<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 GPyTorch regression with derivative information
Introduction
In this notebook, we show how to train a GP regression model in GPyTorch of an unknown function given function value and derivative observations. We consider modeling the function
Step1:  Setting up the training data
We use 50 uniformly distributed points in the interval $[0, 5 \pi]$
Step2:  Setting up the model
A GP prior on the function values implies a multi-output GP prior on the function values and the partial derivatives, see 9.4 in http
Step3:  The model training is similar to training a standard GP regression model
Step4:  Model predictions are also similar to GP regression with only function values, butwe need more CG iterations to get accurate estimates of the predictive variance
<ASSISTANT_TASK:>
Python Code:
import torch
import gpytorch
import math
from matplotlib import pyplot as plt
import numpy as np
%matplotlib inline
%load_ext autoreload
%autoreload 2
Explanation: GPyTorch regression with derivative information
Introduction
In this notebook, we show how to train a GP regression model in GPyTorch of an unknown function given function value and derivative observations. We consider modeling the function:
\begin{align}
              y &= \sin(2x) + cos(x) + \epsilon \
  \frac{dy}{dx} &= 2\cos(2x) - \sin(x) + \epsilon \
       \epsilon &\sim \mathcal{N}(0, 0.5)
\end{align}
using 50 value and derivative observations.
End of explanation
lb, ub = 0.0, 5*math.pi
n = 50
train_x = torch.linspace(lb, ub, n).unsqueeze(-1)
train_y = torch.stack([
    torch.sin(2*train_x) + torch.cos(train_x), 
    -torch.sin(train_x) + 2*torch.cos(2*train_x)
], -1).squeeze(1)
train_y += 0.05 * torch.randn(n, 2)
Explanation: Setting up the training data
We use 50 uniformly distributed points in the interval $[0, 5 \pi]$
End of explanation
class GPModelWithDerivatives(gpytorch.models.ExactGP):
    def __init__(self, train_x, train_y, likelihood):
        super(GPModelWithDerivatives, self).__init__(train_x, train_y, likelihood)
        self.mean_module = gpytorch.means.ConstantMeanGrad()
        self.base_kernel = gpytorch.kernels.RBFKernelGrad()
        self.covar_module = gpytorch.kernels.ScaleKernel(self.base_kernel)
    def forward(self, x):
        mean_x = self.mean_module(x)
        covar_x = self.covar_module(x)
        return gpytorch.distributions.MultitaskMultivariateNormal(mean_x, covar_x)
likelihood = gpytorch.likelihoods.MultitaskGaussianLikelihood(num_tasks=2)  # Value + Derivative
model = GPModelWithDerivatives(train_x, train_y, likelihood)
Explanation: Setting up the model
A GP prior on the function values implies a multi-output GP prior on the function values and the partial derivatives, see 9.4 in http://www.gaussianprocess.org/gpml/chapters/RW9.pdf for more details. This allows using a MultitaskMultivariateNormal and MultitaskGaussianLikelihood to train a GP model from both function values and gradients. The resulting RBF kernel that models the covariance between the values and partial derivatives has been implemented in RBFKernelGrad and the extension of a constant mean is implemented in ConstantMeanGrad.
The RBFKernelGrad is generally worse conditioned than the RBFKernel, so we place a lower bound on the noise parameter to keep the smallest eigenvalues of the kernel matrix away from zero.
End of explanation
# this is for running the notebook in our testing framework
import os
smoke_test = ('CI' in os.environ)
training_iter = 2 if smoke_test else 50
# Find optimal model hyperparameters
model.train()
likelihood.train()
# Use the adam optimizer
optimizer = torch.optim.Adam(model.parameters(), lr=0.1)  # Includes GaussianLikelihood parameters
# "Loss" for GPs - the marginal log likelihood
mll = gpytorch.mlls.ExactMarginalLogLikelihood(likelihood, model)
for i in range(training_iter):
    optimizer.zero_grad()
    output = model(train_x)
    loss = -mll(output, train_y)
    loss.backward()
    print('Iter %d/%d - Loss: %.3f   lengthscale: %.3f   noise: %.3f' % (
        i + 1, training_iter, loss.item(),
        model.covar_module.base_kernel.lengthscale.item(),
        model.likelihood.noise.item()
    ))        
    optimizer.step()
Explanation: The model training is similar to training a standard GP regression model
End of explanation
# Set into eval mode
model.train()
model.eval()
likelihood.eval()
# Initialize plots
f, (y1_ax, y2_ax) = plt.subplots(1, 2, figsize=(12, 6))
# Make predictions
with torch.no_grad(), gpytorch.settings.max_cg_iterations(50):
    test_x = torch.linspace(lb, ub, 500)
    predictions = likelihood(model(test_x))
    mean = predictions.mean
    lower, upper = predictions.confidence_region()
    
# Plot training data as black stars
y1_ax.plot(train_x.detach().numpy(), train_y[:, 0].detach().numpy(), 'k*')
# Predictive mean as blue line
y1_ax.plot(test_x.numpy(), mean[:, 0].numpy(), 'b')
# Shade in confidence 
y1_ax.fill_between(test_x.numpy(), lower[:, 0].numpy(), upper[:, 0].numpy(), alpha=0.5)
y1_ax.legend(['Observed Values', 'Mean', 'Confidence'])
y1_ax.set_title('Function values')
# Plot training data as black stars
y2_ax.plot(train_x.detach().numpy(), train_y[:, 1].detach().numpy(), 'k*')
# Predictive mean as blue line
y2_ax.plot(test_x.numpy(), mean[:, 1].numpy(), 'b')
# Shade in confidence 
y2_ax.fill_between(test_x.numpy(), lower[:, 1].numpy(), upper[:, 1].numpy(), alpha=0.5)
y2_ax.legend(['Observed Derivatives', 'Mean', 'Confidence'])
y2_ax.set_title('Derivatives')
None
Explanation: Model predictions are also similar to GP regression with only function values, butwe need more CG iterations to get accurate estimates of the predictive variance
End of explanation
<END_TASK>