code: def analyze(
      problem,
      Y,
      calc_second_order=True,
      num_resamples=100,
      conf_level=0.95,
      print_to_console=False,
      parallel=False,
      n_processors=None,
      keep_resamples=False,
      seed=None,
  ):
      """Perform Sobol Analysis on model outputs.
  
      Returns a dictionary with keys 'S1', 'S1_conf', 'ST', and 'ST_conf', where
      each entry is a list of size D (the number of parameters) containing the
      indices in the same order as the parameter file.  If calc_second_order is
      True, the dictionary also contains keys 'S2' and 'S2_conf'.
  
  
      Notes
      -----
      Compatible with:
          `saltelli` : :func:`SALib.sample.saltelli.sample`
          `sobol` : :func:`SALib.sample.sobol.sample`
  
  
      Examples
      --------
      >>> X = saltelli.sample(problem, 512)
      >>> Y = Ishigami.evaluate(X)
      >>> Si = sobol.analyze(problem, Y, print_to_console=True)
  
  
      Parameters
      ----------
      problem : dict
          The problem definition
      Y : numpy.array
          A NumPy array containing the model outputs
      calc_second_order : bool
          Calculate second-order sensitivities (default True)
      num_resamples : int
          The number of resamples (default 100)
      conf_level : float
          The confidence interval level (default 0.95)
      print_to_console : bool
          Print results directly to console (default False)
      parallel : bool
          Perform analysis in parallel if True
      n_processors : int
          Number of parallel processes (only used if parallel is True)
      keep_resamples : bool
          Whether or not to store intermediate resampling results (default False)
      seed : int
          Seed to generate a random number
  
      References
      ----------
      1. Sobol, I. M. (2001).
         Global sensitivity indices for nonlinear
         mathematical models and their Monte Carlo estimates.
         Mathematics and Computers in Simulation, 55(1-3):271-280,
         doi:10.1016/S0378-4754(00)00270-6.
  
      2. Saltelli, A. (2002).
         Making best use of model evaluations to compute sensitivity indices.
         Computer Physics Communications, 145(2):280-297
         doi:10.1016/S0010-4655(02)00280-1.
  
      3. Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and
         S. Tarantola (2010).
         Variance based sensitivity analysis of model output. Design and
         estimator for the total sensitivity index.
         Computer Physics Communications, 181(2):259-270,
         doi:10.1016/j.cpc.2009.09.018.
      """
      if seed:
          # Set seed to ensure CIs are the same
          rng = np.random.default_rng(seed).integers
      else:
          rng = np.random.randint
  
      # determining if groups are defined and adjusting the number
      # of rows in the cross-sampled matrix accordingly
      groups = _check_groups(problem)
      if not groups:
          D = problem["num_vars"]
      else:
          _, D = extract_group_names(groups)
  
      if calc_second_order and Y.size % (2 * D + 2) == 0:
          N = int(Y.size / (2 * D + 2))
      elif not calc_second_order and Y.size % (D + 2) == 0:
          N = int(Y.size / (D + 2))
      else:
          raise RuntimeError(
              """
          Incorrect number of samples in model output file.
          Confirm that calc_second_order matches option used during sampling."""
          )
  
      if not 0 < conf_level < 1:
          raise RuntimeError("Confidence level must be between 0-1.")
  
      # Normalize the model output.
      # Estimates of the Sobol' indices can be biased for non-centered outputs
      # so we center here by normalizing with the standard deviation.
      # Other approaches opt to subtract the mean.
      Y = (Y - Y.mean()) / Y.std()
  
      A, B, AB, BA = separate_output_values(Y, D, N, calc_second_order)
      r = rng(N, size=(N, num_resamples))
      Z = norm.ppf(0.5 + conf_level / 2)
  
      if not parallel:
          S = create_Si_dict(
              D, num_resamples, keep_resamples, calc_second_order
          )
  
          for j in range(D):
              S["S1"][j] = first_order(A, AB[:, j], B)
              S1_conf_j = first_order(A[r], AB[r, j], B[r])
  
              if keep_resamples:
                  S["S1_conf_all"][:, j] = S1_conf_j
  
              var_diff = np.r_[A[r], B[r]].ptp()
              if var_diff != 0.0:
                  S["S1_conf"][j] = Z * S1_conf_j.std(ddof=1)
              else:
                  S["S1_conf"][j] = 0.0
  
              S["ST"][j] = total_order(A, AB[:, j], B)
  
              ST_conf_j = total_order(A[r], AB[r, j], B[r])
  
              if keep_resamples:
                  S["ST_conf_all"][:, j] = ST_conf_j
  
              if var_diff != 0.0:
                  S["ST_conf"][j] = Z * ST_conf_j.std(ddof=1)
              else:
                  S["ST_conf"][j] = 0.0
  
          # Second order (+conf.)
          if calc_second_order:
              for j in range(D):
                  for k in range(j + 1, D):
                      S["S2"][j, k] = second_order(
                          A, AB[:, j], AB[:, k], BA[:, j], B
                      )
                      S["S2_conf"][j, k] = Z * second_order(
                          A[r], AB[r, j], AB[r, k], BA[r, j], B[r]
                      ).std(ddof=1)
      else:
          tasks, n_processors = create_task_list(
              D, calc_second_order, n_processors
          )
  
          func = partial(sobol_parallel, Z, A, AB, BA, B, r)
          pool = Pool(n_processors)
          S_list = pool.map_async(func, tasks)
          pool.close()
          pool.join()
  
          S = Si_list_to_dict(
              S_list.get(),
              D,
              num_resamples,
              keep_resamples,
              calc_second_order,
          )
  
      # Add problem context and override conversion method for special case
      S.problem = problem
      S.to_df = MethodType(to_df, S)
  
      # Print results to console
      if print_to_console:
          res = S.to_df()
          for df in res:
              print(df)
  
      return S
  
imports: from types import MethodType
  from warnings import warn
  from scipy.stats import norm
  import numpy
  import pandas
  from .None import common_args
  from ..util import read_param_file, ResultDict, extract_group_names, _check_groups
  from multiprocessing import Pool, cpu_count
  from functools import partial
  from itertools import combinations, zip_longest
label_desc: Use image node_id 1 for calling a global function with example usage: analyze(problem, Y, calc_second_order, num_resamples, conf_level, print_to_console, parallel, n_processors, keep_resamples, seed) and returns: S