View source for Covariance

{{About|the degree to which random variables vary similarly}}

In [[probability theory]] and [[statistics]], '''covariance''' is a measure of the joint variability of two [[random variable]]s.<ref>{{Cite book|title=Mathematical Statistics and Data Analysis|last=Rice|first=John|publisher=Brooks/Cole Cengage Learning|year=2007|isbn=978-0534-39942-9|location=Belmont, CA|pages=138|quote=|via=}}</ref> If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, (i.e., the variables tend to show similar behavior), the covariance is positive.<ref>{{MathWorld|urlname=Covariance|title=Covariance}}</ref> In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (i.e., the variables tend to show opposite behavior), the covariance is negative. The sign of the covariance therefore shows the tendency in the [[linear relationship]] between the variables. The magnitude of the covariance is not easy to interpret because it is not normalized and hence depends on the magnitudes of the variables. The [[Covariance and correlation|normalized version of the covariance]], the [[Pearson product-moment correlation coefficient|correlation coefficient]], however, shows by its magnitude the strength of the linear relation.

A distinction must be made between (1) the covariance of two random variables, which is a [[Statistical population|population]] [[Statistical parameter|parameter]] that can be seen as a property of the [[joint probability distribution]], and (2) the [[sample (statistics)|sample]] covariance, which in addition to serving as a descriptor of the sample, also serves as an [[Statistical estimation|estimated]] value of the population parameter.

== Definition ==
For two [[Joint distribution|jointly distributed]] [[real number|real]]-valued [[random variable]]s <math>X</math> and <math>Y</math> with finite [[second moment]]s, the covariance is defined as the expected value (or mean) of the product of their deviations from their individual expected values:<ref>Oxford Dictionary of Statistics, Oxford University Press, 2002, p. 104.</ref><ref name=KunIlPark>{{cite book | author=Park,Kun Il| title=Fundamentals of Probability and Stochastic Processes with Applications to Communications| publisher=Springer | year=2018 | isbn=978-3-319-68074-3}}</ref>{{rp|p. 119}}

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>\operatorname{cov}(X,Y) = \operatorname{E}{\big[(X - \operatorname{E}[X])(Y - \operatorname{E}[Y])\big]},</math>|{{EquationRef|Eq.1}}}}
|cellpadding= 6
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|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>\operatorname{E}[X]</math> is the [[expected value]] of <math>X</math>, also known as the mean of <math>X</math>. The covariance is also sometimes denoted <math>\sigma_{XY}</math> or <math>\sigma(X,Y)</math>, 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:
:<math>
\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}
</math>
:but this equation is susceptible to [[Loss of significance|catastrophic cancellation]] (see the section on [[Covariance#Numerical computation|numerical computation]] below).

The [[unit of measurement|units of measurement]] of the covariance <math>\operatorname{cov}(X,Y)</math> are those of <math>X</math> times those of <math>Y</math>. By contrast, [[correlation|correlation coefficients]], which depend on the covariance, are a [[dimensionless number|dimensionless]] measure of linear dependence. (In fact, correlation coefficients can simply be understood as a normalized version of covariance.)

===Definition for complex random variables===
The covariance between two complex random variables <math>Z,W</math> is defined as<ref name=KunIlPark/>{{rp|p. 119}}

:<math>\operatorname{cov}(Z,W) = \operatorname{E}[(Z-\operatorname{E}[Z])\overline{(W-\operatorname{E}[W])}] = \operatorname{E}[Z\overline{W}]-\operatorname{E}[Z]\operatorname{E}[\overline{W}]</math>

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

===Discrete random variables===
If the random variable pair <math>(X,Y)</math> can take on the values <math>(x_i,y_i)</math> for <math>i=1,\ldots,n</math>, with equal probabilities <math>p_i=1/n</math>, then the covariance can be equivalently written in terms of the means <math>\operatorname{E}[X]</math> and <math>\operatorname{E}[Y]</math> as

:<math>\operatorname{cov} (X,Y)=\frac{1}{n}\sum_{i=1}^n (x_i-E(X))(y_i-E(Y)).</math>

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

:<math> \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). </math>

More generally, if there are <math>n</math> possible realizations of <math>(X,Y)</math>, namely <math>(x_i,y_i)</math> but with possibly equal probabilities <math>p_i</math> for <math>i=1,\ldots,n</math>, then the covariance is

:<math>\operatorname{cov} (X,Y)=\sum_{i=1}^n p_i (x_i-E(X)) (y_i-E(Y)).</math>

==Example==
Suppose that <math>X</math> and <math>Y</math> have the following [[joint probability distribution|joint probability mass function]],<ref>{{Cite web|url=https://onlinecourses.science.psu.edu/stat414/node/109|title=Covariance of X and Y {{!}} STAT 414/415|last=|first=|website=|publisher=The Pennsylvania State University|access-date=August 4, 2019}}</ref> in which the six central cells give the discrete joint probabilities <math>f(x,y)</math> of the six hypothetical realizations <math>(x,y) \in S = \left\{ (1, 1), (1, 2), (1, 3), (2, 1), (2,2), (2, 3) \right\}</math>:
{| class="wikitable"
!
!
!
!y
!
!
|-
!
!<math>f(x,y)</math>
!1
!2
!3
!<math>f_X(x)</math>
|-
!
!1
|1/4
|1/4
|0
|1/2
|-
!x
!2
|0
|1/4
|1/4
|1/2
|-
!
!<math>f_Y(y)</math>
|1/4
|1/2
|1/4
|1
|}
<math>X</math> can take on two values (1 and 2) while <math>Y</math> can take on three (1, 2, and 3). Their means are <math>\mu_X = 3/2</math> and <math>\mu_Y = 2.</math>  The population standard deviations of <math>X</math> and <math>Y</math> are <math>\sigma_X = 1/2</math> and <math>\sigma_Y = \sqrt{2/3}.</math> Then:

:<math>
\begin{align}
\operatorname{cov}(X,Y) = {} & \sigma_{XY} = \sum_{(x,y)\in S}f(x,y) (x - \mu_X)(y - \mu_Y) \\[4pt]
= {} & \left( \frac{1}{4} \right)\left(1 - \frac{3}{2} \right)(1 - 2) + \left(\frac{1}{4} \right) \left(1 - \frac{3}{2}\right)(2 - 2)  \\[4pt]
& {} + (0)\left(1 - \frac{3}{2} \right)(3 - 2) + (0) \left(2 - \frac{3}{2} \right)(1 - 2) \\[4pt]
& {} +\left(\frac{1}{4} \right) \left(2 - \frac{3}{2}\right)(2 - 2)  + \left(\frac{1}{4} \right) \left(2 - \frac{3}{2} \right)(3 - 2) \\[4pt]
= {} & \frac{1}{4}.
\end{align}
</math>

== Properties ==
===Covariance with itself===
The [[variance]] is a special case of the covariance in which the two variables are identical (that is, in which one variable always takes the same value as the other):<ref name=KunIlPark/>{{rp|p. 121}}
:<math>\operatorname{cov}(X, X) =\operatorname{var}(X)\equiv\sigma^2(X)\equiv\sigma_X^2.</math>

===Covariance of linear combinations===
If <math>X</math>, <math>Y</math>, <math>W</math>, and <math>V</math> are real-valued random variables and <math>a,b,c,d</math> are real-valued constants, then the following facts are a consequence of the definition of covariance:
: <math>
\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}
</math>

For a sequence <math>X_1,\ldots,X_n</math> of random variables in real-valued, and constants <math>a_1,\ldots,a_n</math>, we have

:<math>\sigma^2\left(\sum_{i=1}^n a_iX_i \right) = \sum_{i=1}^n a_i^2\sigma^2(X_i) + 2\sum_{i,j\,:\,i<j} a_ia_j\operatorname{cov}(X_i,X_j) = \sum_{i,j} {a_ia_j\operatorname{cov}(X_i,X_j)}
</math>

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

:<math> \operatorname{cov}(X, Y) = \int_\mathbb R \int_\mathbb R (F_{(X,Y)}(x, y) - F_X(x)F_Y(y)) \,dx \, dy </math>

where <math> F_{(X,Y)}(x,y) </math> is the joint distribution function of the random vector <math> (X, Y) </math> and <math> F_X(x), F_Y(y) </math> are the [[Marginal distribution|marginals]].

===Uncorrelatedness and independence===
{{main|Correlation and dependence}}
Random variables whose covariance is zero are called [[uncorrelated]]<ref name=KunIlPark/>{{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 <math>X</math> and <math>Y</math> are [[statistical independence|independent random variables]], then their covariance is zero.<ref name=KunIlPark/>{{rp|p. 123}}<ref>{{Cite web|url=http://www.math.uah.edu/stat/expect/Covariance.html|title=Covariance and Correlation|last=Siegrist|first=Kyle|date=|website=|publisher=University of Alabama in Huntsville|access-date=August 4, 2019}}</ref> This follows because under independence,

: <math>\operatorname{E}[XY]=\operatorname{E}[X] \cdot \operatorname{E}[Y]. </math>

The converse, however, is not generally true. For example, let <math>X</math> be uniformly distributed in <math>[-1,1]</math> and let <math>Y=X^2</math>. Clearly, <math>X</math> and <math>Y</math> are not independent, but
: <math>
\begin{align}
 \operatorname{cov}(X, Y) &= \operatorname{cov}(X, X^2) \\
         &= \operatorname{E}[X \cdot X^2] - \operatorname{E}[X] \cdot \operatorname{E}[X^2] \\
         &= \operatorname{E}\left[X^3\right] - \operatorname{E}[X]\operatorname{E}[X^2]  \\
         &= 0 - 0 \cdot \operatorname{E}[X^2]   \\
         &= 0.  
\end{align}
</math>
In this case, the relationship between <math>Y</math> and <math>X</math> 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 [[Multivariate normal distribution|jointly normally distributed]] (but not if they are merely [[Normally distributed and uncorrelated does not imply independent|individually normally distributed]]), uncorrelatedness ''does'' imply independence.

=== Relationship to inner products ===
Many of the properties of covariance can be extracted elegantly by observing that it satisfies similar properties to those of an [[inner product]]:
# [[bilinear operator|bilinear]]:  for constants <math>a</math> and <math>b</math> and random variables <math>X,Y,Z</math>, <math>\operatorname{cov}(aX+bY,Z) = a \operatorname{cov}(X,Z) + b \operatorname{cov}(Y,Z)</math>
# symmetric: <math>\operatorname{cov}(X,Y) = \operatorname{cov}(Y,X)</math>
# [[definite bilinear form|positive semi-definite]]: <math>\sigma^2(X) = \operatorname{cov}(X,X) \ge 0</math> for all random variables <math>X</math>, and <math>\operatorname{cov}(X,X) = 0</math> implies that <math>X</math> is constant [[almost surely]].

In fact these properties imply that the covariance defines an inner product over the [[quotient space (linear algebra)|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 [[Lp space|L<sup>2</sup>]] inner product of real-valued functions on the sample space.

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

: <math>|\operatorname{cov}(X,Y)| \le \sqrt{\sigma^2(X) \sigma^2(Y)} </math>

holds via the [[Cauchy–Schwarz inequality]].

Proof: If <math>\sigma^2(Y) = 0</math>, then it holds trivially. Otherwise, let random variable

: <math> Z = X - \frac{\operatorname{cov}(X,Y)}{\sigma^2(Y)} Y.</math>

Then we have

: <math>
\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)}.
\end{align}
</math>

== Calculating the sample covariance ==
{{Main article|Sample mean and sample covariance}}

The sample covariances among <math>K</math> variables based on <math>N</math> observations of each, drawn from an otherwise unobserved population, are given by the <math>K \times K</math> [[Matrix (mathematics)|matrix]] <math>\textstyle \overline{\mathbf{ q }}=\left[ q_{jk}\right] </math> with the entries

:<math> q_{jk}=\frac{1}{N-1}\sum_{i=1}^{N}\left(  X_{ij}-\bar{X}_j \right)  \left( X_{ik}-\bar{X}_k \right), </math>

which is an estimate of the covariance between variable <math>j</math> and variable <math>k</math>.

The sample mean and the sample covariance matrix are [[Bias of an estimator|unbiased estimates]] of the [[mean]] and the [[covariance matrix]] of the [[random vector]] <math>\textstyle \mathbf{X}</math>, a vector whose ''j''th element <math>(j=1,\ldots,K)</math> is one of the random variables. The reason the sample covariance matrix has <math>\textstyle N-1</math> in the denominator rather than <math>\textstyle N</math> is essentially that the population mean <math>\operatorname{E}(X)</math> is not known and is replaced by the sample mean <math>\mathbf{\bar{X}}</math>. If the population mean <math>\operatorname{E}(X)</math> is known, the analogous unbiased estimate is given by

: <math> q_{jk}=\frac 1 N \sum_{i=1}^N \left(  X_{ij} - \operatorname{E}(X_j)\right)  \left( X_{ik} - \operatorname{E}(X_k)\right)</math>.

== Generalizations ==
=== Auto-covariance matrix of real random vectors ===
For a vector <math>\mathbf{X} = \begin{bmatrix}X_1 & X_2 & \dots & X_m\end{bmatrix}^\mathrm{T}</math> of <math>m</math> jointly distributed random variables with finite second moments, its [[auto-covariance matrix]] (also known as the '''variance–covariance matrix''' or simply the '''covariance matrix''') <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}</math> (also denoted by <math>\Sigma(\mathbf{X})</math>) is defined as<ref name=Gubner>{{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}}</ref>{{rp|p.335}}

:<math>\begin{align}
\operatorname{K}_{\mathbf{X}\mathbf{X}} = \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{X} \mathbf{X}^\mathrm{T}\right] - \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{X}]^\mathrm{T}.
\end{align}</math>

Let <math>\mathbf{X}</math> be a [[random vector]] with covariance matrix {{math|Σ}}, and let {{math|'''A'''}} be a matrix that can act on <math>\mathbf{X}</math> on the left. The covariance matrix of the matrix-vector product {{math|'''A X'''}} is:
:<math>\Sigma(\mathbf{A} \mathbf{X}) = \operatorname{E}[\mathbf{A} \mathbf{X} \mathbf{X}^{\mathrm{T}} \mathbf{A}^{\mathrm{T}}] - \operatorname{E}[\mathbf{A} \mathbf{X}] \operatorname{E}[\mathbf{X}^{\mathrm{T}} \mathbf{A}^{\mathrm{T}}]= \mathbf{A} \Sigma \mathbf{A}^\mathrm{T}.</math>

This is a direct result of the linearity of [[expected value|expectation]] and is useful
when applying a [[linear transformation]], such as a [[whitening transformation]], to a vector.

=== Cross-covariance matrix of real random vectors ===
For real [[random vector]]s <math>\mathbf{X} \in \mathbb{R}^m</math> and <math>\mathbf{Y} \in \mathbb{R}^n</math>, the <math>m \times n</math> [[cross-covariance matrix]] is equal to<ref name=Gubner/>{{rp|p.336}}

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>
\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}
</math>|{{EquationRef|Eq.2}}}}
|cellpadding= 6
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|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>\mathbf{Y}^{\mathrm T}</math> is the [[transpose]] of the vector (or matrix) <math>\mathbf{Y}</math>.

The <math>(i,j)</math>-th element of this matrix is equal to the covariance <math>\operatorname{cov}(X_i,Y_j)</math> between the {{math|''i''}}-th scalar component of <math>\mathbf{X}</math> and the {{math|''j''}}-th scalar component of <math>\mathbf{Y}</math>. In particular, <math>\operatorname{cov}(\mathbf{Y},\mathbf{X})</math> is the [[transpose]]  of <math>\operatorname{cov}(\mathbf{X},\mathbf{Y})</math>.

== Numerical computation ==
When <math>\operatorname{E}[XY] \approx \operatorname{E}[X]\operatorname{E}[Y]</math>, the equation <math>\operatorname{cov}(X, Y) = \operatorname{E}\left[X Y\right] - \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right]</math> is prone to [[catastrophic cancellation]] when computed with [[floating point]] arithmetic and thus should be avoided in computer programs when the data has not been centered before.<ref>[[Donald E. Knuth]] (1998). ''[[The Art of Computer Programming]]'', volume 2: ''Seminumerical Algorithms'', 3rd edn., p. 232. Boston: Addison-Wesley.</ref> [[Algorithms for calculating variance#Covariance|Numerically stable algorithms]] should be preferred in this case.<ref>{{Cite journal|last=Schubert|first=Erich|last2=Gertz|first2=Michael|date=2018|title=Numerically stable parallel computation of (co-)variance|url=http://dl.acm.org/citation.cfm?doid=3221269.3223036|journal=Proceedings of the 30th International Conference on Scientific and Statistical Database Management – SSDBM '18|language=en|location=Bozen-Bolzano, Italy|publisher=ACM Press|pages=1–12|doi=10.1145/3221269.3223036|isbn=9781450365055}}</ref>

== 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 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.

==Applications==

=== In genetics and molecular biology ===
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 [[protein]]s, 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 financial economics===

Covariances play a key role in [[financial economics]], especially in [[Modern portfolio theory|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 economics|normative analysis]]) or are predicted to (in a [[Positive economics|positive analysis]]) choose to hold in a context of [[Diversification (finance)|diversification]].

===In meteorological and oceanographic data assimilation===
The covariance matrix is important in estimating the initial conditions required for running weather forecast models. 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.

===In micrometeorology ===
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.

===In feature extraction===
The covariance matrix is used to capture the spectral variability of a signal.<ref>{{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}}</ref>

== See also ==
{{Div col|colwidth=30em}}
* [[Algorithms for calculating variance#Covariance|Algorithms for calculating covariance]]
* [[Analysis of covariance]]
* [[Autocovariance]]
* [[Correlation and dependence]]
* [[Covariance function]]
* [[Covariance matrix]]
* [[Covariance operator]]
* [[Distance covariance]], or Brownian covariance.
* [[Eddy covariance]]
* [[Law of total covariance]]
* [[Propagation of uncertainty]]
{{Div col end}}

==References==
{{Refimprove|date=December 2010}}
{{More footnotes|date=December 2010}}
{{Reflist}}

== External links ==
{{Wiktionary|covariance}}
* {{springer|title=Covariance|id=p/c026800}}
* [http://mathworld.wolfram.com/Covariance.html MathWorld page on calculating the sample covariance]
* [http://www.r-tutor.com/elementary-statistics/numerical-measures/covariance Covariance Tutorial using R]
* [http://itfeature.com/probability/covariance-and-correlation Covariance and Correlation]

{{statistics}}

[[Category:Covariance and correlation]]
[[Category:Algebra of random variables]]