Covariance#Relationship to inner products

For two jointly distributed real-valued random variables $X$ and $Y$ with finite second moments, the covariance is defined as the expected value (or mean) of the product of their deviations from their individual expected values:Oxford Dictionary of Statistics, Oxford University Press, 2002, p. 104.{{cite book | author=Park, Kun Il| title=Fundamentals of Probability and Stochastic Processes with Applications to Communications| publisher=Springer | year=2018 | isbn=9783319680743 }}{{rp|p = 119}}

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

where $\backslash operatorname\{E\}[X]$ is the expected value of $X$, also known as the mean of $X$. The covariance is also sometimes denoted $\backslash sigma\_\{XY\}$ or $\backslash sigma(X,Y)$, in analogy to variance. 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{align}

\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[X Y - 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[X 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[X\right] \operatorname{E}\left[Y\right] \\

&= \operatorname{E}\left[X Y\right] - \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right].

\end{align}

This identity is useful for mathematical derivations. From the viewpoint of numerical computation, however, it is susceptible to catastrophic cancellation (see the section on numerical computation below).

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

{{main|Complex random variable#Covariance}}

The covariance between two complex random variables $Z,\; W$ is defined as{{rp|p= 119}}

$$\backslash operatorname\{cov\}(Z,\; W)\; =$$

\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 related pseudo-covariance can also be defined.

If the (real) random variable pair $(X,Y)$ can take on the values $(x\_i,y\_i)$ for $i\; =\; 1,\backslash ldots,n$, with equal probabilities $p\_i=1/n$, then the covariance can be equivalently written in terms of the means $\backslash operatorname\{E\}[X]$ and $\backslash operatorname\{E\}[Y]$ as

$$\backslash operatorname\{cov\}\; (X,Y)\; =\; \backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^n\; (x\_i-E(X))\; (y\_i-E(Y)).$$

It can also be equivalently expressed, without directly referring to the means, as{{cite conference |author=Yuli Zhang |author2=Huaiyu Wu |author3=Lei Cheng |title=Some new deformation formulas about variance and covariance|book-title=Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012)|date=June 2012 |pages=987–992}}

$$\backslash operatorname\{cov\}(X,Y)\; =\; \backslash frac\{1\}\{n^2\}\; \backslash sum\_\{i=1\}^n\; \backslash sum\_\{j=1\}^n\; \backslash frac\{1\}\{2\}(x\_i\; -\; x\_j)(y\_i\; -\; y\_j)\; =\; \backslash frac\{1\}\{n^2\}\; \backslash sum\_i\; \backslash sum\_\{j>i\}\; (x\_i-x\_j)(y\_i\; -\; y\_j).$$

More generally, if there are $n$ possible realizations of $(X,Y)$, namely $(x\_i,y\_i)$ but with possibly unequal probabilities $p\_i$ for $i\; =\; 1,\backslash ldots,n$, then the covariance is

$$\backslash operatorname\{cov\}\; (X,Y)\; =\; \backslash sum\_\{i=1\}^n\; p\_i\; (x\_i-E(X))\; (y\_i-E(Y)).$$

In the case where two discrete random variables $X$ and $Y$ 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:

$$\backslash operatorname\{cov\}\; (X,\; Y)\; =\; \backslash sum\_\{i=1\}^\{n\}\backslash sum\_\{j=1\}^\{n\}\; p\_\{i,j\}\; (x\_i\; -\; E[X])(y\_j\; -\; E[Y]).$$

Consider three independent random variables $A,\; B,\; C$ and two constants $q,\; r$.

\begin{align}

X &= qA + B \\

Y &= rA + C \\

\operatorname{cov}(X, Y)

&= qr \operatorname{var}(A)

\end{align}

In the special case, $q=1$ and $r=1$, the covariance between $X$ and $Y$ is just the variance of $A$ and the name covariance is entirely appropriate.

File:Covariance_geometric_visualisation.svg bounding box of its point {{nowrap|(x, y, f (x, y)),}} and the {{nowrap|X and Y means}} (magenta point). {{nowrap|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 $X$ and $Y$ have the following joint probability mass function,{{Cite web| url=https://onlinecourses.science.psu.edu/stat414/node/109| title=Covariance of X and Y {{!}} STAT 414/415|publisher=The Pennsylvania State University | access-date=August 4, 2019 | archive-url=https://web.archive.org/web/20170817034656/https://onlinecourses.science.psu.edu/stat414/node/109 | archive-date=August 17, 2017}} in which the six central cells give the discrete joint probabilities $f(x,\; y)$ of the six hypothetical realizations {{nowrap|$(x,\; y)\; \backslash in\; S\; =\; \backslash left\backslash \{\; (5,\; 8),\; (6,\; 8),\; (7,\; 8),\; (5,\; 9),\; (6,\; 9),\; (7,\; 9)\; \backslash right\backslash \}$:}}

$X$ can take on three values (5, 6 and 7) while $Y$ can take on two (8 and 9). Their means are $\backslash mu\_X\; =\; 5(0.3)\; +\; 6(0.4)\; +\; 7(0.1\; +\; 0.2)\; =\; 6$ and $\backslash mu\_Y\; =\; 8(0.4\; +\; 0.1)\; +\; 9(0.3\; +\; 0.2)\; =\; 8.5$. Then,

$$\backslash begin\{align\}$$

\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{align}

The variance is a special case of the covariance in which the two variables are identical:{{rp|p=121}}

$$\backslash operatorname\{cov\}(X,\; X)\; =\; \backslash operatorname\{var\}(X)\backslash equiv\backslash sigma^2(X)\backslash equiv\backslash sigma\_X^2.$$

If $X$, $Y$, $W$, and $V$ are real-valued random variables and $a,b,c,d$ are real-valued constants, then the following facts are a consequence of the definition of covariance:

\begin{align}

\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{align}

For a sequence $X\_1,\backslash ldots,X\_n$ of random variables in real-valued, and constants $a\_1,\backslash ldots,a\_n$, we have

A useful identity to compute the covariance between two random variables $X,\; Y$ is the Hoeffding's covariance identity:{{cite book| last1=Papoulis| title=Probability, Random Variables and Stochastic Processes| date=1991| publisher=McGraw-Hill}}

$$\backslash operatorname\{cov\}(X,\; Y)\; =\; \backslash int\_\backslash mathbb\; R\; \backslash int\_\backslash mathbb\; R\; \backslash left(F\_\{(X,\; Y)\}(x,\; y)\; -\; F\_X(x)F\_Y(y)\backslash right)\; \backslash ,dx\; \backslash ,dy$$

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 the marginals.

{{main|Correlation and dependence}}

Random variables whose covariance is zero are called uncorrelated.{{rp|p= 121}} Similarly, the components of random vectors whose covariance matrix is zero in every entry outside the main diagonal are also called uncorrelated.

If $X$ and $Y$ are independent random variables, then their covariance is zero.{{rp|p= 123}}{{Cite web | url=http://www.randomservices.org/random/expect/Covariance.html| title=Covariance and Correlation | last=Siegrist|first=Kyle| publisher=University of Alabama in Huntsville |access-date=Oct 3, 2022}} This follows because under independence,

$$\backslash operatorname\{E\}[XY]\; =\; \backslash operatorname\{E\}[X]\; \backslash cdot\; \backslash operatorname\{E\}[Y].$$

The converse, however, is not generally true. For example, let $X$ be uniformly distributed in $[-1,1]$ and let $Y\; =\; X^2$. Clearly, $X$ and $Y$ are not independent, but

$$\backslash begin\{align\}$$

\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{align}

In this case, the relationship between $Y$ and $X$ 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 are jointly normally distributed (but not if they are merely individually normally distributed), uncorrelatedness does imply independence.{{Cite book |title=A modern introduction to probability and statistics: understanding why and how |date=2005 |publisher=Springer |isbn=978-1-85233-896-1 |editor-last=Dekking |editor-first=Michel |series=Springer texts in statistics |location=London [Heidelberg]}}

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

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

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

In fact these properties imply that the covariance defines an inner product over the quotient 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 the L² inner product of real-valued functions on the sample space.

As a result, for random variables with finite variance, the inequality

$$\backslash left|\backslash operatorname\{cov\}(X,\; Y)\backslash right|\; \backslash le\; \backslash sqrt\{\backslash sigma^2(X)\; \backslash sigma^2(Y)\}$$

holds via the Cauchy–Schwarz inequality.

Proof: If $\backslash sigma^2(Y)\; =\; 0$, then it holds trivially. Otherwise, let random variable

$$Z\; =\; X\; -\; \backslash frac\{\backslash operatorname\{cov\}(X,\; Y)\}\{\backslash sigma^2(Y)\}\; Y.$$

Then we have

$$\backslash begin\{align\}$$

0 \le \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 &\le \sigma^2(X)\sigma^2(Y) \\

\left|\operatorname{cov}(X, Y)\right| &\le \sqrt{\sigma^2(X)\sigma^2(Y)}

\end{align}

{{further|Sample mean and sample covariance}}

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

:$q\_\{jk\}\; =\; \backslash frac\{1\}\{N\; -\; 1\}\backslash sum\_\{i=1\}^N\; \backslash left(X\_\{ij\}\; -\; \backslash bar\{X\}\_j\backslash right)\; \backslash left(X\_\{ik\}\; -\; \backslash bar\{X\}\_k\backslash right),$

which is an estimate of the covariance between variable $j$ and variable $k$.

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

: $q\_\{jk\}\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{i=1\}^N\; \backslash left(X\_\{ij\}\; -\; \backslash operatorname\{E\}\backslash left(X\_j\backslash right)\backslash right)\; \backslash left(X\_\{ik\}\; -\; \backslash operatorname\{E\}\backslash left(X\_k\backslash right)\backslash right)$.

{{main|Auto-covariance matrix}}

For a vector of $m$ jointly distributed random variables with finite second moments, its auto-covariance matrix (also known as the variance–covariance matrix or simply the covariance matrix) $\backslash operatorname\{K\}\_\{\backslash mathbf\{X\}\backslash mathbf\{X\}\}$ (also denoted by $\backslash Sigma(\backslash mathbf\{X\})$ or $\backslash operatorname\{cov\}(\backslash mathbf\{X\},\; \backslash mathbf\{X\})$) is defined as{{cite book |first=John A. |last=Gubner |year=2006 |title=Probability and Random Processes for Electrical and Computer Engineers |publisher=Cambridge University Press |isbn=978-0-521-86470-1}}{{rp|p=335}}

$$\backslash begin\{align\}$$

\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{align}

Let $\backslash mathbf\{X\}$ be a random vector with covariance matrix {{math|Σ}}, and let {{math|A}} be a matrix that can act on $\backslash mathbf\{X\}$ on the left. The covariance matrix of the matrix-vector product {{math|A X}} is:

$$\backslash begin\{align\}$$

\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{align}

This is a direct result of the linearity of expectation and is useful

when applying a linear transformation, such as a whitening transformation, to a vector.

{{main|Cross-covariance matrix}}

For real random vectors $\backslash mathbf\{X\}\; \backslash in\; \backslash mathbb\{R\}^m$ and $\backslash mathbf\{Y\}\; \backslash in\; \backslash mathbb\{R\}^n$, the $m\; \backslash times\; n$ cross-covariance matrix is equal to{{rp|p=336}}

{{Equation box 1

|indent = :

|title =

|equation = {{NumBlk||$\backslash begin\{align\}$

\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{align}|{{EquationRef|Eq.2}}}}

|cellpadding = 6

|border

|border colour = #0073CF

|background colour = #F5FFFA}}

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

The $(i,j)$-th element of this matrix is equal to the covariance $\backslash operatorname\{cov\}(X\_i,Y\_j)$ between the {{math|i}}-th scalar component of $\backslash mathbf\{X\}$ and the {{math|j}}-th scalar component of $\backslash mathbf\{Y\}$. In particular, $\backslash operatorname\{cov\}(\backslash mathbf\{Y\},\backslash mathbf\{X\})$ is the transpose of $\backslash operatorname\{cov\}(\backslash mathbf\{X\},\backslash mathbf\{Y\})$.

More generally let $H\_1\; =\; (H\_1,\; \backslash langle\; \backslash ,,\backslash rangle\_1)$ and $H\_2\; =\; (H\_2,\; \backslash langle\; \backslash ,,\backslash rangle\_2)$, be Hilbert spaces over $\backslash mathbb\{R\}$ or $\backslash mathbb\{C\}$ with $\backslash langle\; \backslash ,,\; \backslash rangle$ anti linear in the first variable, and let $\backslash mathbf\{X\},\; \backslash mathbf\{Y\}$ be $H\_1$ resp. $H\_2$ valued random variables.

Then the covariance of $\backslash mathbf\{X\}$ and $\backslash mathbf\{Y\}$ is the sesquilinear form on $H\_1\; \backslash times\; H\_2$

(anti linear in the first variable) given by

$$\backslash begin\{align\}$$

\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{align}

{{main|Algorithms for calculating variance#Covariance}}

When $\backslash operatorname\{E\}[XY]\; \backslash approx\; \backslash operatorname\{E\}[X]\backslash operatorname\{E\}[Y]$, the equation $\backslash operatorname\{cov\}(X,\; Y)\; =\; \backslash operatorname\{E\}\backslash left[X\; Y\backslash right]\; -\; \backslash operatorname\{E\}\backslash left[X\backslash right]\; \backslash operatorname\{E\}\backslash left[Y\backslash right]$ is prone to catastrophic cancellation if $\backslash operatorname\{E\}\backslash left[X\; Y\backslash right]$ and $\backslash operatorname\{E\}\backslash left[X\backslash right]\; \backslash operatorname\{E\}\backslash left[Y\backslash right]$ are not computed exactly and thus should be avoided in computer programs when the data has not been centered before.Donald E. Knuth (1998). The Art of Computer Programming, volume 2: Seminumerical Algorithms, 3rd edn., p. 232. Boston: Addison-Wesley. Numerically stable algorithms should be preferred in this case.{{Cite book|last1=Schubert|first1=Erich|last2=Gertz|first2=Michael|title=Proceedings of the 30th International Conference on Scientific and Statistical Database Management |chapter=Numerically stable parallel computation of (Co-)variance |date=2018|chapter-url=http://dl.acm.org/citation.cfm?doid=3221269.3223036|language=en|location=Bozen-Bolzano, Italy|publisher=ACM Press|pages=1–12|doi=10.1145/3221269.3223036|isbn=978-1-4503-6505-5|s2cid=49665540}}

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 of linear algebra (see linear dependence). When the covariance is normalized, one obtains the Pearson 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.

Covariance is an important measure in biology. Certain sequences of DNA are conserved more than others among species, and thus to study secondary and tertiary structures of proteins, or of RNA structures, sequences are compared in closely related species. If sequence changes are found or no changes at all are found in noncoding RNA (such as microRNA), 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 of evolution and natural selection, the price equation describes how a genetic trait changes in frequency over time. The equation uses a covariance between a trait and fitness, 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.{{cite journal |last1= Price | first1=George |year=1970 |title=Selection and covariance |journal=Nature |volume=227 |issue=5257 |pages=520–521 | doi=10.1038/227520a0 |pmid=5428476| bibcode=1970Natur.227..520P | s2cid=4264723 }}{{cite journal |last1= Harman |first1=Oren |year=2020 | title=When science mirrors life: on the origins of the Price equation |publisher=royalsocietypublishing.org |journal= Philosophical Transactions of the Royal Society B: Biological Sciences|volume=375 |issue=1797 |pages=1–7 | doi=10.1098/rstb.2019.0352 |pmid=32146891 |pmc=7133509 |doi-access=free }}

Covariances play a key role in financial economics, especially in modern portfolio theory and in the capital 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 a normative analysis) or are predicted to (in a positive analysis) choose to hold in a context of diversification.

{{unsourced section|date=May 2025}}

The covariance matrix is important in estimating the initial conditions required for running weather forecast models, a procedure known as data 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 to Kalman filtering and more general state estimation for time-varying systems.

The eddy 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.

The covariance matrix is used to capture the spectral variability of a signal.{{cite journal|last=Sahidullah|first=Md.|author2=Kinnunen, Tomi|title=Local spectral variability features for speaker verification|journal=Digital Signal Processing|date=March 2016|volume=50|pages=1–11|doi=10.1016/j.dsp.2015.10.011|bibcode=2016DSP....50....1S |url=https://erepo.uef.fi/handle/123456789/4375}}

The covariance matrix is used in principal component analysis to reduce feature dimensionality in data preprocessing.

{{Div col|colwidth=30em}}

• Algorithms for calculating covariance
• Analysis of covariance
• Autocovariance
• Covariance function
• Covariance matrix
• Covariance operator
• Distance covariance, or Brownian covariance.
• Law of total covariance
• Propagation of uncertainty

{{Div col end}}

{{Reflist}}

{{statistics}}

Category:Covariance and correlation

Category:Algebra of random variables