.1016/j.dsp.2015.10.011.

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Descriptive statistics

Continuous data

Center	Mean Arithmetic Arithmetic-Geometric Contraharmonic Cubic Generalized/power Geometric Harmonic Heronian Heinz Lehmer Median Mode
Dispersion	Average absolute deviation Coefficient of variation Interquartile range Percentile Range Standard deviation Variance
Shape	Central limit theorem Moments Kurtosis L-moments Skewness

Index of dispersion

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Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife
Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons
Parametric tests	Likelihood-ratio Score/Lagrange multiplier Wald

Z-test(normal) Student'st-test F-test
Goodness of fit	Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality(Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC
Rank statistics	Sign Sample median Signed rank(Wilcoxon) Hodges–Lehmann estimator Rank sum(Mann–Whitney) Nonparametric anova 1-way(Kruskal–Wallis) 2-way(Friedman) Ordered alternative(Jonckheere–Terpstra) Van der Waerden test

Bayesian inference

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis (see alsoTemplate:Least squares and regression analysis	Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)
Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression
Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Homoscedasticity and Heteroscedasticity
Generalized linear model	Exponential families Logistic(Bernoulli) / Binomial / Poisson regressions
Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / multivariate / time-series / survival analysis

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality
Specific tests	Dickey–Fuller Johansen Q-statistic(Ljung–Box) Durbin–Watson Breusch–Godfrey
Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model(Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR) (Autoregressive model (AR))
Frequency domain	Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time
Hazard function	Nelson–Aalen estimator
Test	Log-rank test

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics
Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification
Social statistics	Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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