text: """Compatibility fixes for older versions of libraries
  
  If you add content to this file, please give the version of the package
  at which the fix is no longer needed.
  
  # originally copied from scikit-learn
  
  """
  # Authors: Emmanuelle Gouillart <emmanuelle.gouillart@normalesup.org>
  #          Gael Varoquaux <gael.varoquaux@normalesup.org>
  #          Fabian Pedregosa <fpedregosa@acm.org>
  #          Lars Buitinck <L.J.Buitinck@uva.nl>
  # License: BSD
  
  import inspect
  from math import log
  import os
  import warnings
  
  import numpy as np
  
  
  ###############################################################################
  # distutils
  
  # distutils has been deprecated since Python 3.10 and is scheduled for removal
  # from the standard library with the release of Python 3.12. For version
  # comparisons, we use setuptools's `parse_version` if available.
  
  def _compare_version(version_a, operator, version_b):
      """Compare two version strings via a user-specified operator.
  
      Parameters
      ----------
      version_a : str
          First version string.
      operator : '==' | '>' | '<' | '>=' | '<='
          Operator to compare ``version_a`` and ``version_b`` in the form of
          ``version_a operator version_b``.
      version_b : str
          Second version string.
  
      Returns
      -------
      bool
          The result of the version comparison.
      """
      from packaging.version import parse
      with warnings.catch_warnings(record=True):
          warnings.simplefilter('ignore')
          return eval(f'parse("{version_a}") {operator} parse("{version_b}")')
  
  
  ###############################################################################
  # Misc
  
  def _median_complex(data, axis):
      """Compute marginal median on complex data safely.
  
      Can be removed when numpy introduces a fix.
      See: https://github.com/scipy/scipy/pull/12676/.
      """
      # np.median must be passed real arrays for the desired result
      if np.iscomplexobj(data):
          data = (np.median(np.real(data), axis=axis)
                  + 1j * np.median(np.imag(data), axis=axis))
      else:
          data = np.median(data, axis=axis)
      return data
  
  
  # helpers to get function arguments
  def _get_args(function, varargs=False, *,
                exclude=('var_positional', 'var_keyword')):
      params = inspect.signature(function).parameters
      # As of Python 3.10:
      # https://docs.python.org/3/library/inspect.html#inspect.Parameter.kind
      # POSITIONAL_ONLY, POSITIONAL_OR_KEYWORD, VAR_POSITIONAL, KEYWORD_ONLY,
      # VAR_KEYWORD
      exclude = set(getattr(inspect.Parameter, ex.upper()) for ex in exclude)
      args = [key for key, param in params.items() if param.kind not in exclude]
      if varargs:
          varargs = [param.name for param in params.values()
                     if param.kind == param.VAR_POSITIONAL]
          if len(varargs) == 0:
              varargs = None
          return args, varargs
      else:
          return args
  
  
  def _safe_svd(A, **kwargs):
      """Wrapper to get around the SVD did not converge error of death"""
      # Intel has a bug with their GESVD driver:
      #     https://software.intel.com/en-us/forums/intel-distribution-for-python/topic/628049  # noqa: E501
      # For SciPy 0.18 and up, we can work around it by using
      # lapack_driver='gesvd' instead.
      from scipy import linalg
      if kwargs.get('overwrite_a', False):
          raise ValueError('Cannot set overwrite_a=True with this function')
      try:
          return linalg.svd(A, **kwargs)
      except np.linalg.LinAlgError as exp:
          from .utils import warn
          warn('SVD error (%s), attempting to use GESVD instead of GESDD'
                  % (exp,))
          return linalg.svd(A, lapack_driver='gesvd', **kwargs)
  
  
  def _csc_matrix_cast(x):
      from scipy.sparse import csc_matrix
      return csc_matrix(x)
  
  
  ###############################################################################
  # Backporting nibabel's read_geometry
  
  def _get_read_geometry():
      """Get the geometry reading function."""
      try:
          import nibabel as nib
          has_nibabel = True
      except ImportError:
          has_nibabel = False
      if has_nibabel:
          from nibabel.freesurfer import read_geometry
      else:
          read_geometry = _read_geometry
      return read_geometry
  
  
  def _read_geometry(filepath, read_metadata=False, read_stamp=False):
      """Backport from nibabel."""
      from .surface import _fread3, _fread3_many
      volume_info = dict()
  
      TRIANGLE_MAGIC = 16777214
      QUAD_MAGIC = 16777215
      NEW_QUAD_MAGIC = 16777213
      with open(filepath, "rb") as fobj:
          magic = _fread3(fobj)
          if magic in (QUAD_MAGIC, NEW_QUAD_MAGIC):  # Quad file
              nvert = _fread3(fobj)
              nquad = _fread3(fobj)
              (fmt, div) = (">i2", 100.) if magic == QUAD_MAGIC else (">f4", 1.)
              coords = np.fromfile(fobj, fmt, nvert * 3).astype(np.float64) / div
              coords = coords.reshape(-1, 3)
              quads = _fread3_many(fobj, nquad * 4)
              quads = quads.reshape(nquad, 4)
              #
              #   Face splitting follows
              #
              faces = np.zeros((2 * nquad, 3), dtype=np.int64)
              nface = 0
              for quad in quads:
                  if (quad[0] % 2) == 0:
                      faces[nface] = quad[0], quad[1], quad[3]
                      nface += 1
                      faces[nface] = quad[2], quad[3], quad[1]
                      nface += 1
                  else:
                      faces[nface] = quad[0], quad[1], quad[2]
                      nface += 1
                      faces[nface] = quad[0], quad[2], quad[3]
                      nface += 1
  
          elif magic == TRIANGLE_MAGIC:  # Triangle file
              create_stamp = fobj.readline().rstrip(b'\n').decode('utf-8')
              fobj.readline()
              vnum = np.fromfile(fobj, ">i4", 1)[0]
              fnum = np.fromfile(fobj, ">i4", 1)[0]
              coords = np.fromfile(fobj, ">f4", vnum * 3).reshape(vnum, 3)
              faces = np.fromfile(fobj, ">i4", fnum * 3).reshape(fnum, 3)
  
              if read_metadata:
                  volume_info = _read_volume_info(fobj)
          else:
              raise ValueError("File does not appear to be a Freesurfer surface")
  
      coords = coords.astype(np.float64)
  
      ret = (coords, faces)
      if read_metadata:
          if len(volume_info) == 0:
              warnings.warn('No volume information contained in the file')
          ret += (volume_info,)
      if read_stamp:
          ret += (create_stamp,)
  
      return ret
  
  
  ###############################################################################
  # NumPy Generator (NumPy 1.17)
  
  
  def rng_uniform(rng):
      """Get the unform/randint from the rng."""
      # prefer Generator.integers, fall back to RandomState.randint
      return getattr(rng, 'integers', getattr(rng, 'randint', None))
  
  
  def _validate_sos(sos):
      """Helper to validate a SOS input"""
      sos = np.atleast_2d(sos)
      if sos.ndim != 2:
          raise ValueError('sos array must be 2D')
      n_sections, m = sos.shape
      if m != 6:
          raise ValueError('sos array must be shape (n_sections, 6)')
      if not (sos[:, 3] == 1).all():
          raise ValueError('sos[:, 3] should be all ones')
      return sos, n_sections
  
  
  ###############################################################################
  # Misc utilities
  
  # get_fdata() requires knowing the dtype ahead of time, so let's triage on our
  # own instead
  def _get_img_fdata(img):
      data = np.asanyarray(img.dataobj)
      dtype = np.complex128 if np.iscomplexobj(data) else np.float64
      return data.astype(dtype)
  
  
  def _read_volume_info(fobj):
      """An implementation of nibabel.freesurfer.io._read_volume_info, since old
      versions of nibabel (<=2.1.0) don't have it.
      """
      volume_info = dict()
      head = np.fromfile(fobj, '>i4', 1)
      if not np.array_equal(head, [20]):  # Read two bytes more
          head = np.concatenate([head, np.fromfile(fobj, '>i4', 2)])
          if not np.array_equal(head, [2, 0, 20]):
              warnings.warn("Unknown extension code.")
              return volume_info
  
      volume_info['head'] = head
      for key in ['valid', 'filename', 'volume', 'voxelsize', 'xras', 'yras',
                  'zras', 'cras']:
          pair = fobj.readline().decode('utf-8').split('=')
          if pair[0].strip() != key or len(pair) != 2:
              raise IOError('Error parsing volume info.')
          if key in ('valid', 'filename'):
              volume_info[key] = pair[1].strip()
          elif key == 'volume':
              volume_info[key] = np.array(pair[1].split()).astype(int)
          else:
              volume_info[key] = np.array(pair[1].split()).astype(float)
      # Ignore the rest
      return volume_info
  
  
  def _serialize_volume_info(volume_info):
      """An implementation of nibabel.freesurfer.io._serialize_volume_info, since
      old versions of nibabel (<=2.1.0) don't have it."""
      keys = ['head', 'valid', 'filename', 'volume', 'voxelsize', 'xras', 'yras',
              'zras', 'cras']
      diff = set(volume_info.keys()).difference(keys)
      if len(diff) > 0:
          raise ValueError('Invalid volume info: %s.' % diff.pop())
  
      strings = list()
      for key in keys:
          if key == 'head':
              if not (np.array_equal(volume_info[key], [20]) or np.array_equal(
                      volume_info[key], [2, 0, 20])):
                  warnings.warn("Unknown extension code.")
              strings.append(np.array(volume_info[key], dtype='>i4').tobytes())
          elif key in ('valid', 'filename'):
              val = volume_info[key]
              strings.append('{} = {}\n'.format(key, val).encode('utf-8'))
          elif key == 'volume':
              val = volume_info[key]
              strings.append('{} = {} {} {}\n'.format(
                  key, val[0], val[1], val[2]).encode('utf-8'))
          else:
              val = volume_info[key]
              strings.append('{} = {:0.10g} {:0.10g} {:0.10g}\n'.format(
                  key.ljust(6), val[0], val[1], val[2]).encode('utf-8'))
      return b''.join(strings)
  
  
  ##############################################################################
  # adapted from scikit-learn
  
  
  def is_classifier(estimator):
      """Returns True if the given estimator is (probably) a classifier.
  
      Parameters
      ----------
      estimator : object
          Estimator object to test.
  
      Returns
      -------
      out : bool
          True if estimator is a classifier and False otherwise.
      """
      return getattr(estimator, "_estimator_type", None) == "classifier"
  
  
  def is_regressor(estimator):
      """Returns True if the given estimator is (probably) a regressor.
  
      Parameters
      ----------
      estimator : object
          Estimator object to test.
  
      Returns
      -------
      out : bool
          True if estimator is a regressor and False otherwise.
      """
      return getattr(estimator, "_estimator_type", None) == "regressor"
  
  
  _DEFAULT_TAGS = {
      'non_deterministic': False,
      'requires_positive_X': False,
      'requires_positive_y': False,
      'X_types': ['2darray'],
      'poor_score': False,
      'no_validation': False,
      'multioutput': False,
      "allow_nan": False,
      'stateless': False,
      'multilabel': False,
      '_skip_test': False,
      '_xfail_checks': False,
      'multioutput_only': False,
      'binary_only': False,
      'requires_fit': True,
      'preserves_dtype': [np.float64],
      'requires_y': False,
      'pairwise': False,
  }
  
  
  class BaseEstimator(object):
      """Base class for all estimators in scikit-learn.
  
      Notes
      -----
      All estimators should specify all the parameters that can be set
      at the class level in their ``__init__`` as explicit keyword
      arguments (no ``*args`` or ``**kwargs``).
      """
  
      @classmethod
      def _get_param_names(cls):
          """Get parameter names for the estimator"""
          # fetch the constructor or the original constructor before
          # deprecation wrapping if any
          init = getattr(cls.__init__, 'deprecated_original', cls.__init__)
          if init is object.__init__:
              # No explicit constructor to introspect
              return []
  
          # introspect the constructor arguments to find the model parameters
          # to represent
          init_signature = inspect.signature(init)
          # Consider the constructor parameters excluding 'self'
          parameters = [p for p in init_signature.parameters.values()
                        if p.name != 'self' and p.kind != p.VAR_KEYWORD]
          for p in parameters:
              if p.kind == p.VAR_POSITIONAL:
                  raise RuntimeError("scikit-learn estimators should always "
                                     "specify their parameters in the signature"
                                     " of their __init__ (no varargs)."
                                     " %s with constructor %s doesn't "
                                     " follow this convention."
                                     % (cls, init_signature))
          # Extract and sort argument names excluding 'self'
          return sorted([p.name for p in parameters])
  
      def get_params(self, deep=True):
          """Get parameters for this estimator.
  
          Parameters
          ----------
          deep : bool, optional
              If True, will return the parameters for this estimator and
              contained subobjects that are estimators.
  
          Returns
          -------
          params : dict
              Parameter names mapped to their values.
          """
          out = dict()
          for key in self._get_param_names():
              # We need deprecation warnings to always be on in order to
              # catch deprecated param values.
              # This is set in utils/__init__.py but it gets overwritten
              # when running under python3 somehow.
              warnings.simplefilter("always", DeprecationWarning)
              try:
                  with warnings.catch_warnings(record=True) as w:
                      value = getattr(self, key, None)
                  if len(w) and w[0].category == DeprecationWarning:
                      # if the parameter is deprecated, don't show it
                      continue
              finally:
                  warnings.filters.pop(0)
  
              # XXX: should we rather test if instance of estimator?
              if deep and hasattr(value, 'get_params'):
                  deep_items = value.get_params().items()
                  out.update((key + '__' + k, val) for k, val in deep_items)
              out[key] = value
          return out
  
      def set_params(self, **params):
          """Set the parameters of this estimator.
  
          The method works on simple estimators as well as on nested objects
          (such as pipelines). The latter have parameters of the form
          ``<component>__<parameter>`` so that it's possible to update each
          component of a nested object.
  
          Parameters
          ----------
          **params : dict
              Parameters.
  
          Returns
          -------
          inst : instance
              The object.
          """
          if not params:
              # Simple optimisation to gain speed (inspect is slow)
              return self
          valid_params = self.get_params(deep=True)
          for key, value in params.items():
              split = key.split('__', 1)
              if len(split) > 1:
                  # nested objects case
                  name, sub_name = split
                  if name not in valid_params:
                      raise ValueError('Invalid parameter %s for estimator %s. '
                                       'Check the list of available parameters '
                                       'with `estimator.get_params().keys()`.' %
                                       (name, self))
                  sub_object = valid_params[name]
                  sub_object.set_params(**{sub_name: value})
              else:
                  # simple objects case
                  if key not in valid_params:
                      raise ValueError('Invalid parameter %s for estimator %s. '
                                       'Check the list of available parameters '
                                       'with `estimator.get_params().keys()`.' %
                                       (key, self.__class__.__name__))
                  setattr(self, key, value)
          return self
  
      def __repr__(self):
          from sklearn.base import _pprint
          class_name = self.__class__.__name__
          return '%s(%s)' % (class_name, _pprint(self.get_params(deep=False),
                                                 offset=len(class_name),),)
  
      # __getstate__ and __setstate__ are omitted because they only contain
      # conditionals that are not satisfied by our objects (e.g.,
      # ``if type(self).__module__.startswith('sklearn.')``.
  
      def _more_tags(self):
          return _DEFAULT_TAGS
  
      def _get_tags(self):
          collected_tags = {}
          for base_class in reversed(inspect.getmro(self.__class__)):
              if hasattr(base_class, '_more_tags'):
                  # need the if because mixins might not have _more_tags
                  # but might do redundant work in estimators
                  # (i.e. calling more tags on BaseEstimator multiple times)
                  more_tags = base_class._more_tags(self)
                  collected_tags.update(more_tags)
          return collected_tags
  
  
  # newer sklearn deprecates importing from sklearn.metrics.scoring,
  # but older sklearn does not expose check_scoring in sklearn.metrics.
  def _get_check_scoring():
      try:
          from sklearn.metrics import check_scoring  # noqa
      except ImportError:
          from sklearn.metrics.scorer import check_scoring  # noqa
      return check_scoring
  
  
  def _check_fit_params(X, fit_params, indices=None):
      """Check and validate the parameters passed during `fit`.
  
      Parameters
      ----------
      X : array-like of shape (n_samples, n_features)
          Data array.
  
      fit_params : dict
          Dictionary containing the parameters passed at fit.
  
      indices : array-like of shape (n_samples,), default=None
          Indices to be selected if the parameter has the same size as
          `X`.
  
      Returns
      -------
      fit_params_validated : dict
          Validated parameters. We ensure that the values support
          indexing.
      """
      try:
          from sklearn.utils.validation import \
              _check_fit_params as _sklearn_check_fit_params
          return _sklearn_check_fit_params(X, fit_params, indices)
      except ImportError:
          from sklearn.model_selection import _validation
  
          fit_params_validated = \
              {k: _validation._index_param_value(X, v, indices)
               for k, v in fit_params.items()}
          return fit_params_validated
  
  
  ###############################################################################
  # Copied from sklearn to simplify code paths
  
  def empirical_covariance(X, assume_centered=False):
      """Computes the Maximum likelihood covariance estimator
  
  
      Parameters
      ----------
      X : ndarray, shape (n_samples, n_features)
          Data from which to compute the covariance estimate
  
      assume_centered : Boolean
          If True, data are not centered before computation.
          Useful when working with data whose mean is almost, but not exactly
          zero.
          If False, data are centered before computation.
  
      Returns
      -------
      covariance : 2D ndarray, shape (n_features, n_features)
          Empirical covariance (Maximum Likelihood Estimator).
  
      """
      X = np.asarray(X)
      if X.ndim == 1:
          X = np.reshape(X, (1, -1))
  
      if X.shape[0] == 1:
          warnings.warn("Only one sample available. "
                        "You may want to reshape your data array")
  
      if assume_centered:
          covariance = np.dot(X.T, X) / X.shape[0]
      else:
          covariance = np.cov(X.T, bias=1)
  
      if covariance.ndim == 0:
          covariance = np.array([[covariance]])
      return covariance
  
  
  class EmpiricalCovariance(BaseEstimator):
      """Maximum likelihood covariance estimator
  
      Read more in the :ref:`User Guide <covariance>`.
  
      Parameters
      ----------
      store_precision : bool
          Specifies if the estimated precision is stored.
  
      assume_centered : bool
          If True, data are not centered before computation.
          Useful when working with data whose mean is almost, but not exactly
          zero.
          If False (default), data are centered before computation.
  
      Attributes
      ----------
      covariance_ : 2D ndarray, shape (n_features, n_features)
          Estimated covariance matrix
  
      precision_ : 2D ndarray, shape (n_features, n_features)
          Estimated pseudo-inverse matrix.
          (stored only if store_precision is True)
  
      """
      def __init__(self, store_precision=True, assume_centered=False):
          self.store_precision = store_precision
          self.assume_centered = assume_centered
  
      def _set_covariance(self, covariance):
          """Saves the covariance and precision estimates
  
          Storage is done accordingly to `self.store_precision`.
          Precision stored only if invertible.
  
          Parameters
          ----------
          covariance : 2D ndarray, shape (n_features, n_features)
              Estimated covariance matrix to be stored, and from which precision
              is computed.
  
          """
          from scipy import linalg
          # covariance = check_array(covariance)
          # set covariance
          self.covariance_ = covariance
          # set precision
          if self.store_precision:
              self.precision_ = linalg.pinvh(covariance)
          else:
              self.precision_ = None
  
      def get_precision(self):
          """Getter for the precision matrix.
  
          Returns
          -------
          precision_ : array-like,
              The precision matrix associated to the current covariance object.
  
          """
          from scipy import linalg
          if self.store_precision:
              precision = self.precision_
          else:
              precision = linalg.pinvh(self.covariance_)
          return precision
  
      def fit(self, X, y=None):
          """Fit the Maximum Likelihood Estimator covariance model.
  
          Parameters
          ----------
          X : array-like, shape = [n_samples, n_features]
            Training data, where n_samples is the number of samples and
            n_features is the number of features.
          y : ndarray | None
              Not used, present for API consistency.
  
          Returns
          -------
          self : object
              Returns self.
          """  # noqa: E501
          # X = check_array(X)
          if self.assume_centered:
              self.location_ = np.zeros(X.shape[1])
          else:
              self.location_ = X.mean(0)
          covariance = empirical_covariance(
              X, assume_centered=self.assume_centered)
          self._set_covariance(covariance)
  
          return self
  
      def score(self, X_test, y=None):
          """Compute the log-likelihood of a Gaussian dataset.
  
          Uses ``self.covariance_`` as an estimator of its covariance matrix.
  
          Parameters
          ----------
          X_test : array-like, shape = [n_samples, n_features]
              Test data of which we compute the likelihood, where n_samples is
              the number of samples and n_features is the number of features.
              X_test is assumed to be drawn from the same distribution than
              the data used in fit (including centering).
          y : ndarray | None
              Not used, present for API consistency.
  
          Returns
          -------
          res : float
              The likelihood of the data set with `self.covariance_` as an
              estimator of its covariance matrix.
          """
          # compute empirical covariance of the test set
          test_cov = empirical_covariance(
              X_test - self.location_, assume_centered=True)
          # compute log likelihood
          res = log_likelihood(test_cov, self.get_precision())
  
          return res
  
      def error_norm(self, comp_cov, norm='frobenius', scaling=True,
                     squared=True):
          """Computes the Mean Squared Error between two covariance estimators.
  
          Parameters
          ----------
          comp_cov : array-like, shape = [n_features, n_features]
              The covariance to compare with.
          norm : str
              The type of norm used to compute the error. Available error types:
              - 'frobenius' (default): sqrt(tr(A^t.A))
              - 'spectral': sqrt(max(eigenvalues(A^t.A))
              where A is the error ``(comp_cov - self.covariance_)``.
          scaling : bool
              If True (default), the squared error norm is divided by n_features.
              If False, the squared error norm is not rescaled.
          squared : bool
              Whether to compute the squared error norm or the error norm.
              If True (default), the squared error norm is returned.
              If False, the error norm is returned.
  
          Returns
          -------
          The Mean Squared Error (in the sense of the Frobenius norm) between
          `self` and `comp_cov` covariance estimators.
          """
          from scipy import linalg
          # compute the error
          error = comp_cov - self.covariance_
          # compute the error norm
          if norm == "frobenius":
              squared_norm = np.sum(error ** 2)
          elif norm == "spectral":
              squared_norm = np.amax(linalg.svdvals(np.dot(error.T, error)))
          else:
              raise NotImplementedError(
                  "Only spectral and frobenius norms are implemented")
          # optionally scale the error norm
          if scaling:
              squared_norm = squared_norm / error.shape[0]
          # finally get either the squared norm or the norm
          if squared:
              result = squared_norm
          else:
              result = np.sqrt(squared_norm)
  
          return result
  
      def mahalanobis(self, observations):
          """Computes the squared Mahalanobis distances of given observations.
  
          Parameters
          ----------
          observations : array-like, shape = [n_observations, n_features]
              The observations, the Mahalanobis distances of the which we
              compute. Observations are assumed to be drawn from the same
              distribution than the data used in fit.
  
          Returns
          -------
          mahalanobis_distance : array, shape = [n_observations,]
              Squared Mahalanobis distances of the observations.
  
          """
          precision = self.get_precision()
          # compute mahalanobis distances
          centered_obs = observations - self.location_
          mahalanobis_dist = np.sum(
              np.dot(centered_obs, precision) * centered_obs, 1)
  
          return mahalanobis_dist
  
  
  def log_likelihood(emp_cov, precision):
      """Computes the sample mean of the log_likelihood under a covariance model
  
      computes the empirical expected log-likelihood (accounting for the
      normalization terms and scaling), allowing for universal comparison (beyond
      this software package)
  
      Parameters
      ----------
      emp_cov : 2D ndarray (n_features, n_features)
          Maximum Likelihood Estimator of covariance
  
      precision : 2D ndarray (n_features, n_features)
          The precision matrix of the covariance model to be tested
  
      Returns
      -------
      sample mean of the log-likelihood
      """
      p = precision.shape[0]
      log_likelihood_ = - np.sum(emp_cov * precision) + _logdet(precision)
      log_likelihood_ -= p * np.log(2 * np.pi)
      log_likelihood_ /= 2.
      return log_likelihood_
  
  
  # sklearn uses np.linalg for this, but ours is more robust to zero eigenvalues
  
  def _logdet(A):
      """Compute the log det of a positive semidefinite matrix."""
      from scipy import linalg
      vals = linalg.eigvalsh(A)
      # avoid negative (numerical errors) or zero (semi-definite matrix) values
      tol = vals.max() * vals.size * np.finfo(np.float64).eps
      vals = np.where(vals > tol, vals, tol)
      return np.sum(np.log(vals))
  
  
  def _infer_dimension_(spectrum, n_samples, n_features):
      """Infers the dimension of a dataset of shape (n_samples, n_features)
      The dataset is described by its spectrum `spectrum`.
      """
      n_spectrum = len(spectrum)
      ll = np.empty(n_spectrum)
      for rank in range(n_spectrum):
          ll[rank] = _assess_dimension_(spectrum, rank, n_samples, n_features)
      return ll.argmax()
  
  
  def _assess_dimension_(spectrum, rank, n_samples, n_features):
      from scipy.special import gammaln
      if rank > len(spectrum):
          raise ValueError("The tested rank cannot exceed the rank of the"
                           " dataset")
  
      pu = -rank * log(2.)
      for i in range(rank):
          pu += (gammaln((n_features - i) / 2.) -
                 log(np.pi) * (n_features - i) / 2.)
  
      pl = np.sum(np.log(spectrum[:rank]))
      pl = -pl * n_samples / 2.
  
      if rank == n_features:
          pv = 0
          v = 1
      else:
          v = np.sum(spectrum[rank:]) / (n_features - rank)
          pv = -np.log(v) * n_samples * (n_features - rank) / 2.
  
      m = n_features * rank - rank * (rank + 1.) / 2.
      pp = log(2. * np.pi) * (m + rank + 1.) / 2.
  
      pa = 0.
      spectrum_ = spectrum.copy()
      spectrum_[rank:n_features] = v
      for i in range(rank):
          for j in range(i + 1, len(spectrum)):
              pa += log((spectrum[i] - spectrum[j]) *
                        (1. / spectrum_[j] - 1. / spectrum_[i])) + log(n_samples)
  
      ll = pu + pl + pv + pp - pa / 2. - rank * log(n_samples) / 2.
  
      return ll
  
  
  def svd_flip(u, v, u_based_decision=True):
      if u_based_decision:
          # columns of u, rows of v
          max_abs_cols = np.argmax(np.abs(u), axis=0)
          signs = np.sign(u[max_abs_cols, np.arange(u.shape[1])])
          u *= signs
          v *= signs[:, np.newaxis]
      else:
          # rows of v, columns of u
          max_abs_rows = np.argmax(np.abs(v), axis=1)
          signs = np.sign(v[np.arange(v.shape[0]), max_abs_rows])
          u *= signs
          v *= signs[:, np.newaxis]
      return u, v
  
  
  def stable_cumsum(arr, axis=None, rtol=1e-05, atol=1e-08):
      """Use high precision for cumsum and check that final value matches sum
  
      Parameters
      ----------
      arr : array-like
          To be cumulatively summed as flat
      axis : int, optional
          Axis along which the cumulative sum is computed.
          The default (None) is to compute the cumsum over the flattened array.
      rtol : float
          Relative tolerance, see ``np.allclose``
      atol : float
          Absolute tolerance, see ``np.allclose``
      """
      out = np.cumsum(arr, axis=axis, dtype=np.float64)
      expected = np.sum(arr, axis=axis, dtype=np.float64)
      if not np.all(np.isclose(out.take(-1, axis=axis), expected, rtol=rtol,
                               atol=atol, equal_nan=True)):
          warnings.warn('cumsum was found to be unstable: '
                        'its last element does not correspond to sum',
                        RuntimeWarning)
      return out
  
  
  # This shim can be removed once NumPy 1.19.0+ is required (1.18.4 has sign bug)
  def svd(a, hermitian=False):
      if hermitian:  # faster
          s, u = np.linalg.eigh(a)
          sgn = np.sign(s)
          s = np.abs(s)
          sidx = np.argsort(s)[..., ::-1]
          sgn = np.take_along_axis(sgn, sidx, axis=-1)
          s = np.take_along_axis(s, sidx, axis=-1)
          u = np.take_along_axis(u, sidx[..., None, :], axis=-1)
          # singular values are unsigned, move the sign into v
          vt = (u * sgn[..., np.newaxis, :]).swapaxes(-2, -1).conj()
          np.abs(s, out=s)
          return u, s, vt
      else:
          return np.linalg.svd(a)
  
  
  ###############################################################################
  # From nilearn
  
  
  def _crop_colorbar(cbar, cbar_vmin, cbar_vmax):
      """
      crop a colorbar to show from cbar_vmin to cbar_vmax
      Used when symmetric_cbar=False is used.
      """
      import matplotlib
      if (cbar_vmin is None) and (cbar_vmax is None):
          return
      cbar_tick_locs = cbar.locator.locs
      if cbar_vmax is None:
          cbar_vmax = cbar_tick_locs.max()
      if cbar_vmin is None:
          cbar_vmin = cbar_tick_locs.min()
      new_tick_locs = np.linspace(cbar_vmin, cbar_vmax,
                                  len(cbar_tick_locs))
  
      # matplotlib >= 3.2.0 no longer normalizes axes between 0 and 1
      # See https://matplotlib.org/3.2.1/api/prev_api_changes/api_changes_3.2.0.html
      # _outline was removed in
      # https://github.com/matplotlib/matplotlib/commit/03a542e875eba091a027046d5ec652daa8be6863
      # so we use the code from there
      if _compare_version(matplotlib.__version__, '>=', '3.2.0'):
          cbar.ax.set_ylim(cbar_vmin, cbar_vmax)
          X = cbar._mesh()[0]
          X = np.array([X[0], X[-1]])
          Y = np.array([[cbar_vmin, cbar_vmin], [cbar_vmax, cbar_vmax]])
          N = X.shape[0]
          ii = [0, 1, N - 2, N - 1, 2 * N - 1, 2 * N - 2, N + 1, N, 0]
          x = X.T.reshape(-1)[ii]
          y = Y.T.reshape(-1)[ii]
          xy = (np.column_stack([y, x])
                if cbar.orientation == 'horizontal' else
                np.column_stack([x, y]))
          cbar.outline.set_xy(xy)
      else:
          cbar.ax.set_ylim(cbar.norm(cbar_vmin), cbar.norm(cbar_vmax))
          outline = cbar.outline.get_xy()
          outline[:2, 1] += cbar.norm(cbar_vmin)
          outline[2:6, 1] -= (1. - cbar.norm(cbar_vmax))
          outline[6:, 1] += cbar.norm(cbar_vmin)
          cbar.outline.set_xy(outline)
  
      cbar.set_ticks(new_tick_locs)
      cbar.update_ticks()
  
  
  ###############################################################################
  # Numba (optional requirement)
  
  # Here we choose different defaults to speed things up by default
  try:
      import numba
      if _compare_version(numba.__version__, '<', '0.48'):
          raise ImportError
      prange = numba.prange
      def jit(nopython=True, nogil=True, fastmath=True, cache=True,
              **kwargs):  # noqa
          return numba.jit(nopython=nopython, nogil=nogil, fastmath=fastmath,
                           cache=cache, **kwargs)
  except ImportError:
      has_numba = False
  else:
      has_numba = (os.getenv('MNE_USE_NUMBA', 'true').lower() == 'true')
  
  
  if not has_numba:
      def jit(**kwargs):  # noqa
          def _jit(func):
              return func
          return _jit
      prange = range
      bincount = np.bincount
      mean = np.mean
  
  else:
      @jit()
      def bincount(x, weights, minlength):  # noqa: D103
          out = np.zeros(minlength)
          for idx, w in zip(x, weights):
              out[idx] += w
          return out
  
      # fix because Numba does not support axis kwarg for mean
      @jit()
      def _np_apply_along_axis(func1d, axis, arr):
          assert arr.ndim == 2
          assert axis in [0, 1]
          if axis == 0:
              result = np.empty(arr.shape[1])
              for i in range(len(result)):
                  result[i] = func1d(arr[:, i])
          else:
              result = np.empty(arr.shape[0])
              for i in range(len(result)):
                  result[i] = func1d(arr[i, :])
          return result
  
      @jit()
      def mean(array, axis):
          return _np_apply_along_axis(np.mean, axis, array)
  
  
  ###############################################################################
  # Matplotlib
  
  # workaround: plt.close() doesn't spawn close_event on Agg backend
  # https://github.com/matplotlib/matplotlib/issues/18609
  # scheduled to be fixed by MPL 3.6
  def _close_event(fig):
      """Force calling of the MPL figure close event."""
      from .utils import logger
      try:
          fig.canvas.close_event()
          logger.debug(f'Called {fig!r}.canvas.close_event()')
      except ValueError:  # old mpl with Qt
          logger.debug(f'Calling {fig!r}.canvas.close_event() failed')
          pass  # pragma: no cover
  
  
  def _is_last_row(ax):
      try:
          return ax.get_subplotspec().is_last_row()  # 3.4+
      except AttributeError:
          return ax.is_last_row()
      return ax.get_subplotspec().is_last_row()
  
  
  ###############################################################################
  # SciPy deprecation of pinv + pinvh rcond (never worked properly anyway) in 1.7
  
  def pinvh(a, rtol=None):
      """Compute a pseudo-inverse of a Hermitian matrix."""
      s, u = np.linalg.eigh(a)
      del a
      if rtol is None:
          rtol = s.size * np.finfo(s.dtype).eps
      maxS = np.max(np.abs(s))
      above_cutoff = (abs(s) > maxS * rtol)
      psigma_diag = 1.0 / s[above_cutoff]
      u = u[:, above_cutoff]
      return (u * psigma_diag) @ u.conj().T
  
  
  def pinv(a, rtol=None):
      """Compute a pseudo-inverse of a matrix."""
      u, s, vh = np.linalg.svd(a, full_matrices=False)
      del a
      maxS = np.max(s)
      if rtol is None:
          rtol = max(vh.shape + u.shape) * np.finfo(u.dtype).eps
      rank = np.sum(s > maxS * rtol)
      u = u[:, :rank]
      u /= s[:rank]
      return (u @ vh[:rank]).conj().T
  
  
  ###############################################################################
  # PyVista
  
  # Deal with pyvista deprecation of point_data and cell_data
  # (can be removed once we require 0.31+)
  
  def _point_data(obj):
      try:
          return obj.point_data
      except AttributeError:
          return obj.point_arrays
  
  
  def _cell_data(obj):
      try:
          return obj.cell_data
      except AttributeError:
          return obj.cell_arrays
  