text: """
  Linear Discriminant Analysis and Quadratic Discriminant Analysis
  """
  
  # Authors: Clemens Brunner
  #          Martin Billinger
  #          Matthieu Perrot
  #          Mathieu Blondel
  
  # License: BSD 3-Clause
  
  import warnings
  import numpy as np
  from scipy import linalg
  from scipy.special import expit
  
  from .base import BaseEstimator, TransformerMixin, ClassifierMixin
  from .linear_model._base import LinearClassifierMixin
  from .covariance import ledoit_wolf, empirical_covariance, shrunk_covariance
  from .utils.multiclass import unique_labels
  from .utils.validation import check_is_fitted
  from .utils.multiclass import check_classification_targets
  from .utils.extmath import softmax
  from .preprocessing import StandardScaler
  
  
  __all__ = ["LinearDiscriminantAnalysis", "QuadraticDiscriminantAnalysis"]
  
  
  def _cov(X, shrinkage=None, covariance_estimator=None):
      """Estimate covariance matrix (using optional covariance_estimator).
      Parameters
      ----------
      X : array-like of shape (n_samples, n_features)
          Input data.
  
      shrinkage : {'empirical', 'auto'} or float, default=None
          Shrinkage parameter, possible values:
            - None or 'empirical': no shrinkage (default).
            - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
            - float between 0 and 1: fixed shrinkage parameter.
  
          Shrinkage parameter is ignored if  `covariance_estimator`
          is not None.
  
      covariance_estimator : estimator, default=None
          If not None, `covariance_estimator` is used to estimate
          the covariance matrices instead of relying on the empirical
          covariance estimator (with potential shrinkage).
          The object should have a fit method and a ``covariance_`` attribute
          like the estimators in :mod:`sklearn.covariance``.
          if None the shrinkage parameter drives the estimate.
  
          .. versionadded:: 0.24
  
      Returns
      -------
      s : ndarray of shape (n_features, n_features)
          Estimated covariance matrix.
      """
      if covariance_estimator is None:
          shrinkage = "empirical" if shrinkage is None else shrinkage
          if isinstance(shrinkage, str):
              if shrinkage == "auto":
                  sc = StandardScaler()  # standardize features
                  X = sc.fit_transform(X)
                  s = ledoit_wolf(X)[0]
                  # rescale
                  s = sc.scale_[:, np.newaxis] * s * sc.scale_[np.newaxis, :]
              elif shrinkage == "empirical":
                  s = empirical_covariance(X)
              else:
                  raise ValueError("unknown shrinkage parameter")
          elif isinstance(shrinkage, float) or isinstance(shrinkage, int):
              if shrinkage < 0 or shrinkage > 1:
                  raise ValueError("shrinkage parameter must be between 0 and 1")
              s = shrunk_covariance(empirical_covariance(X), shrinkage)
          else:
              raise TypeError("shrinkage must be a float or a string")
      else:
          if shrinkage is not None and shrinkage != 0:
              raise ValueError(
                  "covariance_estimator and shrinkage parameters "
                  "are not None. Only one of the two can be set."
              )
          covariance_estimator.fit(X)
          if not hasattr(covariance_estimator, "covariance_"):
              raise ValueError(
                  "%s does not have a covariance_ attribute"
                  % covariance_estimator.__class__.__name__
              )
          s = covariance_estimator.covariance_
      return s
  
  
  def _class_means(X, y):
      """Compute class means.
  
      Parameters
      ----------
      X : array-like of shape (n_samples, n_features)
          Input data.
  
      y : array-like of shape (n_samples,) or (n_samples, n_targets)
          Target values.
  
      Returns
      -------
      means : array-like of shape (n_classes, n_features)
          Class means.
      """
      classes, y = np.unique(y, return_inverse=True)
      cnt = np.bincount(y)
      means = np.zeros(shape=(len(classes), X.shape[1]))
      np.add.at(means, y, X)
      means /= cnt[:, None]
      return means
  
  
  def _class_cov(X, y, priors, shrinkage=None, covariance_estimator=None):
      """Compute weighted within-class covariance matrix.
  
      The per-class covariance are weighted by the class priors.
  
      Parameters
      ----------
      X : array-like of shape (n_samples, n_features)
          Input data.
  
      y : array-like of shape (n_samples,) or (n_samples, n_targets)
          Target values.
  
      priors : array-like of shape (n_classes,)
          Class priors.
  
      shrinkage : 'auto' or float, default=None
          Shrinkage parameter, possible values:
            - None: no shrinkage (default).
            - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
            - float between 0 and 1: fixed shrinkage parameter.
  
          Shrinkage parameter is ignored if `covariance_estimator` is not None.
  
      covariance_estimator : estimator, default=None
          If not None, `covariance_estimator` is used to estimate
          the covariance matrices instead of relying the empirical
          covariance estimator (with potential shrinkage).
          The object should have a fit method and a ``covariance_`` attribute
          like the estimators in sklearn.covariance.
          If None, the shrinkage parameter drives the estimate.
  
          .. versionadded:: 0.24
  
      Returns
      -------
      cov : array-like of shape (n_features, n_features)
          Weighted within-class covariance matrix
      """
      classes = np.unique(y)
      cov = np.zeros(shape=(X.shape[1], X.shape[1]))
      for idx, group in enumerate(classes):
          Xg = X[y == group, :]
          cov += priors[idx] * np.atleast_2d(_cov(Xg, shrinkage, covariance_estimator))
      return cov
  
  
  class LinearDiscriminantAnalysis(
      LinearClassifierMixin, TransformerMixin, BaseEstimator
  ):
      """Linear Discriminant Analysis.
  
      A classifier with a linear decision boundary, generated by fitting class
      conditional densities to the data and using Bayes' rule.
  
      The model fits a Gaussian density to each class, assuming that all classes
      share the same covariance matrix.
  
      The fitted model can also be used to reduce the dimensionality of the input
      by projecting it to the most discriminative directions, using the
      `transform` method.
  
      .. versionadded:: 0.17
         *LinearDiscriminantAnalysis*.
  
      Read more in the :ref:`User Guide <lda_qda>`.
  
      Parameters
      ----------
      solver : {'svd', 'lsqr', 'eigen'}, default='svd'
          Solver to use, possible values:
            - 'svd': Singular value decomposition (default).
              Does not compute the covariance matrix, therefore this solver is
              recommended for data with a large number of features.
            - 'lsqr': Least squares solution.
              Can be combined with shrinkage or custom covariance estimator.
            - 'eigen': Eigenvalue decomposition.
              Can be combined with shrinkage or custom covariance estimator.
  
      shrinkage : 'auto' or float, default=None
          Shrinkage parameter, possible values:
            - None: no shrinkage (default).
            - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
            - float between 0 and 1: fixed shrinkage parameter.
  
          This should be left to None if `covariance_estimator` is used.
          Note that shrinkage works only with 'lsqr' and 'eigen' solvers.
  
      priors : array-like of shape (n_classes,), default=None
          The class prior probabilities. By default, the class proportions are
          inferred from the training data.
  
      n_components : int, default=None
          Number of components (<= min(n_classes - 1, n_features)) for
          dimensionality reduction. If None, will be set to
          min(n_classes - 1, n_features). This parameter only affects the
          `transform` method.
  
      store_covariance : bool, default=False
          If True, explicitly compute the weighted within-class covariance
          matrix when solver is 'svd'. The matrix is always computed
          and stored for the other solvers.
  
          .. versionadded:: 0.17
  
      tol : float, default=1.0e-4
          Absolute threshold for a singular value of X to be considered
          significant, used to estimate the rank of X. Dimensions whose
          singular values are non-significant are discarded. Only used if
          solver is 'svd'.
  
          .. versionadded:: 0.17
  
      covariance_estimator : covariance estimator, default=None
          If not None, `covariance_estimator` is used to estimate
          the covariance matrices instead of relying on the empirical
          covariance estimator (with potential shrinkage).
          The object should have a fit method and a ``covariance_`` attribute
          like the estimators in :mod:`sklearn.covariance`.
          if None the shrinkage parameter drives the estimate.
  
          This should be left to None if `shrinkage` is used.
          Note that `covariance_estimator` works only with 'lsqr' and 'eigen'
          solvers.
  
          .. versionadded:: 0.24
  
      Attributes
      ----------
      coef_ : ndarray of shape (n_features,) or (n_classes, n_features)
          Weight vector(s).
  
      intercept_ : ndarray of shape (n_classes,)
          Intercept term.
  
      covariance_ : array-like of shape (n_features, n_features)
          Weighted within-class covariance matrix. It corresponds to
          `sum_k prior_k * C_k` where `C_k` is the covariance matrix of the
          samples in class `k`. The `C_k` are estimated using the (potentially
          shrunk) biased estimator of covariance. If solver is 'svd', only
          exists when `store_covariance` is True.
  
      explained_variance_ratio_ : ndarray of shape (n_components,)
          Percentage of variance explained by each of the selected components.
          If ``n_components`` is not set then all components are stored and the
          sum of explained variances is equal to 1.0. Only available when eigen
          or svd solver is used.
  
      means_ : array-like of shape (n_classes, n_features)
          Class-wise means.
  
      priors_ : array-like of shape (n_classes,)
          Class priors (sum to 1).
  
      scalings_ : array-like of shape (rank, n_classes - 1)
          Scaling of the features in the space spanned by the class centroids.
          Only available for 'svd' and 'eigen' solvers.
  
      xbar_ : array-like of shape (n_features,)
          Overall mean. Only present if solver is 'svd'.
  
      classes_ : array-like of shape (n_classes,)
          Unique class labels.
  
      n_features_in_ : int
          Number of features seen during :term:`fit`.
  
          .. versionadded:: 0.24
  
      feature_names_in_ : ndarray of shape (`n_features_in_`,)
          Names of features seen during :term:`fit`. Defined only when `X`
          has feature names that are all strings.
  
          .. versionadded:: 1.0
  
      See Also
      --------
      QuadraticDiscriminantAnalysis : Quadratic Discriminant Analysis.
  
      Examples
      --------
      >>> import numpy as np
      >>> from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
      >>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
      >>> y = np.array([1, 1, 1, 2, 2, 2])
      >>> clf = LinearDiscriminantAnalysis()
      >>> clf.fit(X, y)
      LinearDiscriminantAnalysis()
      >>> print(clf.predict([[-0.8, -1]]))
      [1]
      """
  
      def __init__(
          self,
          solver="svd",
          shrinkage=None,
          priors=None,
          n_components=None,
          store_covariance=False,
          tol=1e-4,
          covariance_estimator=None,
      ):
          self.solver = solver
          self.shrinkage = shrinkage
          self.priors = priors
          self.n_components = n_components
          self.store_covariance = store_covariance  # used only in svd solver
          self.tol = tol  # used only in svd solver
          self.covariance_estimator = covariance_estimator
  
      def _solve_lsqr(self, X, y, shrinkage, covariance_estimator):
          """Least squares solver.
  
          The least squares solver computes a straightforward solution of the
          optimal decision rule based directly on the discriminant functions. It
          can only be used for classification (with any covariance estimator),
          because
          estimation of eigenvectors is not performed. Therefore, dimensionality
          reduction with the transform is not supported.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Training data.
  
          y : array-like of shape (n_samples,) or (n_samples, n_classes)
              Target values.
  
          shrinkage : 'auto', float or None
              Shrinkage parameter, possible values:
                - None: no shrinkage.
                - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
                - float between 0 and 1: fixed shrinkage parameter.
  
              Shrinkage parameter is ignored if  `covariance_estimator` i
              not None
  
          covariance_estimator : estimator, default=None
              If not None, `covariance_estimator` is used to estimate
              the covariance matrices instead of relying the empirical
              covariance estimator (with potential shrinkage).
              The object should have a fit method and a ``covariance_`` attribute
              like the estimators in sklearn.covariance.
              if None the shrinkage parameter drives the estimate.
  
              .. versionadded:: 0.24
  
          Notes
          -----
          This solver is based on [1]_, section 2.6.2, pp. 39-41.
  
          References
          ----------
          .. [1] R. O. Duda, P. E. Hart, D. G. Stork. Pattern Classification
             (Second Edition). John Wiley & Sons, Inc., New York, 2001. ISBN
             0-471-05669-3.
          """
          self.means_ = _class_means(X, y)
          self.covariance_ = _class_cov(
              X, y, self.priors_, shrinkage, covariance_estimator
          )
          self.coef_ = linalg.lstsq(self.covariance_, self.means_.T)[0].T
          self.intercept_ = -0.5 * np.diag(np.dot(self.means_, self.coef_.T)) + np.log(
              self.priors_
          )
  
      def _solve_eigen(self, X, y, shrinkage, covariance_estimator):
          """Eigenvalue solver.
  
          The eigenvalue solver computes the optimal solution of the Rayleigh
          coefficient (basically the ratio of between class scatter to within
          class scatter). This solver supports both classification and
          dimensionality reduction (with any covariance estimator).
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Training data.
  
          y : array-like of shape (n_samples,) or (n_samples, n_targets)
              Target values.
  
          shrinkage : 'auto', float or None
              Shrinkage parameter, possible values:
                - None: no shrinkage.
                - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
                - float between 0 and 1: fixed shrinkage constant.
  
              Shrinkage parameter is ignored if  `covariance_estimator` i
              not None
  
          covariance_estimator : estimator, default=None
              If not None, `covariance_estimator` is used to estimate
              the covariance matrices instead of relying the empirical
              covariance estimator (with potential shrinkage).
              The object should have a fit method and a ``covariance_`` attribute
              like the estimators in sklearn.covariance.
              if None the shrinkage parameter drives the estimate.
  
              .. versionadded:: 0.24
  
          Notes
          -----
          This solver is based on [1]_, section 3.8.3, pp. 121-124.
  
          References
          ----------
          .. [1] R. O. Duda, P. E. Hart, D. G. Stork. Pattern Classification
             (Second Edition). John Wiley & Sons, Inc., New York, 2001. ISBN
             0-471-05669-3.
          """
          self.means_ = _class_means(X, y)
          self.covariance_ = _class_cov(
              X, y, self.priors_, shrinkage, covariance_estimator
          )
  
          Sw = self.covariance_  # within scatter
          St = _cov(X, shrinkage, covariance_estimator)  # total scatter
          Sb = St - Sw  # between scatter
  
          evals, evecs = linalg.eigh(Sb, Sw)
          self.explained_variance_ratio_ = np.sort(evals / np.sum(evals))[::-1][
              : self._max_components
          ]
          evecs = evecs[:, np.argsort(evals)[::-1]]  # sort eigenvectors
  
          self.scalings_ = evecs
          self.coef_ = np.dot(self.means_, evecs).dot(evecs.T)
          self.intercept_ = -0.5 * np.diag(np.dot(self.means_, self.coef_.T)) + np.log(
              self.priors_
          )
  
      def _solve_svd(self, X, y):
          """SVD solver.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Training data.
  
          y : array-like of shape (n_samples,) or (n_samples, n_targets)
              Target values.
          """
          n_samples, n_features = X.shape
          n_classes = len(self.classes_)
  
          self.means_ = _class_means(X, y)
          if self.store_covariance:
              self.covariance_ = _class_cov(X, y, self.priors_)
  
          Xc = []
          for idx, group in enumerate(self.classes_):
              Xg = X[y == group, :]
              Xc.append(Xg - self.means_[idx])
  
          self.xbar_ = np.dot(self.priors_, self.means_)
  
          Xc = np.concatenate(Xc, axis=0)
  
          # 1) within (univariate) scaling by with classes std-dev
          std = Xc.std(axis=0)
          # avoid division by zero in normalization
          std[std == 0] = 1.0
          fac = 1.0 / (n_samples - n_classes)
  
          # 2) Within variance scaling
          X = np.sqrt(fac) * (Xc / std)
          # SVD of centered (within)scaled data
          U, S, Vt = linalg.svd(X, full_matrices=False)
  
          rank = np.sum(S > self.tol)
          # Scaling of within covariance is: V' 1/S
          scalings = (Vt[:rank] / std).T / S[:rank]
  
          # 3) Between variance scaling
          # Scale weighted centers
          X = np.dot(
              (
                  (np.sqrt((n_samples * self.priors_) * fac))
                  * (self.means_ - self.xbar_).T
              ).T,
              scalings,
          )
          # Centers are living in a space with n_classes-1 dim (maximum)
          # Use SVD to find projection in the space spanned by the
          # (n_classes) centers
          _, S, Vt = linalg.svd(X, full_matrices=0)
  
          if self._max_components == 0:
              self.explained_variance_ratio_ = np.empty((0,), dtype=S.dtype)
          else:
              self.explained_variance_ratio_ = (S ** 2 / np.sum(S ** 2))[
                  : self._max_components
              ]
  
          rank = np.sum(S > self.tol * S[0])
          self.scalings_ = np.dot(scalings, Vt.T[:, :rank])
          coef = np.dot(self.means_ - self.xbar_, self.scalings_)
          self.intercept_ = -0.5 * np.sum(coef ** 2, axis=1) + np.log(self.priors_)
          self.coef_ = np.dot(coef, self.scalings_.T)
          self.intercept_ -= np.dot(self.xbar_, self.coef_.T)
  
      def fit(self, X, y):
          """Fit the Linear Discriminant Analysis model.
  
             .. versionchanged:: 0.19
                *store_covariance* has been moved to main constructor.
  
             .. versionchanged:: 0.19
                *tol* has been moved to main constructor.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Training data.
  
          y : array-like of shape (n_samples,)
              Target values.
  
          Returns
          -------
          self : object
              Fitted estimator.
          """
          X, y = self._validate_data(
              X, y, ensure_min_samples=2, estimator=self, dtype=[np.float64, np.float32]
          )
          self.classes_ = unique_labels(y)
          n_samples, _ = X.shape
          n_classes = len(self.classes_)
  
          if n_samples == n_classes:
              raise ValueError(
                  "The number of samples must be more than the number of classes."
              )
  
          if self.priors is None:  # estimate priors from sample
              _, y_t = np.unique(y, return_inverse=True)  # non-negative ints
              self.priors_ = np.bincount(y_t) / float(len(y))
          else:
              self.priors_ = np.asarray(self.priors)
  
          if (self.priors_ < 0).any():
              raise ValueError("priors must be non-negative")
          if not np.isclose(self.priors_.sum(), 1.0):
              warnings.warn("The priors do not sum to 1. Renormalizing", UserWarning)
              self.priors_ = self.priors_ / self.priors_.sum()
  
          # Maximum number of components no matter what n_components is
          # specified:
          max_components = min(len(self.classes_) - 1, X.shape[1])
  
          if self.n_components is None:
              self._max_components = max_components
          else:
              if self.n_components > max_components:
                  raise ValueError(
                      "n_components cannot be larger than min(n_features, n_classes - 1)."
                  )
              self._max_components = self.n_components
  
          if self.solver == "svd":
              if self.shrinkage is not None:
                  raise NotImplementedError("shrinkage not supported")
              if self.covariance_estimator is not None:
                  raise ValueError(
                      "covariance estimator "
                      "is not supported "
                      "with svd solver. Try another solver"
                  )
              self._solve_svd(X, y)
          elif self.solver == "lsqr":
              self._solve_lsqr(
                  X,
                  y,
                  shrinkage=self.shrinkage,
                  covariance_estimator=self.covariance_estimator,
              )
          elif self.solver == "eigen":
              self._solve_eigen(
                  X,
                  y,
                  shrinkage=self.shrinkage,
                  covariance_estimator=self.covariance_estimator,
              )
          else:
              raise ValueError(
                  "unknown solver {} (valid solvers are 'svd', "
                  "'lsqr', and 'eigen').".format(self.solver)
              )
          if self.classes_.size == 2:  # treat binary case as a special case
              self.coef_ = np.array(
                  self.coef_[1, :] - self.coef_[0, :], ndmin=2, dtype=X.dtype
              )
              self.intercept_ = np.array(
                  self.intercept_[1] - self.intercept_[0], ndmin=1, dtype=X.dtype
              )
          return self
  
      def transform(self, X):
          """Project data to maximize class separation.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Input data.
  
          Returns
          -------
          X_new : ndarray of shape (n_samples, n_components)
              Transformed data.
          """
          if self.solver == "lsqr":
              raise NotImplementedError(
                  "transform not implemented for 'lsqr' solver (use 'svd' or 'eigen')."
              )
          check_is_fitted(self)
  
          X = self._validate_data(X, reset=False)
          if self.solver == "svd":
              X_new = np.dot(X - self.xbar_, self.scalings_)
          elif self.solver == "eigen":
              X_new = np.dot(X, self.scalings_)
  
          return X_new[:, : self._max_components]
  
      def predict_proba(self, X):
          """Estimate probability.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Input data.
  
          Returns
          -------
          C : ndarray of shape (n_samples, n_classes)
              Estimated probabilities.
          """
          check_is_fitted(self)
  
          decision = self.decision_function(X)
          if self.classes_.size == 2:
              proba = expit(decision)
              return np.vstack([1 - proba, proba]).T
          else:
              return softmax(decision)
  
      def predict_log_proba(self, X):
          """Estimate log probability.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Input data.
  
          Returns
          -------
          C : ndarray of shape (n_samples, n_classes)
              Estimated log probabilities.
          """
          prediction = self.predict_proba(X)
          prediction[prediction == 0.0] += np.finfo(prediction.dtype).tiny
          return np.log(prediction)
  
      def decision_function(self, X):
          """Apply decision function to an array of samples.
  
          The decision function is equal (up to a constant factor) to the
          log-posterior of the model, i.e. `log p(y = k | x)`. In a binary
          classification setting this instead corresponds to the difference
          `log p(y = 1 | x) - log p(y = 0 | x)`. See :ref:`lda_qda_math`.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Array of samples (test vectors).
  
          Returns
          -------
          C : ndarray of shape (n_samples,) or (n_samples, n_classes)
              Decision function values related to each class, per sample.
              In the two-class case, the shape is (n_samples,), giving the
              log likelihood ratio of the positive class.
          """
          # Only override for the doc
          return super().decision_function(X)
  
  
  class QuadraticDiscriminantAnalysis(ClassifierMixin, BaseEstimator):
      """Quadratic Discriminant Analysis.
  
      A classifier with a quadratic decision boundary, generated
      by fitting class conditional densities to the data
      and using Bayes' rule.
  
      The model fits a Gaussian density to each class.
  
      .. versionadded:: 0.17
         *QuadraticDiscriminantAnalysis*
  
      Read more in the :ref:`User Guide <lda_qda>`.
  
      Parameters
      ----------
      priors : ndarray of shape (n_classes,), default=None
          Class priors. By default, the class proportions are inferred from the
          training data.
  
      reg_param : float, default=0.0
          Regularizes the per-class covariance estimates by transforming S2 as
          ``S2 = (1 - reg_param) * S2 + reg_param * np.eye(n_features)``,
          where S2 corresponds to the `scaling_` attribute of a given class.
  
      store_covariance : bool, default=False
          If True, the class covariance matrices are explicitly computed and
          stored in the `self.covariance_` attribute.
  
          .. versionadded:: 0.17
  
      tol : float, default=1.0e-4
          Absolute threshold for a singular value to be considered significant,
          used to estimate the rank of `Xk` where `Xk` is the centered matrix
          of samples in class k. This parameter does not affect the
          predictions. It only controls a warning that is raised when features
          are considered to be colinear.
  
          .. versionadded:: 0.17
  
      Attributes
      ----------
      covariance_ : list of len n_classes of ndarray \
              of shape (n_features, n_features)
          For each class, gives the covariance matrix estimated using the
          samples of that class. The estimations are unbiased. Only present if
          `store_covariance` is True.
  
      means_ : array-like of shape (n_classes, n_features)
          Class-wise means.
  
      priors_ : array-like of shape (n_classes,)
          Class priors (sum to 1).
  
      rotations_ : list of len n_classes of ndarray of shape (n_features, n_k)
          For each class k an array of shape (n_features, n_k), where
          ``n_k = min(n_features, number of elements in class k)``
          It is the rotation of the Gaussian distribution, i.e. its
          principal axis. It corresponds to `V`, the matrix of eigenvectors
          coming from the SVD of `Xk = U S Vt` where `Xk` is the centered
          matrix of samples from class k.
  
      scalings_ : list of len n_classes of ndarray of shape (n_k,)
          For each class, contains the scaling of
          the Gaussian distributions along its principal axes, i.e. the
          variance in the rotated coordinate system. It corresponds to `S^2 /
          (n_samples - 1)`, where `S` is the diagonal matrix of singular values
          from the SVD of `Xk`, where `Xk` is the centered matrix of samples
          from class k.
  
      classes_ : ndarray of shape (n_classes,)
          Unique class labels.
  
      n_features_in_ : int
          Number of features seen during :term:`fit`.
  
          .. versionadded:: 0.24
  
      feature_names_in_ : ndarray of shape (`n_features_in_`,)
          Names of features seen during :term:`fit`. Defined only when `X`
          has feature names that are all strings.
  
          .. versionadded:: 1.0
  
      See Also
      --------
      LinearDiscriminantAnalysis : Linear Discriminant Analysis.
  
      Examples
      --------
      >>> from sklearn.discriminant_analysis import QuadraticDiscriminantAnalysis
      >>> import numpy as np
      >>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
      >>> y = np.array([1, 1, 1, 2, 2, 2])
      >>> clf = QuadraticDiscriminantAnalysis()
      >>> clf.fit(X, y)
      QuadraticDiscriminantAnalysis()
      >>> print(clf.predict([[-0.8, -1]]))
      [1]
      """
  
      def __init__(
          self, *, priors=None, reg_param=0.0, store_covariance=False, tol=1.0e-4
      ):
          self.priors = np.asarray(priors) if priors is not None else None
          self.reg_param = reg_param
          self.store_covariance = store_covariance
          self.tol = tol
  
      def fit(self, X, y):
          """Fit the model according to the given training data and parameters.
  
              .. versionchanged:: 0.19
                 ``store_covariances`` has been moved to main constructor as
                 ``store_covariance``
  
              .. versionchanged:: 0.19
                 ``tol`` has been moved to main constructor.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Training vector, where `n_samples` is the number of samples and
              `n_features` is the number of features.
  
          y : array-like of shape (n_samples,)
              Target values (integers).
  
          Returns
          -------
          self : object
              Fitted estimator.
          """
          X, y = self._validate_data(X, y)
          check_classification_targets(y)
          self.classes_, y = np.unique(y, return_inverse=True)
          n_samples, n_features = X.shape
          n_classes = len(self.classes_)
          if n_classes < 2:
              raise ValueError(
                  "The number of classes has to be greater than one; got %d class"
                  % (n_classes)
              )
          if self.priors is None:
              self.priors_ = np.bincount(y) / float(n_samples)
          else:
              self.priors_ = self.priors
  
          cov = None
          store_covariance = self.store_covariance
          if store_covariance:
              cov = []
          means = []
          scalings = []
          rotations = []
          for ind in range(n_classes):
              Xg = X[y == ind, :]
              meang = Xg.mean(0)
              means.append(meang)
              if len(Xg) == 1:
                  raise ValueError(
                      "y has only 1 sample in class %s, covariance is ill defined."
                      % str(self.classes_[ind])
                  )
              Xgc = Xg - meang
              # Xgc = U * S * V.T
              _, S, Vt = np.linalg.svd(Xgc, full_matrices=False)
              rank = np.sum(S > self.tol)
              if rank < n_features:
                  warnings.warn("Variables are collinear")
              S2 = (S ** 2) / (len(Xg) - 1)
              S2 = ((1 - self.reg_param) * S2) + self.reg_param
              if self.store_covariance or store_covariance:
                  # cov = V * (S^2 / (n-1)) * V.T
                  cov.append(np.dot(S2 * Vt.T, Vt))
              scalings.append(S2)
              rotations.append(Vt.T)
          if self.store_covariance or store_covariance:
              self.covariance_ = cov
          self.means_ = np.asarray(means)
          self.scalings_ = scalings
          self.rotations_ = rotations
          return self
  
      def _decision_function(self, X):
          # return log posterior, see eq (4.12) p. 110 of the ESL.
          check_is_fitted(self)
  
          X = self._validate_data(X, reset=False)
          norm2 = []
          for i in range(len(self.classes_)):
              R = self.rotations_[i]
              S = self.scalings_[i]
              Xm = X - self.means_[i]
              X2 = np.dot(Xm, R * (S ** (-0.5)))
              norm2.append(np.sum(X2 ** 2, axis=1))
          norm2 = np.array(norm2).T  # shape = [len(X), n_classes]
          u = np.asarray([np.sum(np.log(s)) for s in self.scalings_])
          return -0.5 * (norm2 + u) + np.log(self.priors_)
  
      def decision_function(self, X):
          """Apply decision function to an array of samples.
  
          The decision function is equal (up to a constant factor) to the
          log-posterior of the model, i.e. `log p(y = k | x)`. In a binary
          classification setting this instead corresponds to the difference
          `log p(y = 1 | x) - log p(y = 0 | x)`. See :ref:`lda_qda_math`.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Array of samples (test vectors).
  
          Returns
          -------
          C : ndarray of shape (n_samples,) or (n_samples, n_classes)
              Decision function values related to each class, per sample.
              In the two-class case, the shape is (n_samples,), giving the
              log likelihood ratio of the positive class.
          """
          dec_func = self._decision_function(X)
          # handle special case of two classes
          if len(self.classes_) == 2:
              return dec_func[:, 1] - dec_func[:, 0]
          return dec_func
  
      def predict(self, X):
          """Perform classification on an array of test vectors X.
  
          The predicted class C for each sample in X is returned.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Vector to be scored, where `n_samples` is the number of samples and
              `n_features` is the number of features.
  
          Returns
          -------
          C : ndarray of shape (n_samples,)
              Estimated probabilities.
          """
          d = self._decision_function(X)
          y_pred = self.classes_.take(d.argmax(1))
          return y_pred
  
      def predict_proba(self, X):
          """Return posterior probabilities of classification.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Array of samples/test vectors.
  
          Returns
          -------
          C : ndarray of shape (n_samples, n_classes)
              Posterior probabilities of classification per class.
          """
          values = self._decision_function(X)
          # compute the likelihood of the underlying gaussian models
          # up to a multiplicative constant.
          likelihood = np.exp(values - values.max(axis=1)[:, np.newaxis])
          # compute posterior probabilities
          return likelihood / likelihood.sum(axis=1)[:, np.newaxis]
  
      def predict_log_proba(self, X):
          """Return log of posterior probabilities of classification.
  
          Parameters
          ----------
          X : array-like of shape (n_samples, n_features)
              Array of samples/test vectors.
  
          Returns
          -------
          C : ndarray of shape (n_samples, n_classes)
              Posterior log-probabilities of classification per class.
          """
          # XXX : can do better to avoid precision overflows
          probas_ = self.predict_proba(X)
          return np.log(probas_)
  