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Covariance

Covarianceinprobability theory and statistics is a measure of the joint variability of two random variables.^[1]

The sign of the covariance, therefore, shows the tendency in the linear 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. The magnitude of the covariance is the geometric mean of the variances that are in common for the two random variables. The correlation coefficient normalizes the covariance by dividing by the geometric mean of the total variances for the two random variables.

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

Mathematical definition[edit]

For two jointly distributed real-valued random variables ${\displaystyle X}$ and ${\displaystyle Y}$ 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:^{[3][4]: 119}

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

The units of measurement of the covariance ${\displaystyle \operatorname {cov} (X,Y)}$ are those of ${\displaystyle X}$ times those of ${\displaystyle Y}$. 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.)

Complex random variables[edit]

The covariance between two complex random variables ${\displaystyle Z,W}$ is defined as^[4]: 119

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

A related pseudo-covariance can also be defined.

Discrete random variables[edit]

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

It can also be equivalently expressed, without directly referring to the means, as^[5]

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

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

Examples[edit]

Consider 3 independent random variables ${\displaystyle A,B,C}$ and two constants ${\displaystyle q,r}$.

${\displaystyle q=1}$

${\displaystyle r=1}$

${\displaystyle X}$

${\displaystyle Y}$

${\displaystyle A}$

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

${\displaystyle f(x,y)}$		x				${\displaystyle f_{Y}($y)}
		5	6	7
y	8	0	0.4	0.1		0.5
	9	0.3	0	0.2		0.5
${\displaystyle f_{X}($x)}		0.3	0.4	0.3	1

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

Properties[edit]

Covariance with itself[edit]

The variance is a special case of the covariance in which the two variables are identical:^[4]: 121

Covariance of linear combinations[edit]

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

For a sequence ${\displaystyle X_{1},\ldots ,X_{n}}$ of random variables in real-valued, and constants ${\displaystyle a_{1},\ldots ,a_{n}}$, we have

Hoeffding's covariance identity[edit]

A useful identity to compute the covariance between two random variables ${\displaystyle X,Y}$ is the Hoeffding's covariance identity:^[7]

${\displaystyle F_{(X,Y)}(x,y)}$

${\displaystyle (X,Y)}$

${\displaystyle F_{X}($x),F_{Y}(y)}

Uncorrelatedness and independence[edit]

Random variables whose covariance is zero are called uncorrelated.^[4]: 121 Similarly, the components of random vectors whose covariance matrix is zero in every entry outside the main diagonal are also called uncorrelated.

If${\displaystyle X}$ and ${\displaystyle Y}$ are independent random variables, then their covariance is zero.^{[4]: 123 [8]} This follows because under independence,

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

In this case, the relationship between ${\displaystyle Y}$ and ${\displaystyle X}$ 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 are jointly normally distributed (but not if they are merely individually normally distributed), uncorrelatedness does imply independence.^[9]

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

Relationship to inner products[edit]

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

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

Then we have

Calculating the sample covariance[edit]

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

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

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

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

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

Generalizations[edit]

Auto-covariance matrix of real random vectors[edit]

For a vector ${\displaystyle \mathbf {X} ={\begin{bmatrix}X_{1}&X_{2}&\dots &X_{m}\end{bmatrix}}^{\mathrm {T} }}$of${\displaystyle m}$ jointly distributed random variables with finite second moments, its auto-covariance matrix (also known as the variance–covariance matrix or simply the covariance matrix) ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}$ (also denoted by ${\displaystyle \Sigma (\mathbf {X} )}$or${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {X} )}$) is defined as^[10]: 335

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

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.

Cross-covariance matrix of real random vectors[edit]

For real random vectors ${\displaystyle \mathbf {X} \in \mathbb {R} ^{m}}$ and ${\displaystyle \mathbf {Y} \in \mathbb {R} ^{n}}$, the ${\displaystyle m\times n}$ cross-covariance matrix is equal to^[10]: 336

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

(Eq.2)

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

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