In probability theory and statistics, covariance

Covariance

In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

 is a measure of how much two variables change together and the covariance function describes the variance of a random variable process or field. For a random field

Random field

A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....

 or stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 Z(x) on a domain D, a covariance function C(x, y) gives the covariance

Covariance

In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

 of the values of the random field at the two locations x and y:


The same C(x, y) is called autocovariance

Autocovariance

In statistics, given a real stochastic process X, the autocovariance is the covariance of the variable with itself, i.e. the variance of the variable against a time-shifted version of itself...

 in two instances: in time series

Time series

In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...

 (to denote exactly the same concept, where x is time), and in multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance

Cross covariance

In statistics, the term cross-covariance is sometimes used to refer to the covariance cov between two random vectors X and Y, in order to distinguish that concept from the "covariance" of a random vector X, which is understood to be the matrix of covariances between the scalar components of X.In...

 between two different variables at different locations, Cov(Z(x₁), Y(x₂))).

Admissibility

For locations x₁, x₂, …, x_N ∈ D the variance of every linear combination


can be computed as


A function is a valid covariance function if and only if this variance is non-negative for all possible choices of N and weights w₁, …, w_N. A function with this property is called positive definite.

Simplifications with stationarity

In case of a weakly stationary

Stationary process

In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...

 random field

Random field

A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....

, where


for any lag h, the covariance function can be represented by a one-parameter function


which is called a covariogram and also a covariance function. Implicitly the C(x_i, x_j) can be computed from C_s(h) by:


The positive definiteness

Positive definiteness

Positive definiteness is a property of the following mathematical objects:* Positive-definite bilinear form* Positive-definite matrix* Positive-definite function* Positive-definite kernel* Positive-definite function on a group...

 of this single-argument version of the covariance function can be checked by Bochner's theorem

Bochner's theorem

In mathematics, Bochner's theorem characterizes the Fourier transform of a positive finite Borel measure on the real line.- Background :...

.

Parametric families of covariance functions

A simple stationary parametric covariance function is the "exponential covariance function"


where V is a scaling parameter, and d=d(x,y) is the distance between two points. Sample paths of a Gaussian process

Gaussian process

In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...

 with the exponential covariance function are not smooth. The "squared exponential covariance function"


is a stationary covariance function with smooth sample paths.

The Matérn covariance function

Matérn covariance function

In statistics, the Matérn covariance is a covariance function used in spatial statistics, geostatistics, machine learning, image analysis, and other applications of multivariate statistical analysis on metric spaces...

 and rational quadratic covariance function

Rational quadratic covariance function

In statistics, the rational quadratic covariance function is used in spatial statistics, geostatistics, machine learning, image analysis, and other fields where multivariate statistical analysis is conducted on metric spaces. It is commonly used to define the statistical covariance between...

 are two parametric families of stationary covariance functions. The Matérn family includes the exponential and squared exponential covariance functions as special cases.

See also

• Variogram
  Variogram
  In spatial statistics the theoretical variogram 2\gamma is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z...
  
  
• Random field
  Random field
  A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....
  
  
• Stochastic process
  Stochastic process
  In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
  
  
• Kriging
  Kriging
  Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
  
  
• Autocorrelation function
• Correlation function
  Correlation function
  A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...