Covariance

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 Physics, a covariant transformation is a rule (specified below that describes how certain physical entities change under a change of Coordinate system. In Mathematics, especially in Category theory and Homotopy theory In discussion of the Type system of a Programming language, an operator from types to types is covariant if it preserves the ordering ≤ of types, which For other uses of "covariant" or "contravariant" see Covariance and contravariance. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

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). Probability theory is the branch of Mathematics concerned with analysis of random phenomena Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of

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

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. In Mathematics, the real numbers may be described informally in several different ways A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way 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. In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other 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. In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units

Random variables whose covariance is zero are called uncorrelated. In Probability theory and Statistics, two real-valued Random variables are said to be uncorrelated if their Covariance is zero

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

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

Incremental Computation

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

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

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). In Probability theory, the computational formula for the Variance Var( X) of a Random variable X is the formula \operatorname{Var}(X In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, a bilinear map is a function of two arguments that is linear in each In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed

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.

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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Statistics and Probability theory, the covariance matrix is a matrix of Covariances between elements of a vector This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a

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. A probability space, in Probability theory, is the conventional Mathematical model of Randomness. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. 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). There are several well-known theorems in Functional analysis known as the Riesz representation theorem. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where 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). In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix When P is a centred Gaussian measure, C is also a nuclear operator. In Mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R n, closely related to the In Mathematics, a nuclear operator is roughly a Compact operator for which a trace may be defined such that the trace is finite and independent of the In particular, it is a compact operator of trace class, that is, it has finite trace. In Functional analysis, a branch of Mathematics, a compact operator is a Linear operator L from a Banach space X to another In Mathematics, a trace class operator is a Compact operator for which a trace may be defined such that the trace is finite and independent of the choice In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal

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. A probability space, in Probability theory, is the conventional Mathematical model of Randomness. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals

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. For a Random field or Stochastic process Z ( x) on a domain D, a covariance function C ( x, y) gives the The term random element was introduced by Maurice Frechet in 1948 to refer to a Random variable that takes values in spaces more general than had previously been widely A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real but can instead be a multidimensional vector space or even a manifold e. , the value of the linear functional $u \mapsto u(x)$ evaluated at z. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional

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). Linear algebra is the branch of Mathematics concerned with In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors When the covariance is normalized, one obtains the correlation matrix. In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear 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 Statistics, the Pearson product-moment correlation coefficient (sometimes referred to as the MCV or PMCC, and typically denoted by r In this sense covariance is a linear gauge of dependence.

External links

MathWorld page on calculating the sample covariance

For a Random field or Stochastic process Z ( x) on a domain D, a covariance function C ( x, y) gives the In Statistics and Probability theory, the covariance matrix is a matrix of Covariances between elements of a vector In Statistics, given a real Stochastic process X ( t) the autocovariance is simply the Covariance of the signal against a time-shifted Analysis of covariance (ANCOVA is a General linear model with one continuous outcome variable and one or more factors

Dictionary

-noun

(statistics) A statistical measure defined as <math>\scriptstyle\operatorname{Cov}(X, Y) = \operatorname{E}((X - \mu) (Y - \nu))</math> given two real-valued random variables X and Y, with expected values <math>\scriptstyle E(X)\,=\,\mu</math> and <math>\scriptstyle E(Y)\,=\,\nu</math>.

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