code: def test_lcm_icm_equivalence(self):
      # Training points are every 0.1 in [0,1] (note that they're the same for both tasks)
      train_x = torch.linspace(0, 1, 100)
      # y1 function is sin(2*pi*x) with noise N(0, 0.04)
      train_y1 = (
          torch.sin(train_x.data * (2 * math.pi))
          + torch.randn(train_x.size()) * 0.2
      )
      # y2 function is cos(2*pi*x) with noise N(0, 0.04)
      train_y2 = (
          torch.cos(train_x.data * (2 * math.pi))
          + torch.randn(train_x.size()) * 0.2
      )
      # Create a train_y which interleaves the two
      train_y = torch.stack([train_y1, train_y2], -1)
  
      likelihood = MultitaskGaussianLikelihood(num_tasks=2)
      model = MultitaskGPModel(train_x, train_y, likelihood)
  
      # Use the adam optimizer
      optimizer = torch.optim.Adam(
          model.parameters(), lr=0.1
      )  # Includes GaussianL^ikelihood parameters
      model.train()
      likelihood.train()
      mll = gpytorch.mlls.ExactMarginalLogLikelihood(likelihood, model)
      n_iter = 50
      for _ in range(n_iter):
          optimizer.zero_grad()
          output = model(train_x)
          loss = -mll(output, train_y)
          loss.backward()
          optimizer.step()
      model.eval()
      likelihood.eval()
  
      # Make predictions for LCM
      with torch.no_grad():
          test_x = torch.linspace(0, 1, 51)
          observed_pred = likelihood(model(test_x))
          mean = observed_pred.mean
  
      model_icm = MultitaskGPModel_ICM(train_x, train_y, likelihood)
      likelihood = MultitaskGaussianLikelihood(num_tasks=2)
      model_icm.train()
      likelihood.train()
      mll = gpytorch.mlls.ExactMarginalLogLikelihood(
          likelihood, model_icm
      )
      optimizer = torch.optim.Adam(
          model_icm.parameters(), lr=0.1
      )  # Includes GaussianLikelihood parameters
      for _ in range(n_iter):
          optimizer.zero_grad()
          output = model_icm(train_x)
          loss = -mll(output, train_y)
          loss.backward()
          optimizer.step()
      model_icm.eval()
      likelihood.eval()
  
      # Make predictions for ICM
      with torch.no_grad():
          test_x = torch.linspace(0, 1, 51)
          observed_pred_icm = likelihood(model_icm(test_x))
          mean_icm = observed_pred_icm.mean
  
      # Make sure predictions from LCM with one base kernel and ICM are the same.
      self.assertLess((mean - mean_icm).pow(2).mean(), 1e-2)
  
file_path: gpytorch/test/examples/test_lcm_kernel_regression.py
imports: import math
  import unittest
  import torch
  import gpytorch
  from gpytorch.distributions import MultitaskMultivariateNormal
  from gpytorch.kernels import LCMKernel, MultitaskKernel, RBFKernel
  from gpytorch.likelihoods import MultitaskGaussianLikelihood
  from gpytorch.means import ConstantMean, MultitaskMean
label_desc: Use image node_id 1 for calling the TestLCMKernelRegression obj's underlying member method code with example usage: obj.test_lcm_icm_equivalence() without return types