In probability theory and statistics, covariance is a measure of how much two variables change together, and the covariance function, or kernel.

**COVARIANCE FUNCTIONS:**

In
probability theory and statistics, covariance is a measure of how much two
variables change together, and the covariance function, or kernel, describes
the spatial covariance of a random variable process or field. For a random
field or stochastic process Z(x) on a domain D, a covariance function C(x, y)
gives the covariance of the values of the random field at the two locations x
and y:

The same
C(x, y) is called the auto covariance function in two instances: in time series
(to denote exactly the same concept except that x and y refer to locations in
time rather than in space), and in multivariate random fields (to refer to the
covariance of a variable with itself, as opposed to the cross covariance
between two different variables at different locations, Cov(Z(x_{1}), Y(x** _{2}**)))

^{ü} **Mean & Variance of covariance
functions:**

For
locations x_{1}, x_{2}, …, xN ∈ D the variance of every linear combination

A
function is a valid covariance function if and only if this variance is
non-negative for all possible choices of N and weights w_{1}, …, w_{N}. A function
with this property is called positive definite.

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Communication Theory : Random Process : Covariance Functions |

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