code: def map_irls_eqn(
      key: Tuple,
      key_pattern: List[str],
      pdf: pd.DataFrame,
      labeldf: pd.DataFrame,
      sample_blocks: Dict[str, List[str]],
      covdf: pd.DataFrame,
      beta_cov_dict: Dict[str, NDArray],
      maskdf: pd.DataFrame,
      alphas: Dict[str, Float],
  ) -> pd.DataFrame:
      """
      This function constructs matrices X and Y, and computes transpose(X)*diag(p(1-p)*X, beta, and transpose(X)*Y, by
      fitting a logistic model logit(p(Y|X)) ~ X*beta.
  
      Each block X is uniquely identified by a header_block ID, which maps to a set of contiguous columns in the overall
      block matrix, and a sample_block ID, which maps to a set of rows in the overall block matrix (and likewise a set
      of rows in the label matrix Y).  Additionally, each block is associated with a particular label and alpha,
      so the key for the group is (header_block, sample_block, label, alpha).
  
      Args:
          key : unique key identifying the group of rows emitted by a groupBy statement.
          key_pattern : pattern of columns used in the groupBy statement that emitted this group of rows
          pdf : starting Pandas DataFrame used to build X and Y for block X identified by :key:.
              schema:
               |-- header: string
               |-- size: integer
               |-- indices: array (Required only if the matrix is sparse)
               |    |-- element: integer
               |-- values: array
               |    |-- element: double
               |-- header_block: string
               |-- sample_block: string
               |-- sort_key: integer
               |-- mu: double
               |-- sig: double
               |-- label: string
               |-- alpha_name: string
          labeldf : Pandas DataFrame containing label values (i. e., the Y in the normal equation above).
          sample_index : sample_index: dict containing a mapping of sample_block ID to a list of corresponding sample IDs
          covdf : Pandas DataFrame containing covariates that should be included with every block X above (can be empty).
          beta_cov_dict : dict of [label: str, beta: NDArray[Float]] that maps each label to the covariate parameter
              values estimated from the entire population.
          maskdf : Pandas DataFrame mirroring labeldf containing Boolean values flagging samples with missing labels as
              True and others as False.
          alphas : dict of [alpha_name: str, alpha_value: float]
  
      Returns:
          transformed Pandas DataFrame containing beta, XtgX, and XtY corresponding to a particular block X.
              schema (specified by the irls_eqn_struct):
               |-- header_block: string
               |-- sample_block: string
               |-- label: string
               |-- alpha_name : string
               |-- header: string
               |-- sort_key: integer
               |-- beta : double
               |-- xtgx: array
               |    |-- element: double
               |-- xty: array
               |    |-- element: double
      """
      (
          header_block,
          sample_block,
          label,
          alpha_name,
      ) = parse_header_block_sample_block_label_alpha_name(
          key, key_pattern
      )
      sort_in_place(pdf, ["sort_key", "header"])
      n_rows = pdf["size"][0]
      n_cols = len(pdf)
      sample_list = sample_blocks[sample_block]
      beta_cov = beta_cov_dict[label]
  
      if maskdf.empty:
          row_mask = np.array([])
      else:
          row_mask = slice_label_rows(
              maskdf, label, sample_list, np.array([])
          ).ravel()
  
      alpha_value = alphas[alpha_name]
      cov_matrix = (
          np.array([])
          if covdf.empty
          else slice_label_rows(covdf, "all", sample_list, np.array([]))
      )
      n_cov = len(covdf.columns)
      header_col = np.concatenate([covdf.columns, pdf["header"]])
      # Add new sort_keys for covariates, starting from -n_cov up to 0 to ensure they come ahead of the headers.
      sort_key_col = np.concatenate(
          (np.arange(-n_cov, 0), pdf["sort_key"])
      )
      X = assemble_block(n_rows, n_cols, pdf, cov_matrix, row_mask)
  
      Y = slice_label_rows(labeldf, label, sample_list, row_mask)
  
      beta, XtGX, XtY = get_irls_pieces(
          X, Y.ravel(), alpha_value, beta_cov
      )
  
      data = {
          "header_block": header_block,
          "sample_block": sample_block,
          "label": label,
          "alpha_name": alpha_name,
          "header": header_col,
          "sort_key": sort_key_col,
          "beta": list(beta),
          "xtgx": list(XtGX),
          "xty": list(XtY),
      }
  
      return pd.DataFrame(data)
  
imports: from .ridge_udfs import *
label_desc: Use image node_id 1 for calling a global function with example usage: map_irls_eqn(key, key_pattern, pdf, labeldf, sample_blocks, covdf, beta_cov_dict, maskdf, alphas) and returns: pd