<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def incremental_fit(self, train_x, train_y):
 """ Incrementally fit the regressor. """

        if not self._first_fitted:
            raise ValueError("The first_fit function needs to be called first.")

        train_x, train_y = np.array(train_x), np.array(train_y)

        # Incrementally compute K
        up_right_k = edit_distance_matrix(self._x, train_x)
        down_left_k = np.transpose(up_right_k)
        down_right_k = edit_distance_matrix(train_x)
        up_k = np.concatenate((self._distance_matrix, up_right_k), axis=1)
        down_k = np.concatenate((down_left_k, down_right_k), axis=1)
        temp_distance_matrix = np.concatenate((up_k, down_k), axis=0)
        k_matrix = bourgain_embedding_matrix(temp_distance_matrix)
        diagonal = np.diag_indices_from(k_matrix)
        diagonal = (diagonal[0][-len(train_x) :], diagonal[1][-len(train_x) :])
        k_matrix[diagonal] += self.alpha

        try:
            self._l_matrix = cholesky(k_matrix, lower=True)  # Line 2
        except LinAlgError:
            return self

        self._x = np.concatenate((self._x, train_x), axis=0)
        self._y = np.concatenate((self._y, train_y), axis=0)
        self._distance_matrix = temp_distance_matrix

        self._alpha_vector = cho_solve((self._l_matrix, True), self._y)  # Line 3

        return self