# Source code for pyro.contrib.gp.models.model

# Copyright (c) 2017-2019 Uber Technologies, Inc.

from pyro.contrib.gp.parameterized import Parameterized

def _zero_mean_function(x):
return 0

[docs]class GPModel(Parameterized):
r"""
Base class for Gaussian Process models.

The core of a Gaussian Process is a covariance function :math:k which governs
the similarity between input points. Given :math:k, we can establish a
distribution over functions :math:f by a multivarite normal distribution

.. math:: p(f(X)) = \mathcal{N}(0, k(X, X)),

where :math:X is any set of input points and :math:k(X, X) is a covariance
matrix whose entries are outputs :math:k(x, z) of :math:k over input pairs
:math:(x, z). This distribution is usually denoted by

.. math:: f \sim \mathcal{GP}(0, k).

.. note:: Generally, beside a covariance matrix :math:k, a Gaussian Process can
also be specified by a mean function :math:m (which is a zero-value function
by default). In that case, its distribution will be

.. math:: p(f(X)) = \mathcal{N}(m(X), k(X, X)).

Gaussian Process models are :class:~pyro.contrib.gp.util.Parameterized
subclasses. So its parameters can be learned, set priors, or fixed by using
corresponding methods from :class:~pyro.contrib.gp.util.Parameterized. A typical
way to define a Gaussian Process model is

>>> X = torch.tensor([[1., 5, 3], [4, 3, 7]])
>>> y = torch.tensor([2., 1])
>>> kernel = gp.kernels.RBF(input_dim=3)
>>> kernel.variance = pyro.nn.PyroSample(dist.Uniform(torch.tensor(0.5), torch.tensor(1.5)))
>>> kernel.lengthscale = pyro.nn.PyroSample(dist.Uniform(torch.tensor(1.0), torch.tensor(3.0)))
>>> gpr = gp.models.GPRegression(X, y, kernel)

There are two ways to train a Gaussian Process model:

+ Using an MCMC algorithm (in module :mod:pyro.infer.mcmc) on :meth:model to
get posterior samples for the Gaussian Process's parameters. For example:

>>> hmc_kernel = HMC(gpr.model)
>>> mcmc = MCMC(hmc_kernel, num_samples=10)
>>> mcmc.run()
>>> ls_name = "kernel.lengthscale"
>>> posterior_ls = mcmc.get_samples()[ls_name]

+ Using a variational inference on the pair :meth:model, :meth:guide:

>>> loss_fn = pyro.infer.TraceMeanField_ELBO().differentiable_loss
>>>
>>> for i in range(1000):
...     svi.step()  # doctest: +SKIP
...     loss = loss_fn(gpr.model, gpr.guide)  # doctest: +SKIP
...     loss.backward()  # doctest: +SKIP
...     optimizer.step()

To give a prediction on new dataset, simply use :meth:forward like any PyTorch
:class:torch.nn.Module:

>>> Xnew = torch.tensor([[2., 3, 1]])
>>> f_loc, f_cov = gpr(Xnew, full_cov=True)

Reference:

[1] Gaussian Processes for Machine Learning,
Carl E. Rasmussen, Christopher K. I. Williams

:param torch.Tensor X: A input data for training. Its first dimension is the number
of data points.
:param torch.Tensor y: An output data for training. Its last dimension is the
number of data points.
:param ~pyro.contrib.gp.kernels.kernel.Kernel kernel: A Pyro kernel object, which
is the covariance function :math:k.
:param callable mean_function: An optional mean function :math:m of this Gaussian
process. By default, we use zero mean.
:param float jitter: A small positive term which is added into the diagonal part of
a covariance matrix to help stablize its Cholesky decomposition.
"""
def __init__(self, X, y, kernel, mean_function=None, jitter=1e-6):
super().__init__()
self.set_data(X, y)
self.kernel = kernel
self.mean_function = (mean_function if mean_function is not None else
_zero_mean_function)
self.jitter = jitter

[docs]    def model(self):
"""
A "model" stochastic function. If self.y is None, this method returns
mean and variance of the Gaussian Process prior.
"""
raise NotImplementedError

[docs]    def guide(self):
"""
A "guide" stochastic function to be used in variational inference methods. It
also gives posterior information to the method :meth:forward for prediction.
"""
raise NotImplementedError

[docs]    def forward(self, Xnew, full_cov=False):
r"""
Computes the mean and covariance matrix (or variance) of Gaussian Process
posterior on a test input data :math:X_{new}:

.. math:: p(f^* \mid X_{new}, X, y, k, \theta),

where :math:\theta are parameters of this model.

.. note:: Model's parameters :math:\theta together with kernel's parameters
have been learned from a training procedure (MCMC or SVI).

:param torch.Tensor Xnew: A input data for testing. Note that
Xnew.shape[1:] must be the same as X.shape[1:].
:param bool full_cov: A flag to decide if we want to predict full covariance
matrix or just variance.
:returns: loc and covariance matrix (or variance) of :math:p(f^*(X_{new}))
:rtype: tuple(torch.Tensor, torch.Tensor)
"""
raise NotImplementedError

[docs]    def set_data(self, X, y=None):
"""
Sets data for Gaussian Process models.

Some examples to utilize this method are:

.. doctest::
:hide:

>>> X = torch.tensor([[1., 5, 3], [4, 3, 7]])
>>> y = torch.tensor([2., 1])
>>> kernel = gp.kernels.RBF(input_dim=3)
>>> kernel.variance = pyro.nn.PyroSample(dist.Uniform(torch.tensor(0.5), torch.tensor(1.5)))
>>> kernel.lengthscale = pyro.nn.PyroSample(dist.Uniform(torch.tensor(1.0), torch.tensor(3.0)))

+ Batch training on a sparse variational model:

>>> Xu = torch.tensor([[1., 0, 2]])  # inducing input
>>> likelihood = gp.likelihoods.Gaussian()
>>> vsgp = gp.models.VariationalSparseGP(X, y, kernel, Xu, likelihood)
>>> loss_fn = pyro.infer.TraceMeanField_ELBO().differentiable_loss
>>> batched_X, batched_y = X.split(split_size=10), y.split(split_size=10)
>>> for Xi, yi in zip(batched_X, batched_y):
...     vsgp.set_data(Xi, yi)
...     svi.step()  # doctest: +SKIP
...     loss = loss_fn(vsgp.model, vsgp.guide)  # doctest: +SKIP
...     loss.backward()  # doctest: +SKIP
...     optimizer.step()

+ Making a two-layer Gaussian Process stochastic function:

>>> gpr1 = gp.models.GPRegression(X, None, kernel)
>>> Z, _ = gpr1.model()
>>> gpr2 = gp.models.GPRegression(Z, y, kernel)
>>> def two_layer_model():
...     Z, _ = gpr1.model()
...     gpr2.set_data(Z, y)
...     return gpr2.model()

References:

[1] Scalable Variational Gaussian Process Classification,
James Hensman, Alexander G. de G. Matthews, Zoubin Ghahramani

[2] Deep Gaussian Processes,
Andreas C. Damianou, Neil D. Lawrence

:param torch.Tensor X: A input data for training. Its first dimension is the
number of data points.
:param torch.Tensor y: An output data for training. Its last dimension is the
number of data points.
"""
if y is not None and X.size(0) != y.size(-1):
raise ValueError("Expected the number of input data points equal to the "
"number of output data points, but got {} and {}."
.format(X.size(0), y.size(-1)))
self.X = X
self.y = y

def _check_Xnew_shape(self, Xnew):
"""
Checks the correction of the shape of new data.

:param torch.Tensor Xnew: A input data for testing. Note that
Xnew.shape[1:] must be the same as self.X.shape[1:].
"""
if Xnew.dim() != self.X.dim():
raise ValueError("Train data and test data should have the same "
"number of dimensions, but got {} and {}."
.format(self.X.dim(), Xnew.dim()))
if self.X.shape[1:] != Xnew.shape[1:]:
raise ValueError("Train data and test data should have the same "
"shape of features, but got {} and {}."
.format(self.X.shape[1:], Xnew.shape[1:]))