.. _FitVariable:


Least square fit variable
-------------------------


 

This variable fits a parameter vector :math:`\mathbf{p}` consisting of  :math:`M` parameters :math:`p_1, p_2,\dots, p_M` to a model function :math:`f(\mathbf{x}, \mathbf{p})` that depends on the model variables :math:`x_1, x_2,\dots, x_D`. The output of the variable is a least-square estimate of the fit parameters to data points  :math:`(\mathbf{x}_i, y_i), i=1,\dots,N`, where :math:`y_1,\dots,y_n` are the mean values of the input. That is, the variable locally minimizes  

.. math::   \chi^2(\mathbf{p}) = \sum_{i=1}^N \frac{\left(f\left(\mathbf{x}_i, \mathbf{p}\right) - y_i\right)^2}{\eta_i^2},  

where :math:`\eta_i^2` is the variance of input :math:`i` to the variable.

.. _ActiveLearning.Fit.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.Fit.output_dim:


output_dim (int)
""""""""""""""""
   Output dimension of the fit variable. The variables outputs the values of the fitted parameters in the first :math:`M` entries and the minimum value of the mean squared error :math:`{\rm MSE} = \chi^2/(N-M)` found after the  least-square fit in the last entry. Hence, the `output_dim` is :math:`M+1`. 

   Default: This value has no default and must be provided

.. _ActiveLearning.Fit.output_names:


output_names (list[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.Fit.input:


input (str)
"""""""""""
   Names of a multi-output surrogate or variable. 

   Default: This value has no default and must be provided

.. _ActiveLearning.Fit.initial_parameters:


initial_parameters (list[float])
""""""""""""""""""""""""""""""""
   Initial values of the fit parameters to perform the least-square fit. 

   Default: This value has no default and must be provided

   .. admonition:: Example

      The initial value of :math:`M=2` parameters.

      .. code-block:: python

         [0.5,0.2]


.. _ActiveLearning.Fit.prior_uncertainties:


prior_uncertainties (list[float])
"""""""""""""""""""""""""""""""""
   Prior uncertainties of the fit parameters. The posterior covariance of the parameters is determined as the likelihood times the prior. 

   Default: Each parameter has a uncertainty of 100.

.. _ActiveLearning.Fit.model_variables:


model_variables (list[str])
"""""""""""""""""""""""""""
   The names of the variables :math:`x_1, x_2,\dots, x_D` of the model function. The number of variables :math:`D` must  correspond to the output dimensionality of the input surrogate  or variable. 

   Default: By default, the variables are named 'x0', 'x1', ... 'x{D-1}'.

.. _ActiveLearning.Fit.variable_values:


variable_values (list[list[float]])
"""""""""""""""""""""""""""""""""""
    
   
   The values of the model variables  :math:`\mathbf{x}_1,\dots,\mathbf{x}_N`.  This is a list of :math:`N` vectors where each entry is a list of  :math:`D` values. 

   Default: This value has no default and must be provided

   .. admonition:: Example

      A list with of six values for two variables.

      .. code-block:: python

         [[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.Fit.expression:


expression (str)
""""""""""""""""
   The expression describing the model function  :math:`f(\mathbf{x}, \mathbf{p})`. The names of the fit parameters correspond to the first :math:`M` output names. 

   Default: This value has no default and must be provided

   .. admonition:: Example

      Fit of the amplitude, frequency and phase of an oscillating input.

      .. code-block:: python

         'p0*sin(p1*x0 + p2)'


.. _ActiveLearning.Fit.ftol:


ftol (float)
""""""""""""
   The least square problem is solved using  scipy.optimize.least_squares. The parameter `ftol` is one of the convergence criteria. Smaller values correspond to more accurate fitting parameters, but longer fitting times that lead to longer sample computation times. 

   Default: ``1e-05``

.. _ActiveLearning.Fit.diff_step:


diff_step (float)
"""""""""""""""""
   Absolute step size for the finite difference  approximation of the Jacobian. 

   Default: ``1e-10``

.. _ActiveLearning.Fit.max_nfev:


max_nfev (int)
""""""""""""""
   Maximum number of function evaluations before the termination of the least-square fit. 

   Default: :math:`100\cdot M`.