code: def test_lower_bounds_inequalities():
      """Test that the expected inequalities are verified."""
      # Toy dataset
      rng = np.random.RandomState(42)
      n_samples_train, n_samples_test, n_timestamps = 20, 30, 60
      window_size = 0.1
      X_train = rng.randn(n_samples_train, n_timestamps)
      X_test = rng.randn(n_samples_test, n_timestamps)
  
      # DTW
      X_dtw = pairwise_distances(X_test, X_train, dtw)
      region = sakoe_chiba_band(n_timestamps, window_size=window_size)
      X_dtw_window = pairwise_distances(
          X_test, X_train, _dtw_sakoechiba, window_size=window_size
      )
  
      # Lower bounds
      lb_yi = lower_bound_yi(X_train, X_test)
      lb_kim = lower_bound_kim(X_train, X_test)
      lb_keogh = lower_bound_keogh(X_train, X_test, region)
      lb_improved = lower_bound_improved(X_train, X_test, region)
  
      # Sanity check
      EPS = 1e-8
      np.testing.assert_array_less(lb_yi, X_dtw + EPS)
      np.testing.assert_array_less(lb_kim, X_dtw + EPS)
      np.testing.assert_array_less(lb_keogh, X_dtw_window + EPS)
      np.testing.assert_array_less(lb_improved, X_dtw_window + EPS)
      np.testing.assert_array_less(lb_keogh, lb_improved + EPS)
  
imports: import numpy
  import pytest
  import re
  from math import sqrt
  from pyts.metrics.lower_bounds import _lower_bound_yi_x_y, _lower_bound_yi_X_Y, _warping_envelope, _clip
  from pyts.metrics import lower_bound_improved, lower_bound_keogh, lower_bound_kim, lower_bound_yi
  from pyts.metrics import dtw, sakoe_chiba_band
  from pyts.metrics.dtw import _dtw_sakoechiba
  from sklearn.metrics import pairwise_distances
label_desc: Use image node_id 12 for calling a global function with example usage: test_lower_bounds_inequalities() without return types