code: def projection_mat(
      S: np.ndarray,
      D: Optional[np.ndarray] = None,
  ) -> np.ndarray:
      r"""
      Computes the projection matrix :math: P for projecting base forecasts
      :math: \bar{y} on to the space of coherent forecasts: :math: P \bar{y}.
  
      More precisely,
  
      .. math::
          P = S (S^T S)^{-1} S^T,      if D is None,\\
          P = S (S^T D S)^{-1} S^T D,   otherwise.
  
      Parameters
      ----------
      S
          The summation or the aggregation matrix. Shape:
          (total_num_time_series, num_bottom_time_series)
      D
          Symmetric positive definite matrix (typically a diagonal matrix).
          Shape: (total_num_time_series, total_num_time_series)
          Optional.
          If provided then the distance between the reconciled and unreconciled
          forecasts is calculated based on the norm induced by D. Useful for
          weighing the distances differently for each level of the hierarchy.
          By default Euclidean distance is used.
  
      Returns
      -------
      Numpy ND array
          Projection matrix, shape (total_num_time_series, total_num_time_series)
      """
      if D is None:
          return S @ np.linalg.pinv(S.T @ S) @ S.T
      else:
          assert np.all(D == D.T), "`D` must be symmetric."
          assert np.all(
              np.linalg.eigvals(D) > 0
          ), "`D` must be positive definite."
  
          return S @ np.linalg.pinv(S.T @ D @ S) @ S.T @ D
  
imports: import logging
  from typing import List, Optional
  import numpy
  from mxnet.gluon import HybridBlock
  import mxnet
  from gluonts.core.component import validated
  from gluonts.mx import DeepVAREstimator
  from gluonts.mx.model.predictor import Predictor
  from gluonts.mx.model.predictor import RepresentableBlockPredictor
  from gluonts.mx.distribution import LowrankMultivariateGaussianOutput
  from gluonts.mx.trainer import Trainer
  from gluonts.mx.util import copy_parameters
  from gluonts.time_feature import TimeFeature
  from gluonts.transform import Transformation
  from ._network import DeepVARHierarchicalPredictionNetwork, DeepVARHierarchicalTrainingNetwork
label_desc: Use image node_id 1 for calling a global function with example usage: projection_mat(S, D) and returns: unknown, unknown