https://github.com/GPflow/GPflow
Tip revision: b819db324fb3c64cab4db52c8f618ab8ff0f5778 authored by st-- on 14 September 2020, 17:03:08 UTC
Merge pull request #1565 from GPflow/develop
Merge pull request #1565 from GPflow/develop
Tip revision: b819db3
conditionals.py
# Copyright 2017-2020 The GPflow Contributors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
import tensorflow as tf
from ..covariances import Kuf, Kuu
from ..inducing_variables import InducingVariables
from ..kernels import Kernel
from ..utilities.ops import eye
from ..config import default_jitter
from .dispatch import conditional
from .util import base_conditional, expand_independent_outputs
@conditional.register(object, InducingVariables, Kernel, object)
def _conditional(
Xnew: tf.Tensor,
inducing_variable: InducingVariables,
kernel: Kernel,
f: tf.Tensor,
*,
full_cov=False,
full_output_cov=False,
q_sqrt=None,
white=False,
):
"""
Single-output GP conditional.
The covariance matrices used to calculate the conditional have the following shape:
- Kuu: [M, M]
- Kuf: [M, N]
- Kff: [N, N]
Further reference
-----------------
- See `gpflow.conditionals._conditional` (below) for a detailed explanation of
conditional in the single-output case.
- See the multiouput notebook for more information about the multiouput framework.
Parameters
----------
:param Xnew: data matrix, size [N, D].
:param f: data matrix, [M, R]
:param full_cov: return the covariance between the datapoints
:param full_output_cov: return the covariance between the outputs.
NOTE: as we are using a single-output kernel with repetitions
these covariances will be zero.
:param q_sqrt: matrix of standard-deviations or Cholesky matrices,
size [M, R] or [R, M, M].
:param white: boolean of whether to use the whitened representation
:return:
- mean: [N, R]
- variance: [N, R], [R, N, N], [N, R, R] or [N, R, N, R]
Please see `gpflow.conditional._expand_independent_outputs` for more information
about the shape of the variance, depending on `full_cov` and `full_output_cov`.
"""
Kmm = Kuu(inducing_variable, kernel, jitter=default_jitter()) # [M, M]
Kmn = Kuf(inducing_variable, kernel, Xnew) # [M, N]
Knn = kernel(Xnew, full_cov=full_cov)
fmean, fvar = base_conditional(
Kmn, Kmm, Knn, f, full_cov=full_cov, q_sqrt=q_sqrt, white=white
) # [N, R], [R, N, N] or [N, R]
return fmean, expand_independent_outputs(fvar, full_cov, full_output_cov)
@conditional.register(object, object, Kernel, object)
def _conditional(
Xnew: tf.Tensor,
X: tf.Tensor,
kernel: Kernel,
f: tf.Tensor,
*,
full_cov=False,
full_output_cov=False,
q_sqrt=None,
white=False,
):
"""
Given f, representing the GP at the points X, produce the mean and
(co-)variance of the GP at the points Xnew.
Additionally, there may be Gaussian uncertainty about f as represented by
q_sqrt. In this case `f` represents the mean of the distribution and
q_sqrt the square-root of the covariance.
Additionally, the GP may have been centered (whitened) so that
p(v) = ๐ฉ(๐, ๐)
f = ๐v
thus
p(f) = ๐ฉ(๐, ๐๐แต) = ๐ฉ(๐, ๐).
In this case `f` represents the values taken by v.
The method can either return the diagonals of the covariance matrix for
each output (default) or the full covariance matrix (full_cov=True).
We assume R independent GPs, represented by the columns of f (and the
first dimension of q_sqrt).
:param Xnew: data matrix, size [N, D]. Evaluate the GP at these new points
:param X: data points, size [M, D].
:param kernel: GPflow kernel.
:param f: data matrix, [M, R], representing the function values at X,
for R functions.
:param q_sqrt: matrix of standard-deviations or Cholesky matrices,
size [M, R] or [R, M, M].
:param white: boolean of whether to use the whitened representation as
described above.
:return:
- mean: [N, R]
- variance: [N, R] (full_cov = False), [R, N, N] (full_cov = True)
"""
Kmm = kernel(X) + eye(tf.shape(X)[-2], value=default_jitter(), dtype=X.dtype) # [..., M, M]
Kmn = kernel(X, Xnew) # [M, ..., N]
Knn = kernel(Xnew, full_cov=full_cov) # [..., N] (full_cov = False) or [..., N, N] (True)
mean, var = base_conditional(Kmn, Kmm, Knn, f, full_cov=full_cov, q_sqrt=q_sqrt, white=white)
return mean, var # [N, R], [N, R] or [R, N, N]