%feature("docstring") OT::AbsoluteExponential
"Absolute exponential covariance function.

Available constructors:
    AbsoluteExponential(*spatialDim=1*)

    AbsoluteExponential(*scale*)

    AbsoluteExponential(*scale, amplitude*)

Parameters
----------
spatialDim : int
    Spatial dimension :math:`n`.
    When not fulfilled, the spatial dimension is equal to the  size of the parameter :math:`\vect{\theta}`.
    By default, equal to 1.
scale : sequence of floats
    Scale coefficient :math:`\vect{\theta}\in \Rset^n`.
    The size of :math:`\vect{\theta}` is the spatial dimension.
amplitude : sequence of positive floats
    Amplitude  :math:`\vect{\sigma}\in \Rset^d`.
    Must be of size equal to 1.
    By default, equal to :math:`[1]`.


Notes
-----
The *absolute exponential* function is a stationary covariance function whith dimension :math:`d=1`.

We consider the scalar stochastic process :math:`X: \Omega \times\cD \mapsto \Rset`, where :math:`\omega \in \Omega` is an event, :math:`\cD` is a domain of :math:`\Rset^n`.

The  *absolute exponential* function is defined by:

.. math::

    C(\vect{s}, \vect{t}) = \sigma^2 e^{- \left\|\dfrac{\vect{s}-\vect{t}}{\vect{\theta}}\right\|_{1}}, \quad \forall (\vect{s}, \vect{t}) \in \cD

The correlation function :math:`\rho` writes:

.. math::

    \rho(\vect{s}, \vect{t}) = e^{- \left\| \vect{s}- \vect{t} \right\|_{1}}, \quad \forall (\vect{s}, \vect{t}) \in \cD


See Also
--------
CovarianceModel

Examples
--------
Create a standard absolute exponential covariance function:

>>> import openturns as ot
>>> covModel = ot.AbsoluteExponential(2)
>>> t = [0.1, 0.3]
>>> s = [0.2, 0.4]
>>> print(covModel(s, t))
[[ 0.818731 ]]
>>> tau = [0.1, 0.3]
>>> print(covModel(tau))
[[ 0.67032 ]]

Create an absolute exponential covariance function specifying only the scale vector (amplitude is fixed to 1):

>>> covModel2 = ot.AbsoluteExponential([1.5, 2.5])
>>> covModel2bis = ot.AbsoluteExponential([1.5]*3)

Create an absolute exponential covariance function specifying the scale vector and the amplitude :

>>> covModel3 = ot.AbsoluteExponential([1.5, 2.5], [3.5])"
