%feature("docstring") OT::TensorizedCovarianceModel
"Multivariate covariance function defined as a tensorization of covariance models.

Parameters
----------
coll : sequence of :class:`~openturns.CovarianceModel`
    Collection of covariance models :math:`(C_k)_{1 \leq k \leq K}` of dimension :math:`d_k`.

Notes
-----
The tensorized covariance model defines a multivariate covariance model of dimension :math:`d\geq 1` 
from the tensorization of a given collection of covariance models. 
This allows to create a higher dimension covariance model by combining models of smaller dimensions. 
The dimension of each model in the collection does not necessarily have to be equal to 1. 
The input dimension of each covariance model in the collection must be the same. 

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

Its covariance function :math:`C : \cD \times \cD \rightarrow  \cS_d^+(\Rset)` is defined from the collection of covariance functions :math:`C_k: \cD \times \cD \mapsto  \cS_{d_k}^+(\Rset)` where :math:`d_1+\dots+d_k=d`, as follows:

.. math::
    C(\vect{s}, \vect{t}) =    \left(
        \begin{array}{cccc}
        C_1(\vect{s}, \vect{t}) & 0 & \dots & 0 \\
        0 & C_2(\vect{s}, \vect{t}) & 0 &  \\
        \dots & \dots & \ddots & \dots \\
        0 & \dots & \dots &  C_K(\vect{s}, \vect{t})
        \end{array} \right)

The amplitude of the covariance function is :math:`\Tr{\sigma} =(\Tr{\sigma}_{1}, \dots, \Tr{\sigma}_{K})` and each model :math:`C_k` is parameterized by its scale :math:`\vect{\theta}_k \in \Rset^n`.

The method :math:`setScale(\vect{\theta})` updates the scale the following way. Let :math:`\vect{\theta}_k^0=(\theta_{k,1}^0,\hdots,\theta_{k,n}^0)` be the initial scale of the covariance model :math:`C_k`. After the update, :math:`C_k` has the scale :math:`\vect{\theta}_k=\left(\theta_1\rho_{k,1}^0,\hdots,\theta_n\rho_{k,n}^0\right)` where :math:`\rho_{k,j}^0=\dfrac{\theta^0_{k,j}}{\theta^0_{1,j}}`.

Examples
--------

Create a tensorized covariance function from the tensorization of an absolute exponential function, a squared exponential function and an exponential function:

>>> import openturns as ot
>>> inputDimension = 2

Create the each covariance models:

>>> myCov1 = ot.AbsoluteExponential([3.0] * inputDimension)
>>> myCov2 = ot.SquaredExponential([2.0] * inputDimension)

Define the scale of the tensorized model:

>>> scale = [0.3, 0.8]

Create the tensorized model:

>>> covarianceModel = ot.TensorizedCovarianceModel([myCov1, myCov2], scale)

Fix the same scale to each model:

>>> covarianceModel.setScale([1.0]*inputDimension)"