.. _InterpolationVariable:


Interpolation variable
----------------------


 

This variable up-samples the output of a surrogate model  by interpolating between the output values. An interpolation  can reduce the cost of obtaining observations at high resolution.

.. _ActiveLearning.Interpolation.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.Interpolation.output_dim:


output_dim (int)
""""""""""""""""
   Output dimension of variable. 

   Default: This value has no default and must be provided

.. _ActiveLearning.Interpolation.output_names:


output_names (cell{str})
""""""""""""""""""""""""
   Allows to assign names to each of the  outputs of the variable. By specifying input names  they can be accessed in variables as  ``{output_names[0]}, {output_names[1]}, ..., {output_names[K-1]}``,  where ``K`` is the output dimension. 

   Default:  
   
   By default, the variables can be accessed as  ``{name}0, {name}1, ...,{name}(K-1)``, where ``{name}``  is the name of the variable and ``K`` is the output  dimension.

.. _ActiveLearning.Interpolation.input_surrogate:


input_surrogate (str)
"""""""""""""""""""""
   The name of a surrogate. 

   Default: This value has no default and must be provided

.. _ActiveLearning.Interpolation.input_positions:


input_positions (cell{cell{float}})
"""""""""""""""""""""""""""""""""""
   Each of the :math:`K` outputs of a surrogate model is assigned to a specific position in an  :math:`M`-dimensional. 

   Default: This value has no default and must be provided

   .. admonition:: Example

      A list of 2D positions (:math:`M=2`)  for each of six output values of a surrogate model  (:math:`K=6`).

      .. code-block:: matlab

         {{1.0,0.0},...
          {1.0,0.2},...
          {1.0,0.4},...
          {2.0,0.0},...
          {2.0,0.2},...
          {2.0,0.4}}


.. _ActiveLearning.Interpolation.output_positions:


output_positions (cell{cell{float}})
""""""""""""""""""""""""""""""""""""
   A list of :math:`M`-dimensional positions at  which the interpolation is evaluated. Typically, the length of the list :math:`K'` is larger than the number :math:`K` of trained  outputs of the surrogate model. 

   Default: This value has no default and must be provided

   .. admonition:: Example

      A list of 10 2D-positions (:math:`K'=10,M=2`).

      .. code-block:: matlab

         {{1.0,0.0},...
          {1.0,0.1},...
          {1.0,0.2},...
          {1.0,0.3},...
          {1.0,0.4},...
          {2.0,0.0},...
          {2.0,0.1},...
          {2.0,0.2},...
          {2.0,0.3},...
          {2.0,0.4}}


.. _ActiveLearning.Interpolation.kernel:


kernel (struct)
"""""""""""""""
    

   Default: ``struct('matern_order', 5)``
   

      
   
    
   
     
   
    
   
   Gaussian processes model the correlation (or covariance) between  two function values :math:`f(x), f(x')` by means of a covariance function  :math:`k(x,x') = k(||x-x'||)`, also called kernel. The kernel is monotonically decreasing for increasing distance :math:`d = ||x-x'||` such that far apart function values are uncorrelated while close-by function values are strongly correlated.      
   
   The distance :math:`d` between function values is defined as the  scaled Euclidean distance       
   
   .. math::   d = ||x-x'||=\sqrt{\sum_{i=1}^{D}\frac{(x_i−x_i^{'})^{2}}{l_i^{2}}},      
   
   where the hyperparameters :math:`l_1,\dots,l_D` determine the characteristic  length scales at which the covariance between separated function values  becomes negligible.      
   
   The Matérn covariance function is defined as      
   
   .. math::      
   
     k_{\nu}(x, x') = \frac{\sigma_0^2}{\Gamma(\nu)2^{\nu-1}}   \left[\sqrt{2\nu} d \right]^\nu    K_\nu\left( \sqrt{2\nu} d\right),      
   
   where  :math:`\sigma_0` is a hyperparameter that determines the  standard deviation of possible function values,   :math:`K_\nu(\cdot)` is a modified Bessel function,  and :math:`\Gamma(\cdot)` is the gamma function.   
   
   See :ref:`matern configuration <MaternKernel1>` for details.

.. toctree::
   :maxdepth: 100 
   :hidden:
   
   MaternKernel1



.. _ActiveLearning.Interpolation.length_scale_multiplier:


length_scale_multiplier (int)
"""""""""""""""""""""""""""""
   The interpolation is based on a Gaussian process regression. The length scale in dimension `i` of the covariance function is chosen to be a multiple  ``length_scale_multiplier`` of the  largest distance between sampling points in dimension `i`. 

   Default: ``6``