Stochastic process definitions¶
In this document, we note:
a multivariate stochastic process of dimension , where is an event, is a domain of , is a multivariate index and ;
the random variable at index defined by ;
a realization of the process , for a given defined by .
its mean function, defined by ,
its covariance function, defined by ,
its correlation function, defined for all , by such that for all , .
We recall here some useful definitions.
Spatial (temporal) and Stochastic Mean
The spatial mean of the process is the function defined by:
If and if the mesh is a regular grid , then the spatial mean corresponds to the temporal mean defined by:
A stochastic process is normal if all its finite dimensional joint distributions are normal, which means that for all and , with , there exist and such that:
where , and and is the symmetric matrix:
A normal process is entirely defined by its mean function and its covariance function (or correlation function ).
Weak stationarity (second order stationarity)
A process is weakly stationary or stationary of second order if its mean function is constant and its covariance function is invariant by translation:
We note for as this quantity does not depend on . In the continuous case, must be equal to as it is invariant by any translation. In the discrete case, is a lattice where .
A process is stationary if its distribution is invariant by translation: , , , we have:
Spectral density function
If is a zero-mean weakly stationary continuous process and if for all , is (ie ), we define the bilateral spectral density function where is the set of -dimensional positive definite hermitian matrices, as the Fourier transform of the covariance function :
Furthermore, if for all , is (ie ), may be evaluated from as follows:
In the discrete case, the spectral density is defined for a zero-mean weakly stationary process, where with and where the previous integrals are replaced by sums.