pastas.objective_functions.GaussianLikelihoodAr1
================================================

.. toctree::
   :hidden:

   /api/pastas/objective_functions/GaussianLikelihoodAr1.get_init_parameters
   /api/pastas/objective_functions/GaussianLikelihoodAr1.compute

.. py:class:: pastas.objective_functions.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 !!
Methods
-------

.. autoapisummary::

   pastas.objective_functions.GaussianLikelihoodAr1.get_init_parameters
   pastas.objective_functions.GaussianLikelihoodAr1.compute