code: def _third_order(b_m3, Y_res, l2_penalty, hdmr, t):
      """Sequential determination of third-order component functions.
      it computes component functions via ridge regression, i.e.
      fitting model inputs/features to the model output.
  
      Parameters
      ----------
      b_m3 : numpy.array
          Basis matrix for third-order component functions
      Y_res : numpy.array
          Residual model output
      l2_penalty : float
          Penalty term for ridge regression
      hdmr : namedtuple
          Core parameters of hdmr expansion
      t : int
          bootstrap iteration
  
      Notes
      -----
      Backfitting algorithm is not used here because it may be
      unstable when residual model, Y_res, is close to arrays of zero.
      """
      # To increase readibility
      n3 = hdmr.nt3
      # L2 Penalty
      lambda_eye = l2_penalty * np.identity(n3)
      for i in range(hdmr.nc3):
          try:
              # Left hand side
              a = (
                  b_m3[:, i * n3 : n3 * (i + 1)].T
                  @ b_m3[:, i * n3 : n3 * (i + 1)]
              ) / hdmr.subset
              # Adding L2 Penalty (Ridge Regression)
              a += lambda_eye
              # Right hand side
              b = (
                  b_m3[:, i * n3 : n3 * (i + 1)].T @ Y_res
              ) / hdmr.subset
              # Solution
              hdmr.x[
                  hdmr.tnt1
                  + hdmr.tnt2
                  + i * n3 : hdmr.tnt1
                  + hdmr.tnt2
                  + n3 * (i + 1),
                  t,
              ] = solve(a, b)
          except LinAlgError:
              raise LinAlgError(
                  "Third Order: Least-square regression did not converge. Try increasing L2 penalty term"
              )
  
imports: import math
  import warnings
  from typing import Dict, Tuple
  from types import MethodType
  from itertools import combinations, product
  from collections import defaultdict, namedtuple
  import numpy
  from pandas import DataFrame
  from numpy.linalg import det, pinv, matrix_rank
  from scipy.linalg import svd, LinAlgError, solve
  from scipy import stats, special
  from .None import common_args
  from ..util import read_param_file, ResultDict
label_desc: Use image node_id 12 for calling a global function with example usage: _third_order(b_m3, Y_res, l2_penalty, hdmr, t) without return types