class_docstr: Fit a Gaussian mean/variance regression model.
  
  This class fits a one-dimensional Gaussian process model with
  parametrized mean and covariance structures to grouped data.  For
  each group, there is an independent realization of a latent
  Gaussian process indexed by an observed real-valued time
  variable..  The data consist of the Gaussian process observed at a
  finite number of `time` values.
  
  The process mean and variance can be lined to covariates.  The
  mean structure is linear in the covariates.  The covariance
  structure is non-stationary, and is defined parametrically through
  'scaling', and 'smoothing' parameters.  The covariance of the
  process between two observations in the same group is a function
  of the distance between the time values of the two observations.
  The scaling and smoothing parameters can be linked to covariates.
  
  The observed data are modeled as the sum of the Gaussian process
  realization and (optionally) independent white noise.  The standard
  deviation of the white noise can be linked to covariates.
  
  The data should be provided in 'long form', with a group label to
  indicate which observations belong to the same group.
  Observations in different groups are always independent.
  
  Parameters
  ----------
  endog : array_like
      The dependent variable.
  exog : array_like
      The design matrix for the mean structure
  exog_scale : array_like
      The design matrix for the scaling structure
  exog_smooth : array_like
      The design matrix for the smoothness structure
  exog_noise : array_like
      The design matrix for the additive white noise. The
      linear predictor is the log of the white noise standard
      deviation.  If None, there is no additive noise (the
      process is observed directly).
  time : array_like (1-dimensional)
      The univariate index values, used to calculate distances
      between observations in the same group, which determines
      their correlations.
  groups : array_like (1-dimensional)
      The group values.
  cov : a ProcessCovariance instance
      Defaults to GaussianCovariance.
docstr: Calculate the score function for the model.
  
  Parameters
  ----------
  params : array_like
      The packed parameters for the model.
  
  Returns
  -------
  The score vector at the given parameter point.
  
  Notes
  -----
  The mean, scaling, and smoothing parameters are packed into
  a vector.  Use `unpack` to access the component vectors.
code: def score(self, params):
      """
      Calculate the score function for the model.
  
      Parameters
      ----------
      params : array_like
          The packed parameters for the model.
  
      Returns
      -------
      The score vector at the given parameter point.
  
      Notes
      -----
      The mean, scaling, and smoothing parameters are packed into
      a vector.  Use `unpack` to access the component vectors.
      """
  
      mnpar, scpar, smpar, nopar = self.unpack(params)
      pm, pv, ps = len(mnpar), len(scpar), len(smpar)
  
      # Residuals
      resid = self.endog - np.dot(self.exog, mnpar)
  
      # Scaling
      sc = np.exp(np.dot(self.exog_scale, scpar))
  
      # Smoothness
      sm = np.exp(np.dot(self.exog_smooth, smpar))
  
      # White noise standard deviation
      if self._has_noise:
          no = np.exp(np.dot(self.exog_noise, nopar))
  
      # Get the log-likelihood
      score = np.zeros(
          len(mnpar) + len(scpar) + len(smpar) + len(nopar)
      )
      for _, ix in self._groups_ix.items():
          sc_i = sc[ix]
          sm_i = sm[ix]
          resid_i = resid[ix]
          time_i = self.time[ix]
          exog_i = self.exog[ix, :]
          exog_scale_i = self.exog_scale[ix, :]
          exog_smooth_i = self.exog_smooth[ix, :]
  
          # Get the covariance matrix for this person.
          cm = self.cov.get_cov(time_i, sc_i, sm_i)
  
          if self._has_noise:
              no_i = no[ix]
              exog_noise_i = self.exog_noise[ix, :]
              cm.flat[:: cm.shape[0] + 1] += no[ix] ** 2
  
          cmi = np.linalg.inv(cm)
  
          jacv, jacs = self.cov.jac(time_i, sc_i, sm_i)
  
          # The derivatives for the mean parameters.
          dcr = np.linalg.solve(cm, resid_i)
          score[0:pm] += np.dot(exog_i.T, dcr)
  
          # The derivatives for the scaling parameters.
          rx = np.outer(resid_i, resid_i)
          qm = np.linalg.solve(cm, rx)
          qm = 0.5 * np.linalg.solve(cm, qm.T)
          scx = sc_i[:, None] * exog_scale_i
          for i, _ in enumerate(ix):
              jq = np.sum(jacv[i] * qm)
              score[pm : pm + pv] += jq * scx[i, :]
              score[pm : pm + pv] -= (
                  0.5 * np.sum(jacv[i] * cmi) * scx[i, :]
              )
  
          # The derivatives for the smoothness parameters.
          smx = sm_i[:, None] * exog_smooth_i
          for i, _ in enumerate(ix):
              jq = np.sum(jacs[i] * qm)
              score[pm + pv : pm + pv + ps] += jq * smx[i, :]
              score[pm + pv : pm + pv + ps] -= (
                  0.5 * np.sum(jacs[i] * cmi) * smx[i, :]
              )
  
          # The derivatives with respect to the standard deviation parameters
          if self._has_noise:
              sno = no_i[:, None] ** 2 * exog_noise_i
              score[pm + pv + ps :] -= np.dot(
                  cmi.flat[:: cm.shape[0] + 1], sno
              )
              bm = np.dot(cmi, np.dot(rx, cmi))
              score[pm + pv + ps :] += np.dot(
                  bm.flat[:: bm.shape[0] + 1], sno
              )
  
      if self.verbose:
          print("|G|=", np.sqrt(np.sum(score * score)))
  
      return score
  
imports: import numpy
  import pandas
  import patsy
  import statsmodels.base.model
  from statsmodels.regression.linear_model import OLS
  import collections
  from scipy.optimize import minimize
  from statsmodels.iolib import summary2
  from statsmodels.tools.numdiff import approx_fprime
  import warnings
label_desc: Use image node_id 7 for calling the ProcessMLE obj's underlying member method code with example usage: obj.score(params) and returns: score