%feature("docstring") OT::SquaredExponential
"Squared exponential covariance function.

Available constructors:
    SquaredExponential(*spatialDim=1*)

    SquaredExponential(*scale*)

    SquaredExponential(*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 of the process :math:`\vect{\sigma}\in \Rset^d`.
    Must be of size equal to 1.
    By default, equal to :math:`[1]`.

Notes
-----
The *squared 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  *squared exponential* function is defined by:

.. math::

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

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

.. math::

    \rho(\vect{s}, \vect{t}) = e^{-\frac{1}{2}  \left\| \vect{s}- \vect{t} \right\|_{2}^{2}}, \quad \forall (\vect{s}, \vect{t}) \in \cD


See Also
--------
CovarianceModel

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

>>> import openturns as ot
>>> covModel = ot.SquaredExponential(2)
>>> t = [0.1, 0.3]
>>> s = [0.2, 0.4]
>>> print(covModel(s, t))
[[ 0.99005 ]]
>>> tau = [0.1, 0.3]
>>> print(covModel(tau))
[[ 0.951229 ]]

Create a squared exponential covariance function specifying the scale vector (amplitude is fixed to 1):

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

Create a  squared exponential covariance function specifying the scale vector and the amplitude :

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