Covariance

In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance shows the tendency in the linear relationship between the variables. Covariance is positive when variables tend to show similar behavior and negative when variables tend to show opposite behavior. The magnitude of the covariance is the geometric mean of the variances that are shared for the two random variables, where a larger magnitude means two variables more strongly depend on each other.

Covariance 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. In those situations, we use the correlation coefficient, which normalizes the covariance to a value between -1 and 1 by dividing by the geometric mean of the total variances (i.e., the product of the standard deviations) for the two random variables.

A distinction is 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.

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:

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

where \(\operatorname{E}[X]\) is the expected value of \(X\), also known as the mean of \(X\). The covariance is also sometimes denoted \(\sigma_{XY}\) or \(\sigma(X,Y)\), 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: \[\begin{aligned} \operatorname{cov}(X, Y) &= \operatorname{E}\left[\left(X - \operatorname{E}\left[X\right]\right) \left(Y - \operatorname{E}\left[Y\right]\right)\right] \\ &= \operatorname{E}\left[X Y - X \operatorname{E}\left[Y\right] - \operatorname{E}\left[X\right] Y + \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right]\right] \\ &= \operatorname{E}\left[X Y\right] - \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right] - \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right] + \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right] \\ &= \operatorname{E}\left[X Y\right] - \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right]. \end{aligned}\] 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 \(\operatorname{cov}(X, Y)\) 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.)

The covariance between two complex random variables \(Z, W\) is defined as \[\operatorname{cov}(Z, W) = \operatorname{E}\left[(Z - \operatorname{E}[Z])\overline{(W - \operatorname{E}[W])}\right] = \operatorname{E}\left[Z\overline{W}\right] - \operatorname{E}[Z]\operatorname{E}\left[\overline{W}\right]\]

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 \(i = 1,\ldots,n\), 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 \[\operatorname{cov}(X,Y) = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \frac{1}{2}(x_i - x_j)(y_i - y_j) = \frac{1}{n^2} \sum_i \sum_{j>i} (x_i-x_j)(y_i - y_j).\]

More generally, if there are \(n\) 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 \(X\) 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:

\[\operatorname{cov} (X, Y) = \sum_{i=1}^{n}\sum_{j=1}^{n} p_{i,j} (x_i - E[X])(y_j - E[Y]).\]

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

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

\(X\) can take on three values (5, 6 and 7) while \(Y\) 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, \[\begin{aligned} \operatorname{cov}(X, Y) ={} &\sigma_{XY} = \sum_{(x,y)\in S}f(x, y) \left(x - \mu_X\right)\left(y - \mu_Y\right) \\[4pt] ={} &(0)(5 - 6)(8 - 8.5) + (0.4)(6 - 6)(8 - 8.5) + (0.1)(7 - 6)(8 - 8.5) +{} \\[4pt] &(0.3)(5 - 6)(9 - 8.5) + (0)(6 - 6)(9 - 8.5) + (0.2)(7 - 6)(9 - 8.5) \\[4pt] ={} &{-0.1} \; . \end{aligned}\]

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

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: \[\begin{aligned} \operatorname{cov}(X, a) &= 0 \\ \operatorname{cov}(X, X) &= \operatorname{var}(X) \\ \operatorname{cov}(X, Y) &= \operatorname{cov}(Y, X) \\ \operatorname{cov}(aX, bY) &= ab\, \operatorname{cov}(X, Y) \\ \operatorname{cov}(X+a, Y+b) &= \operatorname{cov}(X, Y) \\ \operatorname{cov}(aX+bY, cW+dV) &= ac\,\operatorname{cov}(X,W)+ad\,\operatorname{cov}(X,V)+bc\,\operatorname{cov}(Y,W)+bd\,\operatorname{cov}(Y,V) \end{aligned}\]

For a sequence \(X_1,\ldots,X_n\) of real-valued random variables and constants \(a_1,\ldots,a_n\), we have \[\operatorname{var}\left(\sum_{i=1}^n a_iX_i \right) = \sum_{i=1}^n a_i^2\operatorname{var}(X_i) + 2\sum_{i,j\,:\,i

A useful identity to compute the covariance between two random variables \(X, Y\) is the Hoeffding's covariance identity: \[\operatorname{cov}(X, Y) = \int_\mathbb R \int_\mathbb R \left(F_{(X, Y)}(x, y) - F_X(x)F_Y(y)\right) \,dx \,dy\] 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 the marginals.

Random variables whose covariance is zero are called uncorrelated. 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. 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 \(X\) be uniformly distributed in \([-1,1]\) and let \(Y = X^2\). Clearly, \(X\) and \(Y\) 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 \(Y\) 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.

\(X\) 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 \(Y

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 \(X,Y,Z,\) \(\operatorname{cov}(aX+bY,Z) = a \operatorname{cov}(X,Z) + b \operatorname{cov}(Y,Z)\)
2. symmetric: \(\operatorname{cov}(X,Y) = \operatorname{cov}(Y,X)\)
3. positive semi-definite: \(\sigma^2(X) = \operatorname{cov}(X,X) \ge 0\) for all random variables \(X\), and \(\operatorname{cov}(X,X) = 0\) 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 L inner product of real-valued functions on the sample space.

As a result, for random variables with finite variance, the inequality \[\left|\operatorname{cov}(X, Y)\right| \le \sqrt{\sigma^2(X) \sigma^2(Y)}\] is a special case of the Cauchy-Schwarz inequality.

For convenience, here is an explicit proof, which uses only the inner product properties above.

If \(\sigma^2(Y) = 0\), then it holds trivially. Otherwise, define the random variable \[Z = X - \frac{\operatorname{cov}(X, Y)}{\sigma^2(Y)} Y.\]

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

The sample covariances among \(K\) variables based on \(N\) 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 \(j\) 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 \(\textstyle \mathbf{X}\), a vector whose jth 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

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

For a vector \(\mathbf{X} = \begin{bmatrix} X_1 & X_2 & \dots & X_m \end{bmatrix}^\mathrm{T}\) 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) \(\operatorname{K}_{\mathbf{X}\mathbf{X}}\) (also denoted by \(\Sigma(\mathbf{X})\) or \(\operatorname{cov}(\mathbf{X}, \mathbf{X})\)) is defined as \[\begin{aligned} \operatorname{K}_\mathbf{XX} = \operatorname{cov}(\mathbf{X}, \mathbf{X}) &= \operatorname{E}\left[(\mathbf{X} - \operatorname{E}[\mathbf{X}]) (\mathbf{X} - \operatorname{E}[\mathbf{X}])^\mathrm{T}\right] \\ &= \operatorname{E}\left[\mathbf{XX}^\mathrm{T}\right] - \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{X}]^\mathrm{T}. \end{aligned}\]

Let \(\mathbf{X}\) be a random vector with covariance matrix Σ, and let A be a matrix that can act on \(\mathbf{X}\) on the left. The covariance matrix of the matrix-vector product A 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 of expectation and is useful when applying a linear transformation, such as a whitening transformation, to a vector.

For real random vectors \(\mathbf{X} \in \mathbb{R}^m\) and \(\mathbf{Y} \in \mathbb{R}^n\), the \(m \times n\) cross-covariance matrix is equal to

where \(\mathbf{Y}^{\mathrm T}\) is the transpose 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 the i-th scalar component of \(\mathbf{X}\) and the j-th scalar component of \(\mathbf{Y}\). In particular, \(\operatorname{cov}(\mathbf{Y},\mathbf{X})\) is the transpose of \(\operatorname{cov}(\mathbf{X},\mathbf{Y})\).

More generally let \(H_1 = (H_1, \langle \,,\rangle_1)\) and \(H_2 = (H_2, \langle \,,\rangle_2)\), be Hilbert 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 the sesquilinear form on \(H_1 \times H_2\) (anti linear in the first variable) given by \[\begin{aligned} \operatorname{K}_{X,Y}(h_1,h_2) = \operatorname{cov}(\mathbf{X},\mathbf{Y})(h_1,h_2) &= \operatorname{E}\left[\langle h_1,(\mathbf{X} - \operatorname{E}[\mathbf{X}])\rangle_1\langle(\mathbf{Y} - \operatorname{E}[\mathbf{Y}]), h_2 \rangle_2\right] \\ &= \operatorname{E}[\langle h_1,\mathbf{X}\rangle_1\langle\mathbf{Y}, h_2 \rangle_2] - \operatorname{E}[\langle h,\mathbf{X} \rangle_1] \operatorname{E}[\langle \mathbf{Y},h_2 \rangle_2] \\ &= \langle h_1, \operatorname{E}\left[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{Y} - \operatorname{E}[\mathbf{Y}])^\dagger \right]h_2 \rangle_1\\ &= \langle h_1, \left( \operatorname{E}[\mathbf{X}\mathbf{Y}^\dagger] - \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^\dagger \right) h_2 \rangle_1\\ \end{aligned}\]