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
.
If ,
may be interpreted as a time stamp to
recover the classical notation of a stochastic process.
If the process is a second order process, we note:
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)¶
The spatial mean is estimated from one realization of the process (see
the use case on Field or Time series).
The stochastic mean of the process is the function
defined by:
(3)¶
The stochastic mean is estimated from a sample of realizations of the
process (see the use case on the Process sample).
For an ergodic process, the stochastic mean and the spatial mean are
equal and constant (equal to the constant vector noted
):
(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.