Stochastic process definitions¶
Notations¶
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:
(1)¶
If and if the mesh is a regular grid
, then the spatial mean corresponds to the
temporal mean defined by:
(2)¶
(3)¶
(4)¶
Normal process¶
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:
(5)¶
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:
(6)¶
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
.
Stationarity¶
A process is stationary if its
distribution is invariant by translation:
,
,
, we have:
(7)¶
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
:
(8)¶
Furthermore, if for all ,
is
(ie
),
may be evaluated from
as follows:
(9)¶
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.