code: def sample(
      problem: Dict,
      N: int,
      *,
      calc_second_order: bool = True,
      scramble: bool = True,
      skip_values: int = 0,
      seed: Optional[Union[int, np.random.Generator]] = None,
  ):
      """Generates model inputs using Saltelli's extension of the Sobol' sequence.
  
      The Sobol' sequence is a popular quasi-random low-discrepancy sequence used
      to generate uniform samples of parameter space.
  
      Returns a NumPy matrix containing the model inputs using Saltelli's
      sampling scheme.
      Saltelli's scheme extends the Sobol' sequence in a way to reduce
      the error rates in the resulting sensitivity index calculations. If
      `calc_second_order` is False, the resulting matrix has ``N * (D + 2)``
      rows, where ``D`` is the number of parameters.
      If `calc_second_order` is `True`, the resulting matrix has ``N * (2D + 2)``
      rows.
  
      These model inputs are intended to be used with
      :func:`SALib.analyze.sobol.analyze`.
  
      Notes
      -----
      The initial points of the Sobol' sequence has some repetition (see Table 2
      in Campolongo [1]__), which can be avoided by scrambling the sequence.
  
      Another option, not recommended and available for educational purposes,
      is to use the `skip_values` parameter.
      Skipping values reportedly improves the uniformity of samples.
      But, it has been shown that naively skipping values may reduce accuracy,
      increasing the number of samples needed to achieve convergence
      (see Owen [2]__).
  
      Parameters
      ----------
      problem : dict,
          The problem definition.
      N : int
          The number of samples to generate.
          Ideally a power of 2 and <= `skip_values`.
      calc_second_order : bool, optional
          Calculate second-order sensitivities. Default is True.
      scramble : bool, optional
          If True, use LMS+shift scrambling. Otherwise, no scrambling is done.
          Default is True.
      skip_values : int, optional
          Number of points in Sobol' sequence to skip, ideally a value of base 2.
          It's recommended not to change this value and use `scramble` instead.
          `scramble` and `skip_values` can be used together.
          Default is 0.
      seed : {None, int, `numpy.random.Generator`}, optional
          If `seed` is None the `numpy.random.Generator` generator is used.
          If `seed` is an int, a new ``Generator`` instance is used,
          seeded with `seed`.
          If `seed` is already a ``Generator`` instance then that instance is
          used. Default is None.
  
      References
      ----------
      1. Campolongo, F., Saltelli, A., Cariboni, J., 2011.
         From screening to quantitative sensitivity analysis.
         A unified approach.
         Computer Physics Communications 182, 978-988.
         https://doi.org/10.1016/j.cpc.2010.12.039
  
      2. Owen, A. B., 2020.
         On dropping the first Sobol' point.
         arXiv:2008.08051 [cs, math, stat].
         Available at: http://arxiv.org/abs/2008.08051
         (Accessed: 20 April 2021).
  
      3. Saltelli, A., 2002.
         Making best use of model evaluations to compute sensitivity indices.
         Computer Physics Communications 145, 280-297.
         https://doi.org/10.1016/S0010-4655(02)00280-1
  
      4. Sobol', I.M., 2001.
         Global sensitivity indices for nonlinear mathematical models and
         their Monte Carlo estimates.
         Mathematics and Computers in Simulation,
         The Second IMACS Seminar on Monte Carlo Methods 55, 271-280.
         https://doi.org/10.1016/S0378-4754(00)00270-6
      """
      D = problem["num_vars"]
      groups = _check_groups(problem)
  
      # Create base sequence - could be any type of sampling
      qrng = qmc.Sobol(d=2 * D, scramble=scramble, seed=seed)
  
      # fast-forward logic
      if skip_values > 0 and isinstance(skip_values, int):
          M = skip_values
          if not ((M & (M - 1) == 0) and (M != 0 and M - 1 != 0)):
              msg = f"""
              Convergence properties of the Sobol' sequence is only valid if
              `skip_values` ({M}) is a power of 2.
              """
              warnings.warn(msg, stacklevel=2)
  
          # warning when N > skip_values
          # see https://github.com/scipy/scipy/pull/10844#issuecomment-673029539
          n_exp = int(np.log2(N))
          m_exp = int(np.log2(M))
          if n_exp > m_exp:
              msg = (
                  "Convergence may not be valid as the number of "
                  "requested samples is"
                  f" > `skip_values` ({N} > {M})."
              )
              warnings.warn(msg, stacklevel=2)
  
          qrng.fast_forward(M)
      elif skip_values < 0 or not isinstance(skip_values, int):
          raise ValueError("`skip_values` must be a positive integer.")
  
      # sample Sobol' sequence
      base_sequence = qrng.random(N)
  
      if not groups:
          Dg = problem["num_vars"]
      else:
          G, group_names = compute_groups_matrix(groups)
          Dg = len(set(group_names))
  
      if calc_second_order:
          saltelli_sequence = np.zeros([(2 * Dg + 2) * N, D])
      else:
          saltelli_sequence = np.zeros([(Dg + 2) * N, D])
  
      index = 0
  
      for i in range(N):
          # Copy matrix "A"
          for j in range(D):
              saltelli_sequence[index, j] = base_sequence[i, j]
  
          index += 1
  
          # Cross-sample elements of "B" into "A"
          for k in range(Dg):
              for j in range(D):
                  if (not groups and j == k) or (
                      groups and group_names[k] == groups[j]
                  ):
                      saltelli_sequence[index, j] = base_sequence[
                          i, j + D
                      ]
                  else:
                      saltelli_sequence[index, j] = base_sequence[i, j]
  
              index += 1
  
          # Cross-sample elements of "A" into "B"
          # Only needed if you're doing second-order indices (true by default)
          if calc_second_order:
              for k in range(Dg):
                  for j in range(D):
                      if (not groups and j == k) or (
                          groups and group_names[k] == groups[j]
                      ):
                          saltelli_sequence[index, j] = base_sequence[
                              i, j
                          ]
                      else:
                          saltelli_sequence[index, j] = base_sequence[
                              i, j + D
                          ]
  
                  index += 1
  
          # Copy matrix "B"
          for j in range(D):
              saltelli_sequence[index, j] = base_sequence[i, j + D]
  
          index += 1
  
      saltelli_sequence = scale_samples(saltelli_sequence, problem)
      return saltelli_sequence
  
file_path: SALib/src/SALib/sample/sobol.py
imports: import warnings
  from typing import Dict, Optional, Union
  import numpy
  from scipy.stats import qmc
  from .None import common_args
  from ..util import scale_samples, read_param_file, compute_groups_matrix, _check_groups
label_desc: Use image node_id 1 for calling a global function with example usage: sample(problem, N) and returns: saltelli_sequence