"""
Create a stationary covariance model
====================================
"""
# %%
# This use case illustrates how the User can create a covariance function from parametric models.
# The library implements the *multivariate Exponential model* as a parametric model for the covariance function where the spatial covariance function :math:`\rho` writes:
#
# .. math::
#    \rho(\underline{s}, \underline{t} ) = e^{-\left\| \underline{s}- \underline{t} \right\|_2} \quad \forall (\underline{s}, \underline{t}) \in \mathcal{D}
#
# It is possible to define the exponential model from the spatial covariance matrix :math:`\underline{\underline{C}}^{spat}`
# rather than the correlation matrix :math:`\underline{\underline{R}}`:
#
# .. math::
#     \forall \underline{t} \in \mathcal{D},\quad \underline{\underline{C}}^{spat} = \mathbb{E} \left[ X_{\underline{t}} X^t_{\underline{t}} \right]
#     = \underline{\underline{A}}\,\underline{\underline{R}}\, \underline{\underline{A}}
#
# with:
#
# .. math::
#    \underline{\underline{A}} = \mbox{Diag}(a_1, \dots, a_d)
#
# We call :math:`\underline{a}` the amplitude vector and :math:`\underline{\lambda}` the scale vector.
#
# The library implements the multivariate exponential model thanks to the object *ExponentialModel* which is created from:
#
#
# - the scale and amplitude vectors :math:`(\underline{\lambda}, \underline{a})`: in that case, by default :math:`\underline{\underline{R}} = \underline{\underline{I}}`;
# - the scale and amplitude vectors and the spatial correlation matrix  :math:`(\underline{\lambda}, \underline{a},\underline{\underline{R}})`;
# - the scale and amplitude vectors and the spatial covariance matrix  :math:`(\underline{\lambda}, \underline{a},\underline{\underline{C}})`;
#   Then :math:`\underline{\underline{C}}` is mapped into the associated correlation matrix :math:`\underline{\underline{R}}` and the previous constructor is used.
#

# %%
import openturns as ot

ot.Log.Show(ot.Log.NONE)

# %%
# Create the amplitude vector (output dimension 3)
amplitude = [1.0, 2.0, 3.0]

# Scale vector (input dimension 1)
scale = [4.0]

# spatialCorrelation
spatialCorrelation = ot.CorrelationMatrix(3)
spatialCorrelation[0, 1] = 0.8
spatialCorrelation[0, 2] = 0.6
spatialCorrelation[1, 2] = 0.1

# spatialCovariance
spatialCovariance = ot.CovarianceMatrix(3)
spatialCovariance[0, 0] = 4.0
spatialCovariance[1, 1] = 5.0
spatialCovariance[2, 2] = 6.0
spatialCovariance[0, 1] = 1.2
spatialCovariance[0, 2] = 0.9
spatialCovariance[1, 2] = -0.2

# %%
# Create the covariance model from the amplitude and scale, no spatial correlation
ot.ExponentialModel(scale, amplitude)

# %%
# or from the amplitude, scale and spatialCovariance
ot.ExponentialModel(scale, amplitude, spatialCorrelation)

# %%
# or from the scale and spatialCovariance
ot.ExponentialModel(scale, spatialCovariance)
