code: def test_actual_results_lower_bound_keogh():
      """Test that the actual results are the expected ones."""
      # Toy dataset
      X_train = np.asarray([[0, 1, 2, 3], [1, 2, 3, 4]])
      X_test = np.asarray([[0, 2.5, 3.5, 6]])
  
      # Region = Sakoe-Chiba band (w=0)
      region = [[0, 1, 2, 3], [1, 2, 3, 4]]
      arr_actual = lower_bound_keogh(X_train, X_test, region)
      arr_desired = np.sqrt(
          np.sum(
              (X_test[:, None, :] - X_train[None, :, :]) ** 2, axis=-1
          )
      )
      np.testing.assert_allclose(
          arr_actual, arr_desired, atol=1e-5, rtol=0
      )
  
      # Region = Sakoe-Chiba band (w=1)
      region_window = [[0, 0, 1, 2], [2, 3, 4, 4]]
      arr_actual_window = lower_bound_keogh(
          X_train, X_test, region_window
      )
      # lower = [[0, 0, 1, 2], [1, 1, 2, 3]]
      # upper = [[1, 2, 3, 3], [2, 3, 4, 4]]
      # X_proj = [[0, 2, 3, 3], [1, 2.5, 3.5, 4]]
      # LB_Keogh = [[sqrt(0.25 + 0.25 + 9), sqrt(1 + 4)]]
      arr_desired_window = np.sqrt([[9.5, 5]])
      np.testing.assert_allclose(
          arr_actual_window, arr_desired_window, atol=1e-5, rtol=0
      )
  
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 10 for calling a global function with example usage: test_actual_results_lower_bound_keogh() without return types