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# Licensed under the Apache License, Version 2.0 (the "License");
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# http://www.apache.org/licenses/LICENSE-2.0
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from typing import Optional
import tensorflow as tf
import gpflow
from .. import posteriors
from ..base import InputData, MeanAndVariance, RegressionData
from ..kernels import Kernel
from ..logdensities import multivariate_normal
from ..mean_functions import MeanFunction
from ..utilities.model_utils import add_noise_cov
from .model import GPModel
from .training_mixins import InternalDataTrainingLossMixin
from .util import data_input_to_tensor
[docs]class GPR_deprecated(GPModel, InternalDataTrainingLossMixin):
r"""
Gaussian Process Regression.
This is a vanilla implementation of GP regression with a Gaussian
likelihood. Multiple columns of Y are treated independently.
The log likelihood of this model is given by
.. math::
\log p(Y \,|\, \mathbf f) =
\mathcal N(Y \,|\, 0, \sigma_n^2 \mathbf{I})
To train the model, we maximise the log _marginal_ likelihood
w.r.t. the likelihood variance and kernel hyperparameters theta.
The marginal likelihood is found by integrating the likelihood
over the prior, and has the form
.. math::
\log p(Y \,|\, \sigma_n, \theta) =
\mathcal N(Y \,|\, 0, \mathbf{K} + \sigma_n^2 \mathbf{I})
"""
def __init__(
self,
data: RegressionData,
kernel: Kernel,
mean_function: Optional[MeanFunction] = None,
noise_variance: float = 1.0,
):
likelihood = gpflow.likelihoods.Gaussian(noise_variance)
_, Y_data = data
super().__init__(kernel, likelihood, mean_function, num_latent_gps=Y_data.shape[-1])
self.data = data_input_to_tensor(data)
[docs] def maximum_log_likelihood_objective(self) -> tf.Tensor:
return self.log_marginal_likelihood()
def _add_noise_cov(self, K: tf.Tensor) -> tf.Tensor:
"""
Returns K + σ² I, where σ² is the likelihood noise variance (scalar),
and I is the corresponding identity matrix.
"""
return add_noise_cov(K, self.likelihood.variance)
[docs] def log_marginal_likelihood(self) -> tf.Tensor:
r"""
Computes the log marginal likelihood.
.. math::
\log p(Y | \theta).
"""
X, Y = self.data
K = self.kernel(X)
ks = self._add_noise_cov(K)
L = tf.linalg.cholesky(ks)
m = self.mean_function(X)
# [R,] log-likelihoods for each independent dimension of Y
log_prob = multivariate_normal(Y, m, L)
return tf.reduce_sum(log_prob)
[docs] def predict_f(
self, Xnew: InputData, full_cov: bool = False, full_output_cov: bool = False
) -> MeanAndVariance:
r"""
This method computes predictions at X \in R^{N \x D} input points
.. math::
p(F* | Y)
where F* are points on the GP at new data points, Y are noisy observations at training data points.
"""
X, Y = self.data
err = Y - self.mean_function(X)
kmm = self.kernel(X)
knn = self.kernel(Xnew, full_cov=full_cov)
kmn = self.kernel(X, Xnew)
kmm_plus_s = self._add_noise_cov(kmm)
conditional = gpflow.conditionals.base_conditional
f_mean_zero, f_var = conditional(
kmn, kmm_plus_s, knn, err, full_cov=full_cov, white=False
) # [N, P], [N, P] or [P, N, N]
f_mean = f_mean_zero + self.mean_function(Xnew)
return f_mean, f_var
[docs]class GPR_with_posterior(GPR_deprecated):
"""
This is an implementation of GPR that provides a posterior() method that
enables caching for faster subsequent predictions.
"""
[docs] def posterior(self, precompute_cache=posteriors.PrecomputeCacheType.TENSOR):
"""
Create the Posterior object which contains precomputed matrices for
faster prediction.
precompute_cache has three settings:
- `PrecomputeCacheType.TENSOR` (or `"tensor"`): Precomputes the cached
quantities and stores them as tensors (which allows differentiating
through the prediction). This is the default.
- `PrecomputeCacheType.VARIABLE` (or `"variable"`): Precomputes the cached
quantities and stores them as variables, which allows for updating
their values without changing the compute graph (relevant for AOT
compilation).
- `PrecomputeCacheType.NOCACHE` (or `"nocache"` or `None`): Avoids
immediate cache computation. This is useful for avoiding extraneous
computations when you only want to call the posterior's
`fused_predict_f` method.
"""
return posteriors.GPRPosterior(
kernel=self.kernel,
data=self.data,
likelihood_variance=self.likelihood.variance,
mean_function=self.mean_function,
precompute_cache=precompute_cache,
)
[docs] def predict_f(self, Xnew: InputData, full_cov=False, full_output_cov=False) -> MeanAndVariance:
"""
For backwards compatibility, GPR's predict_f uses the fused (no-cache)
computation, which is more efficient during training.
For faster (cached) prediction, predict directly from the posterior object, i.e.,:
model.posterior().predict_f(Xnew, ...)
"""
return self.posterior(posteriors.PrecomputeCacheType.NOCACHE).fused_predict_f(
Xnew, full_cov=full_cov, full_output_cov=full_output_cov
)
[docs]class GPR(GPR_with_posterior):
# subclassed to ensure __class__ == "GPR"
pass