Health Dictionary
Reliable, easy-to-read patient information on common medical conditions — written in plain language, free of jargon, and grounded in current evidence. Browse the most frequent diagnoses, understand what they mean for everyday life, and learn which steps and questions help you have a focused conversation with your doctor.
Covariance function
Introduction
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:
- <math>C(x,y):=Cov(Z(x),Z(y)).\,</math>
The same C(x, y) is called autocovariance in two instances: in time series (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 between two different variables at different locations, Cov(Z(x1), Y(x2))).[1]
Admissibilty
Admissibilty
For locations x1, x2, …, xN ∈ D the variance of every linear combination
- <math>X=\sum_{i=1}^N w_i Z(x_i)</math>
can be computed as
- <math>var(X)=\sum_{i=1}^N \sum_{j=1}^N w_i C(x_i,x_j) w_j</math>
A function is a valid covariance function if and only if[2] this variance is non-negative for all possible choices of N and weights w1, …, wN. A function with this property is called positive definite.
Simplifications with Stationarity
Simplifications with Stationarity
In case of a weakly stationary random field, where
- <math>C(x_i,x_j)=C(x_i+h,x_j+h)\,</math>
for any lag h, the covariance function can represented by a one parameter function
- <math>C_s(h)=C(0,h)=C(x,x+h)\,</math>
which is called a covariogram and also a covariance function. Implicitly the C(xi, xj) can be computed from Cs(h) by:
- <math>C(x,y)=C_s(y-x)\,</math>
The positive definiteness of this single argument version of the covariance function can be checked by Bochner’s theorem.[2]