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.. _covariance_model:
Covariance models
=================
We consider :math:`X: \Omega \times\cD \mapsto \Rset^{\inputDim}` a multivariate
stochastic process of dimension :math:`d`, where :math:`\omega \in \Omega`
is an event, :math:`\cD` is a domain of :math:`\Rset^{\sampleSize}`,
:math:`\vect{t}\in \cD` is a multivariate index and
:math:`X(\omega, \vect{t}) \in \Rset^{\inputDim}`.
We note :math:`X_{\vect{t}}: \Omega \rightarrow \Rset^{\inputDim}` the random variable at
index :math:`\vect{t} \in \cD` defined by
:math:`X_{\vect{t}}(\omega)=X(\omega, \vect{t})` and
:math:`X(\omega): \cD \mapsto \Rset^{\inputDim}` a realization of the process
:math:`X`, for a given :math:`\omega \in \Omega` defined by
:math:`X(\omega)(\vect{t})=X(\omega, \vect{t})`.
If the process is a second order process, we note:
- :math:`m : \cD \mapsto \Rset^{\inputDim}` its *mean function*, defined by
:math:`m(\vect{t})=\Expect{X_{\vect{t}}}`,
- :math:`C : \cD \times \cD \mapsto \cS_{\inputDim}^+(\Rset)` its
*covariance function*, defined by
:math:`C(\vect{s}, \vect{t})=\Expect{(X_{\vect{s}}-m(\vect{s}))\Tr{(X_{\vect{t}}-m(\vect{t}))}}`,
- :math:`R : \cD \times \cD \mapsto \cS_{\inputDim}^+(\Rset)` its
*correlation function*, defined for all :math:`(\vect{s}, \vect{t})`,
by :math:`R(\vect{s}, \vect{t})` such that for all :math:`(i,j)`,
:math:`R_{ij}(\vect{s}, \vect{t})=C_{ij}(\vect{s}, \vect{t})/\sqrt{C_{ii}(\vect{s}, \vect{t})C_{jj}(\vect{s}, \vect{t})}`.
In a general way, the covariance models write:
.. math::
C(\vect{s}, \vect{t}) = \mat{L}_{\rho}\left(\dfrac{\vect{s}}{\theta},
\dfrac{\vect{t}}{\theta}\right)\,
\mbox{Diag}(\vect{\sigma}) \, \mat{R} \,
\mbox{Diag}(\vect{\sigma}) \,
\Tr{\mat{L}}_{\rho}\left(\dfrac{\vect{s}}{\theta},
\dfrac{\vect{t}}{\theta}\right), \quad
\forall (\vect{s}, \vect{t}) \in \cD
where:
- :math:`\vect{\theta} \in \Rset^{\sampleSize}` is the *scale* parameter
- :math:`\vect{\sigma} \in \Rset^{\inputDim}` id the *amplitude* parameter
- :math:`\mat{L}_{\rho}(\vect{s}, \vect{t})` is the Cholesky factor of
:math:`\mat{\rho}(\vect{s}, \vect{t})`:
.. math::
\mat{L}_{\rho}(\vect{s}, \vect{t})\,\Tr{\mat{L}_{\rho}(\vect{s}, \vect{t})}
= \mat{\rho}(\vect{s}, \vect{t})
The correlation function :math:`\mat{\rho}` 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 :math:`X_{\vect{t}}`
which is given by the correlation matrix
:math:`\mat{R} \in \cS_{\inputDim}^+(\Rset)` and the vector of marginal variances
:math:`\vect{\sigma} \in \Rset^{\inputDim}`.
The spatial correlation does not depend on :math:`\vect{t} \in \cD`.
For each :math:`\vect{t}`, it links together the components of
:math:`X_{\vect{t}}`.
- the correlation between :math:`X_{\vect{s}}` and :math:`X_{\vect{t}}`
which is given by :math:`\mat{\rho}(\vect{s}, \vect{t})`.
- In the general case, the correlation links each component
:math:`X^i_{\vect{t}}` to all the components of :math:`X_{\vect{s}}`
and :math:`\mat{\rho}(\vect{s}, \vect{t}) \in \cS_{\inputDim}^+(\Rset)`;
- In some particular cases, the correlation is such that
:math:`X^i_{\vect{t}}` depends only on the component
:math:`X^i_{\vect{s}}` and that link does not depend on the component
:math:`i`. In that case, :math:`\mat{\rho}(\vect{s}, \vect{t})` can be
defined from the scalar function :math:`\rho(\vect{s}, \vect{t})` by
:math:`\mat{\rho}(\vect{s}, \vect{t}) = \rho(\vect{s}, \vect{t})\, \mat{I}_{\inputDim}`.
Then, the covariance model writes:
.. math::
C(\vect{s}, \vect{t}) = \rho\left(\dfrac{\vect{s}}{\theta},
\dfrac{\vect{t}}{\theta}\right)\,
\mbox{Diag}(\vect{\sigma}) \, \mat{R} \,
\mbox{Diag}(\vect{\sigma}), \quad
\forall (\vect{s}, \vect{t}) \in \cD
.. topic:: API:
- See :class:`~openturns.AbsoluteExponential`
- See :class:`~openturns.DiracCovarianceModel`
- See :class:`~openturns.ExponentialModel`
- See :class:`~openturns.ExponentiallyDampedCosineModel`
- See :class:`~openturns.GeneralizedExponential`
- See :class:`~openturns.MaternModel`
- See :class:`~openturns.UserDefinedStationaryCovarianceModel`
- See :class:`~openturns.SquaredExponential`
.. topic:: Examples:
- See :doc:`/auto_probabilistic_modeling/stochastic_processes/plot_create_stationary_covmodel`
- See :doc:`/auto_probabilistic_modeling/stochastic_processes/plot_user_stationary_covmodel`
- See :doc:`/auto_probabilistic_modeling/stochastic_processes/plot_userdefined_covariance_model`
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