In probability theory
is a measure of the joint variability of two random variables
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 (that is, the variables tend to show similar behavior), the covariance is positive.
In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (that is, 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 normalized version of the covariance
, the correlation coefficient
, however, shows by its magnitude the strength of the linear relation.
The sign of the covariance of two random variables X and Y
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.
is the expected value
, also known as the mean of
. The covariance is also sometimes denoted
, 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
are those of
times those of
. 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.)
Definition for complex random variables
The covariance between two complex random variables
is defined as:p. 119
Notice the complex conjugation of the second factor in the definition.
Discrete random variables
If the (real) random variable pair
can take on the values
, with equal probabilities
, then the covariance can be equivalently written in terms of the means
It can also be equivalently expressed, without directly referring to the means, as
More generally, if there are possible realizations of
but with possibly unequal probabilities
, then the covariance is
Geometric interpretation of the covariance example. Each cuboid is the bounding box of its point (x, y, f (x, y)) and the X and Y means (magenta point). The covariance is the sum of the volumes of the red cuboids minus blue cuboids.
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 (5, 6 and 7) while
can take on two (8 and 9). Their means are
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)::p. 121
Covariance of linear combinations
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:
is the joint cumulative distribution function of the random vector
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.
The converse, however, is not generally true. For example, let
be uniformly distributed in
are not independent, but
In this case, the relationship between
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
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 ,
- 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. (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
, 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 j
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
Auto-covariance matrix of real random vectors
For a vector
jointly distributed random variables with finite second moments, its auto-covariance matrix
(also known as the variance–covariance matrix
or simply the covariance matrix
(also denoted by
) is defined as:p.335
be a random vector
with covariance matrix Σ, and let A
be a matrix that can act on
on the left. The covariance matrix of the matrix-vector product A X
Cross-covariance matrix of real random vectors
is the transpose
of the vector (or matrix)
-th element of this matrix is equal to the covariance
between the i
-th scalar component of
and the j
-th scalar component of
. In particular,
is the transpose
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.
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
(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
In meteorological and oceanographic data assimilation
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.
In signal processing
The covariance matrix is used to capture the spectral variability of a signal.
In statistics and image processing
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Last edited on 30 May 2021, at 12:05
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