.. _ChiSquaredVariable:


Chi-squared variable
--------------------


 

This variable describes the chi-squared deviation   

.. math::   \chi^2 = \sum_{i=1}^K \frac{\left(t_i - y_i\right)^2}{\eta_i^2}.  

between `K` outputs :math:`y_i` of a surrogate and `K` targets  :math:`t_i` scaled by the target uncertainties :math:`\eta_i`.   

If the target uncertainties are not independent but correlated, it is also possible to specify the covariance matrix :math:`G` of the targets. In this case, the chi-squared deviation is given as  

.. math::   \chi^2 = \sum_{i,j=1}^K (t_i - y_i) G_{i j}^{-1} (t_j - y_j).  

If the predictions of the surrogate, :math:`y_i`,  follow a Gaussian distribution, the variable follows a  generalized chi-squared distribution.

.. _ActiveLearning.ChiSquaredValue.name:


name (str)
""""""""""
   The name of the variable under which it can be addressed by other variables or objectives. The name must be distinct from any surrogate name. 

   Default: ``'v'``

.. _ActiveLearning.ChiSquaredValue.input:


input (str)
"""""""""""
   The name of a surrogate model or a multi-output variable. 

   Default: This value has no default and must be provided

.. _ActiveLearning.ChiSquaredValue.target_vector:


target_vector (list[float])
"""""""""""""""""""""""""""
   Vector of target values :math:`t_i`. 

   Default: Vector of zeros.

.. _ActiveLearning.ChiSquaredValue.uncertainty_vector:


uncertainty_vector (list[float])
""""""""""""""""""""""""""""""""
   Vector of target uncertainties  :math:`\eta_i` such that :math:`\chi^2 = \sum_{i=1}^K \frac{(t_i - y_i)^2}{\eta_i^2}`. 

   Default: Vector of ones.

.. _ActiveLearning.ChiSquaredValue.covariance_matrix:


covariance_matrix (list[list[float]])
"""""""""""""""""""""""""""""""""""""
   Covariance matrix :math:`G`. 

   Default: Diagonal identity matrix.

.. _ActiveLearning.ChiSquaredValue.approximate:


approximate (bool)
""""""""""""""""""
   If true, the generalized chi-squared  variable with different uncertainties of the predictions  of each channel `K` is approximated by a chi-squared  variable with averaged uncertainties. This allows to analytically compute probability densities and any acquisition function that is directly based  on the variable. 

   Default: ``True``

.. _ActiveLearning.ChiSquaredValue.force_MC_integration:


force_MC_integration (bool)
"""""""""""""""""""""""""""
   If true, the posterior of the variable is based on Monte-Carlo samples. In this case one can avoid to determine a Gaussian distribution of function values, which might be numerically unstable, if the correlation between the inputs is strong. 

   Default: ``False``

.. _ActiveLearning.ChiSquaredValue.effective_DOF:


effective_DOF (float)
"""""""""""""""""""""
   Number of effective degrees of freedom (DOF)  used for stochastic variable of the chi-squared distribution. This number roughly indicates how many output channels of the  forward model are statistically independent. 

   Default: If not specified, the value us determined automatically.

   .. note:: If ``approximate`` is false, this parameter has no  effect.

.. _ActiveLearning.ChiSquaredValue.effective_DOF_bounds:


effective_DOF_bounds (list[float])
""""""""""""""""""""""""""""""""""
   The number of effective degrees of freedom (DOF)  is determined by a maximum likelihood estimate within the given lower and upper bounds. 

   Default: ``[10.0,50.0]``

   .. note:: If ``approximate`` is false or the ``effective_DOF`` is set manually, this parameter has no effect.