Covariance modelsΒΆ
We consider a multivariate
stochastic process of dimension
, where
is an event,
is a domain of
,
is a multivariate index and
.
We note the random variable at
index
defined by
and
a realization of the process
, for a given
defined by
.
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
,
.
In a general way, the covariance models write:
where:
is the scale parameter
id the amplitude parameter
is the Cholesky factor of
:
The correlation function may depend on additional
specific parameters which are not made explicit here.
The global correlation is given by two separate correlations:
the spatial correlation between the components of
which is given by the correlation matrix
and the vector of marginal variances
. The spatial correlation does not depend on
. For each
, it links together the components of
.
the correlation between
and
which is given by
.
In the general case, the correlation links each component
to all the components of
and
;
In some particular cases, the correlation is such that
depends only on the component
and that link does not depend on the component
. In that case,
can be defined from the scalar function
by
. Then, the covariance model writes: