pastas.solver.likelihood.GaussianLikelihoodAr1
==============================================

.. toctree::
   :hidden:

   /api/pastas/solver/likelihood/GaussianLikelihoodAr1.get_init_parameters
   /api/pastas/solver/likelihood/GaussianLikelihoodAr1.compute

.. py:class:: pastas.solver.likelihood.GaussianLikelihoodAr1

   
   Gaussian likelihood function for homoscedastic, autocorrelated residuals.

   .. rubric:: Notes

   The Gaussian log-likelihood function with AR1 autocorrelated residuals
   :cite:p:`smith_modeling_2015` is defined as:

   .. math::

       \\log(L) = -\\frac{N-1}{2}\\log(2\\pi\\sigma^2) -
        \\frac{\\sum_{t=1}^N(\\epsilon_t - \\phi \\epsilon_{t-\\Delta t})^2}
       {2\\sigma^2}

   where :math:`N` is the number of observations, :math:`\\sigma^2` is the
   variance of the residuals, :math:`\\epsilon_t` is the residual at time
   :math:`t`. :math:`\\Delta t` is the time step between the observations.
   :math:`\\phi` is the autoregressive parameter. The parameters :math:`\\phi` and
   :math:`\\sigma^2` need to be estimated.

   The current implementation is valid for equidistant time series only.















   ..
       !! processed by numpydoc !!

   .. py:property:: nparam
      :type: int


      
      Number of parameters in the log-likelihood function.
















      ..
          !! processed by numpydoc !!

Methods
-------

.. autoapisummary::

   pastas.solver.likelihood.GaussianLikelihoodAr1.get_init_parameters
   pastas.solver.likelihood.GaussianLikelihoodAr1.compute