Covariance

In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
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, the covariance is positive. In the opposite case, when greater values of one variable mainly correspond to lesser values of the other, 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 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 by dividing by the geometric mean of the total variances for the two random variables to get a result between -1 and 1 and makes the units irrelevant.
A distinction must be made between the covariance of two random variables, which is a population parameter that can be seen as a property of the joint probability distribution, and the sample covariance, which in addition to serving as a descriptor of the sample, also serves as an estimated value of the population parameter.

Definition

For two jointly distributed real-valued random variables and with finite second moments, the covariance is defined as the expected value of the product of their deviations from their individual expected values:
where is the expected value of, also known as the mean of. The covariance is also sometimes denoted or, 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:
This identity is useful for mathematical derivations. From the viewpoint of [|numerical computation], however, it is susceptible to catastrophic cancellation.
The units of measurement of the covariance are those of times those of. By contrast, correlation coefficients, which depend on the covariance, are a dimensionless measure of linear dependence.

Complex random variables

The covariance between two complex random variables is defined as
Notice the complex conjugation of the second factor in the definition.
A related pseudo-covariance can also be defined.

Discrete random variables

If the random variable pair can take on the values for, with equal probabilities, then the covariance can be equivalently written in terms of the means and as
It can also be equivalently expressed, without directly referring to the means, as
More generally, if there are possible realizations of, namely but with possibly unequal probabilities for, then the covariance is
In the case where two discrete random variables and have a joint probability distribution, represented by elements corresponding to the joint probabilities of, the covariance is calculated using a double summation over the indices of the matrix:

Examples

Consider three independent random variables and two constants.
In the special case, and, the covariance between and is just the variance of and the name covariance is entirely appropriate.
File:Covariance_geometric_visualisation.svg|thumb|300px|Geometric interpretation of the covariance example. axis-aligned bounding box of its point and the . is the sum of the volumes of the cuboids in the 1st and 3rd quadrants and in the 2nd and 4th.
Suppose that and have the following joint probability mass function, in which the six central cells give the discrete joint probabilities of the six hypothetical realizations
can take on three values while can take on two. Their means are and. Then,

Properties

Covariance with itself

The variance is a special case of the covariance in which the two variables are identical:

Covariance of linear combinations

If,,, and are real-valued random variables and are real-valued constants, then the following facts are a consequence of the definition of covariance:
For a sequence of random variables in real-valued, and constants, we have

Hoeffding's covariance identity

A useful identity to compute the covariance between two random variables is the Hoeffding's covariance identity:
where is the joint cumulative distribution function of the random vector and are the marginals.

Uncorrelatedness and independence

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 and are independent random variables, then their covariance is zero. This follows because under independence,
The converse, however, is not generally true. For example, let be uniformly distributed in and let. Clearly, and are not independent, but
In this case, the relationship between and 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, uncorrelatedness does imply independence.
and whose covariance is positive are called positively correlated, which implies if then likely. Conversely, and with negative covariance are negatively correlated, and if then likely.

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: for constants and and random variables
symmetric:
positive semi-definite: for all random variables, and implies that 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. 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
holds via the Cauchy–Schwarz inequality.
Proof: If, then it holds trivially. Otherwise, let random variable
Then we have

Calculating the sample covariance

The sample covariances among variables based on observations of each, drawn from an otherwise unobserved population, are given by the matrix with the entries
which is an estimate of the covariance between variable and variable.
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a vector whose jth element is one of the random variables. The reason the sample covariance matrix has in the denominator rather than is essentially that the population mean is not known and is replaced by the sample mean. If the population mean is known, the analogous unbiased estimate is given by

Generalizations

Auto-covariance matrix of real random vectors

For a vector of jointly distributed random variables with finite second moments, its auto-covariance matrix is defined as
Let be a random vector with covariance matrix, and let be a matrix that can act on on the left. The covariance matrix of the matrix-vector product 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

For real random vectors and, the cross-covariance matrix is equal to
where is the transpose of the vector .
The -th element of this matrix is equal to the covariance between the -th scalar component of and the -th scalar component of. In particular, is the transpose of.

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

More generally let and, be Hilbert spaces over or with anti linear in the first variable, and let be resp. valued random variables.
Then the covariance of and is the sesquilinear form on
given by

Numerical computation

When, the equation is prone to catastrophic cancellation if and are not computed exactly and thus should be avoided in computer programs when the data has not been centered before. Numerically stable algorithms should be preferred in this case.

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. 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 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, 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 , 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.