Covariance

From Wikipedia, the free encyclopedia

For the physics topics, see covariant transformation; about the mathematics example for groupoids, see covariance in special relativity; for the computer science topic see covariance and contravariance (computer science), in mathematics and theoretical physics see covariance and contravariance of vectors, in category theory see covariance and contravariance of functors.

In probability theory and statistics, covariance is the measure of how much two variables change together (as distinct from variance, which measures how much a single variable changes).

If two variables tend to vary together (that is, when one of them is above its expected value, then the other variable tends to be above its expected value too), then the covariance between the two variables will be positive. On the other hand, if one of them is above its expected value and the other variable tends to be below its expected value, then the covariance between the two variables will be negative.

1 Definition
2 Properties
- 2.1 Incremental Computation
- 2.2 Relationship to inner products
3 Covariance matrix, operator, bilinear form, and function
4 Comments
5 See also
6 External links

[edit] Definition

The covariance between two real-valued random variables X and Y, with expected values $\scriptstyle E(X)\,=\,\mu$ and $\scriptstyle E(Y)\,=\,\nu$ is defined as

$\operatorname{Cov}(X, Y) = \operatorname{E}((X - \mu) (Y - \nu)), \,$

where E is the expected value operator. This can also be written:

$\operatorname{Cov}(X, Y) = \operatorname{E}(X \cdot Y - \mu Y - \nu X + \mu \nu), \,$

$\operatorname{Cov}(X, Y) = \operatorname{E}(X \cdot Y) - \mu \operatorname{E}(Y) - \nu \operatorname{E}(X) + \mu \nu, \,$

$\operatorname{Cov}(X, Y) = \operatorname{E}(X \cdot Y) - \mu \nu. \,$

If X and Y are independent, then their covariance is zero. This follows because under independence,

$E(X \cdot Y)=E(X) \cdot E(Y)=\mu\nu.$

Recalling the final form of the covariance derivation given above, and substituting, we get

$\operatorname{Cov}(X, Y) = \mu \nu - \mu \nu = 0.$

The converse, however, is not true: if X and Y have covariance zero, they need not be independent.

The units of measurement of the covariance Cov(X, Y) are those of X times those of Y. By contrast, correlation, which depends on the covariance, is a dimensionless measure of linear dependence.

Random variables whose covariance is zero are called uncorrelated.

[edit] Properties

If X, Y are real-valued random variables and a, b, c, d are constant ("constant" in this context means non-random), then the following facts are a consequence of the definition of covariance:

$\operatorname{Cov}(X, X) = \operatorname{Var}(X)\,$

$\operatorname{Cov}(X, Y) = \operatorname{Cov}(Y, X)\,$

$\operatorname{Cov}(aX, bY) = ab\, \operatorname{Cov}(X, Y)\,$

$\operatorname{Cov}(X+a, Y+b) = \operatorname{Cov}(X, Y)\,$

$\operatorname{Cov}(aX+bY, cW+dV) = ac\,\operatorname{Cov}(X,W)+ad\,\operatorname{Cov}(X,V)+bc\,\operatorname{Cov}(Y,W)+bd\,\operatorname{Cov}(Y,V)\,$

For sequences X₁, ..., X_n and Y₁, ..., Y_m of random variables, we have

$\operatorname{Cov}\left(\sum_{i=1}^n {X_i}, \sum_{j=1}^m{Y_j}\right) = \sum_{i=1}^n{\sum_{j=1}^m{\operatorname{Cov}\left(X_i, Y_j\right)}}.\,$

For a sequence X₁, ..., X_n of random variables, we have

$\operatorname{Var}\left(\sum_{i=1}^n X_i \right) = \sum_{i=1}^n \operatorname{Var}(X_i) + 2\sum_{i,j\,:\,i<j} \operatorname{Cov}(X_i,X_j).$

[edit] Incremental Computation

Covariance can be computed efficiently from incrementally available values using a generalization of the computational formula for the variance:

$\operatorname{Cov}(X_i, X_j) = \operatorname{E}\left((X_i-\operatorname{E}(X_i))(X_j-\operatorname{E}(X_j))\right) = \operatorname{E}(X_iX_j) -\operatorname{E}(X_i)\operatorname{E}(X_j)$

[edit] 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:

(1) bilinear: for constants a and b and random variables X, Y, and U, Cov(aX + bY, U) = a Cov(X, U) + bCov(Y, U)

(2) symmetric: Cov(X, Y) = Cov(Y, X)

(3) positive definite: Var(X) = Cov(X, X) ≥ 0, and Cov(X, X) = 0 implies that X is a constant random variable (K).

It can be shown that the covariance is an inner product over some subspace of the vector space of random variables with finite second moment.

[edit] Covariance matrix, operator, bilinear form, and function

For column-vector valued random variables X and Y with respective expected values μ and ν, and respective scalar components m and n, the covariance is defined to be the m×n matrix called the covariance matrix:

$\operatorname{Cov}(X, Y) = \operatorname{E}((X-\mu)(Y-\nu)^\top).\,$

For vector-valued random variables, Cov(X, Y) and Cov(Y, X) are each other's transposes.

More generally, for a probability measure P on a Hilbert space H with inner product $\langle \cdot,\cdot\rangle$ , the covariance of P is the bilinear form Cov: H × H → H given by

$\mathrm{Cov}(x, y) = \int_{H} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \mathbf{P} (z)$

for all x and y in H. The covariance operator C is then defined by

$\mathrm{Cov}(x, y) = \langle Cx, y \rangle$

(from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is self-adjoint (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case). When P is a centred Gaussian measure, C is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace.

Even more generally, for a probability measure P on a Banach space B, the covariance of P is the bilinear form on the algebraic dual $B^\#$ , defined by

$\mathrm{Cov}(x, y) = \int_{B} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \mathbf{P} (z)$

where $\langle x, z \rangle$ is now the value of the linear fuctional x on the element z.

Quite similarly, the covariance function of a function-valued random element (in special cases called random process or random field) z is

$\mathrm{Cov}(x, y) = \int z(x) z(y) \, \mathrm{d} \mathbf{P} (z) = E(z(x) z(y)),$

where $z (x)$ is now the value of the function z at the point x, i.e., the value of the linear functional $u \mapsto u(x)$ evaluated at z.

[edit] 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 correlation matrix. From it, one can obtain the Pearson coefficient, which gives us 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.