text: from typing import NoReturn
  
  import pandas
  import IMLearn.metrics
  from ...base import BaseEstimator
  import numpy as np
  from numpy.linalg import det, inv
  
  from ...metrics import misclassification_error
  
  
  class LDA(BaseEstimator):
      """
      Linear Discriminant Analysis (LDA) classifier
  
      Attributes
      ----------
      self.classes_ : np.ndarray of shape (n_classes,)
          The different labels classes. To be set in `LDA.fit`
  
      self.mu_ : np.ndarray of shape (n_classes,n_features)
          The estimated features means for each class. To be set in `LDA.fit`
  
      self.cov_ : np.ndarray of shape (n_features,n_features)
          The estimated features covariance. To be set in `LDA.fit`
  
      self._cov_inv : np.ndarray of shape (n_features,n_features)
          The inverse of the estimated features covariance. To be set in `LDA.fit`
  
      self.pi_: np.ndarray of shape (n_classes)
          The estimated class probabilities. To be set in `GaussianNaiveBayes.fit`
      """
  
      def __init__(self):
          """
          Instantiate an LDA classifier
          """
          super().__init__()
          self.classes_, self.mu_, self.cov_, self._cov_inv, self.pi_ = None, None, None, None, None
  
      def _fit(self, X: np.ndarray, y: np.ndarray) -> NoReturn:
          """
          fits an LDA model.
          Estimates gaussian for each label class - Different mean vector, same covariance
          matrix with dependent features.
  
          Parameters
          ----------
          X : ndarray of shape (n_samples, n_features)
              Input data to fit an estimator for
  
          y : ndarray of shape (n_samples, )
              Responses of input data to fit to
          """
  
          # self.mu_ = np.ndarray((len(self.classes_), len(X[0])))
          # self.cov_ = np.ndarray((len(X[0]), len(X[0])))
          m = len(X)
          self.classes_, counts = np.unique(y, return_counts=True)  # classes_
          self.pi_ = counts / m
          self.cov_ = np.zeros((len(X[0]), len(X[0])))
          self.mu_ = np.ndarray((len(self.classes_), len(X[0])))
          for k, class_ in enumerate(self.classes_):
              mu = np.mean(X[y == class_], axis=0)
              self.mu_[k] = np.array(mu)     # mu_
              sigma = np.transpose(X[y == class_] - self.mu_[k]) @ (X[y == class_] - self.mu_[k])
              self.cov_ += sigma  # summing all sigmas - devide after loop
          self.cov_ = self.cov_ / (m - len(self.classes_))  # cov
          self._cov_inv = np.linalg.inv(self.cov_)  # _cov_inv
          self.fitted_ = True
  
      def _predict(self, X: np.ndarray) -> np.ndarray:
          """
          Predict responses for given samples using fitted estimator
  
          Parameters
          ----------
          X : ndarray of shape (n_samples, n_features)
              Input data to predict responses for
  
          Returns
          -------
          responses : ndarray of shape (n_samples, )
              Predicted responses of given samples
          """
  
          return self.classes_[np.argmax(self.likelihood(X), axis=1)]
  
      def likelihood(self, X: np.ndarray) -> np.ndarray:
          """
          Calculate the likelihood of a given data over the estimated model
  
          Parameters
          ----------
          X : np.ndarray of shape (n_samples, n_features)
              Input data to calculate its likelihood over the different classes.
  
          Returns
          -------
          likelihoods : np.ndarray of shape (n_samples, n_classes)
              The likelihood for each sample under each of the classes
  
          """
          if not self.fitted_:
              raise ValueError("Estimator must first be fitted before calling `likelihood` function")
  
          likelihood = np.zeros((len(X), len(self.classes_)))
          for i in range(len(X)):
              cur = np.zeros((len(self.classes_)))
              for k, class_ in enumerate(self.classes_):
                  a = self._cov_inv @ self.mu_[k]
                  b = np.log(self.pi_[k]) - ((self.mu_[k] @ self._cov_inv @ self.mu_[k]) / 2)
                  cur[k] = np.transpose(a) @ X[i] + b
              likelihood[i] = cur
          return likelihood
  
      def _loss(self, X: np.ndarray, y: np.ndarray) -> float:
          """
          Evaluate performance under misclassification loss function
  
          Parameters
          ----------
          X : ndarray of shape (n_samples, n_features)
              Test samples
  
          y : ndarray of shape (n_samples, )
              True labels of test samples
  
          Returns
          -------
          loss : float
              Performance under missclassification loss function
          """
          y_pred = self._predict(X)
          return misclassification_error(y, y_pred)
  