code: def test_incremental_variance_numerical_stability():
      # Test Youngs and Cramer incremental variance formulas.
  
      def np_var(A):
          return A.var(axis=0)
  
      # Naive one pass variance computation - not numerically stable
      # https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
      def one_pass_var(X):
          n = X.shape[0]
          exp_x2 = (X**2).sum(axis=0) / n
          expx_2 = (X.sum(axis=0) / n) ** 2
          return exp_x2 - expx_2
  
      # Two-pass algorithm, stable.
      # We use it as a benchmark. It is not an online algorithm
      # https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Two-pass_algorithm
      def two_pass_var(X):
          mean = X.mean(axis=0)
          Y = X.copy()
          return np.mean((Y - mean) ** 2, axis=0)
  
      # Naive online implementation
      # https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Online_algorithm
      # This works only for chunks for size 1
      def naive_mean_variance_update(
          x, last_mean, last_variance, last_sample_count
      ):
          updated_sample_count = last_sample_count + 1
          samples_ratio = last_sample_count / float(
              updated_sample_count
          )
          updated_mean = (
              x / updated_sample_count + last_mean * samples_ratio
          )
          updated_variance = (
              last_variance * samples_ratio
              + (x - last_mean)
              * (x - updated_mean)
              / updated_sample_count
          )
          return updated_mean, updated_variance, updated_sample_count
  
      # We want to show a case when one_pass_var has error > 1e-3 while
      # _batch_mean_variance_update has less.
      tol = 200
      n_features = 2
      n_samples = 10000
      x1 = np.array(1e8, dtype=np.float64)
      x2 = np.log(1e-5, dtype=np.float64)
      A0 = np.full((n_samples // 2, n_features), x1, dtype=np.float64)
      A1 = np.full((n_samples // 2, n_features), x2, dtype=np.float64)
      A = np.vstack((A0, A1))
  
      # Naive one pass var: >tol (=1063)
      assert np.abs(np_var(A) - one_pass_var(A)).max() > tol
  
      # Starting point for online algorithms: after A0
  
      # Naive implementation: >tol (436)
      mean, var, n = A0[0, :], np.zeros(n_features), n_samples // 2
      for i in range(A1.shape[0]):
          mean, var, n = naive_mean_variance_update(
              A1[i, :], mean, var, n
          )
      assert n == A.shape[0]
      # the mean is also slightly unstable
      assert np.abs(A.mean(axis=0) - mean).max() > 1e-6
      assert np.abs(np_var(A) - var).max() > tol
  
      # Robust implementation: <tol (177)
      mean, var = A0[0, :], np.zeros(n_features)
      n = np.full(n_features, n_samples // 2, dtype=np.int32)
      for i in range(A1.shape[0]):
          mean, var, n = _incremental_mean_and_var(
              A1[i, :].reshape((1, A1.shape[1])), mean, var, n
          )
      assert_array_equal(n, A.shape[0])
      assert_array_almost_equal(A.mean(axis=0), mean)
      assert tol > np.abs(np_var(A) - var).max()
  
imports: import numpy
  import pytest
  from scipy import linalg, sparse
  from scipy.linalg import eigh
  from scipy.sparse.linalg import eigsh
  from scipy.special import expit
  from sklearn.datasets import make_low_rank_matrix, make_sparse_spd_matrix
  from sklearn.utils import gen_batches
  from sklearn.utils._arpack import _init_arpack_v0
  from sklearn.utils._testing import assert_allclose, assert_allclose_dense_sparse, assert_almost_equal, assert_array_almost_equal, assert_array_equal, skip_if_32bit
  from sklearn.utils.extmath import _deterministic_vector_sign_flip, _incremental_mean_and_var, _randomized_eigsh, _safe_accumulator_op, cartesian, density, log_logistic, randomized_svd, row_norms, safe_sparse_dot, softmax, stable_cumsum, svd_flip, weighted_mode
  from sklearn.utils.fixes import COO_CONTAINERS, CSC_CONTAINERS, CSR_CONTAINERS, DOK_CONTAINERS, LIL_CONTAINERS, _mode
label_desc: Use image node_id 28 for calling a global function with example usage: test_incremental_variance_numerical_stability() and returns: A, unknown, np, updated_mean,  updated_variance,  updated_sample_count