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.. _process_transformation:
Process transformation
----------------------
The objective here is to create a process :math:`Y` as the image through
a field function :math:`f_{dyn}` of another process :math:`X`:
.. math::
\begin{aligned}
Y=f_{dyn}(X)\end{aligned}
General case
~~~~~~~~~~~~
In the general case, :math:`X: \Omega \times\cD \rightarrow \Rset^{\inputDim}`
is a multivariate stochastic process of dimension :math:`\inputDim` where
:math:`\cD \in \Rset^n`,
:math:`Y: \Omega \times \cD' \rightarrow \Rset^q` a multivariate
stochastic process of dimension :math:`q` where
:math:`\cD' \in \Rset^p` and
:math:`f_{dyn}:\cD \times \Rset^{\inputDim} \rightarrow \cD' \times \Rset^q` and
:math:`f_{dyn}` is defined in :eq:`dynFct`.
We build the composite process :math:`Y` thanks to function :math:`f_{dyn}`
and the process :math:`X`.
The library proposes two kinds of field function: the value
functions defined in :eq:`spatFunc` and the vertex-value functions defined
in :eq:`tempFunc`.
Trend modifications
~~~~~~~~~~~~~~~~~~~
Very often, we have to remove a trend from a process or to add it. If
we note :math:`f_{trend}: \Rset^n \rightarrow \Rset^{\inputDim}` the function
modelling a trend, then the field function which consists in
adding the trend to a process is the vertex-value function
:math:`f_{temp}: \cD \times \Rset^{\inputDim} \rightarrow \Rset^n \times \Rset^{\inputDim}`
defined by:
.. math::
:label: trendTempFunc
\begin{aligned}
f_{temp}(\vect{t}, \vect{x})=(\vect{t}, \vect{x} + f_{trend}(\vect{t}))\end{aligned}
The library enables to directly convert the function
:math:`f_{trend}` into the vertex-value function :math:`f_{temp}` thanks
to the *TrendTransform* object which maps :math:`f_{trend}` into the
vertex-value function :math:`f_{temp}`.
Then, the process :math:`Y` is built with the object
*CompositeProcess* from the data: :math:`f_{temp}` and the process
:math:`X` such that:
.. math::
\begin{aligned}
\forall \omega \in \Omega, \forall \vect{t} \in \cD, \quad Y(\omega, \vect{t}) = X(\omega, \vect{t}) + f_{trend}(\vect{t})\end{aligned}
Box Cox transformation
~~~~~~~~~~~~~~~~~~~~~~
If the transformation of the process :math:`X` into :math:`Y`
corresponds to the Box Cox transformation
:math:`f_{BoxCox}: \Rset^{\inputDim} \rightarrow \Rset^{\inputDim}` which transforms
:math:`X` into a process :math:`Y` with stabilized variance, then the
corresponding field function is the value function
:math:`f_{spat}: \cD \times \Rset^{\inputDim} \rightarrow \cD \times \Rset^{\inputDim}`
defined by:
.. math::
:label: spatFuncBC
\begin{aligned}
f_{spat}(\vect{t}, \vect{x})=(\vect{t},f_{BoxCox}(\vect{x}))\end{aligned}
The library enables to directly convert the function
:math:`f_{BoxCox}` into the value function :math:`f_{spat}` thanks
to the *ValueFunction* object.
Then, the process :math:`Y` is built with the object
*CompositeProcess* from the data: :math:`f_{spat}` and the process
:math:`X` such that:
.. math::
\begin{aligned}
\forall \omega \in \Omega, \forall \vect{t} \in \cD, \quad Y(\omega, \vect{t}) = f_{BoxCox}(X(\omega, \vect{t}))\end{aligned}
.. topic:: API:
- See :class:`~openturns.CompositeProcess`
- See :class:`~openturns.TrendTransform`
- See :class:`~openturns.BoxCoxFactory`
.. topic:: Examples:
- See :doc:`/auto_stochastic_processes/plot_add_trend`
- See :doc:`/auto_stochastic_processes/plot_trend_transform`
- See :doc:`/auto_stochastic_processes/plot_box_cox_transform`
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