# Copyright 2021-2023 Lawrence Livermore National Security, LLC and other
# MuyGPyS Project Developers. See the top-level COPYRIGHT file for details.
#
# SPDX-License-Identifier: MIT
"""
Multivariate MuyGPs implementation
"""
from copy import deepcopy
import MuyGPyS._src.math as mm
from MuyGPyS._src.gp.muygps import _mmuygps_fast_posterior_mean
from MuyGPyS.gp.muygps import MuyGPS
from MuyGPyS.gp.mean import PosteriorMean
from MuyGPyS.gp.noise import HeteroscedasticNoise
from MuyGPyS.gp.noise.perturbation import select_perturb_fn
from MuyGPyS.gp.sigma_sq import SigmaSq
from MuyGPyS.gp.variance import PosteriorVariance
[docs]class MultivariateMuyGPS:
"""
Multivariate Local Kriging Gaussian Process.
Performs approximate GP inference by locally approximating an observation's
response using its nearest neighbors with a separate kernel allocated for
each response dimension, implemented as individual
:class:`MuyGPyS.gp.muygps.MuyGPS` objects.
This class is similar in interface to :class:`MuyGPyS.gp.muygps.MuyGPS`, but
requires a list of hyperparameter dicts at initialization.
Example:
>>> from MuyGPyS.gp import MultivariateMuyGPS as MMuyGPS
>>> k_kwargs1 = {
... "eps": {"val": 1e-5},
... "nu": {"val": 0.67, "bounds": (0.1, 2.5)},
... "length_scale": {"val": 7.2},
... }
>>> k_kwargs2 = {
... "eps": {"val": 1e-5},
... "nu": {"val": 0.38, "bounds": (0.1, 2.5)},
... "length_scale": {"val": 7.2},
... }
>>> k_args = [k_kwargs1, k_kwargs2]
>>> mmuygps = MMuyGPS("matern", *k_args)
We can realize kernel tensors for each of the models contained within a
`MultivariateMuyGPS` object by iterating over its `models` member. Once we
have computed `pairwise_diffs` and `crosswise_diffs` tensors, it
is straightforward to perform each of these realizations.
Example:
>>> for model in MuyGPyS.models:
>>> K = model.kernel(pairwise_diffs)
>>> Kcross = model.kernel(crosswise_diffs)
>>> # do something with K and Kcross...
Args
model_args:
Dictionaries defining each internal
:class:`MuyGPyS.gp.muygps.MuyGPS` instance.
"""
def __init__(
self,
*model_args,
):
self.models = [MuyGPS(**args) for args in model_args]
self.sigma_sq = SigmaSq(len(self.models))
[docs] def fixed(self) -> bool:
"""
Checks whether all kernel and model parameters are fixed for each model,
excluding :math:`\\sigma^2`.
Returns:
Returns `True` if all parameters in all models are fixed, and
`False` otherwise.
"""
return bool(all([model.fixed() for model in self.models]))
[docs] def posterior_mean(
self,
pairwise_diffs: mm.ndarray,
crosswise_diffs: mm.ndarray,
batch_nn_targets: mm.ndarray,
) -> mm.ndarray:
"""
Performs simultaneous posterior mean inference on provided difference
tensors and the target matrix.
Computes parallelized local solves of systems of linear equations using
the kernel realizations, one for each internal model, of the last two
dimensions of `pairwise_diffs` along with `crosswise_diffs` and
`batch_nn_targets` to predict responses in terms of the posterior mean.
Assumes that difference tensors `pairwise_diffs` and `crosswise_diffs`
are already computed and given as arguments.
Returns the predicted response in the form of a posterior mean for each
element of the batch of observations by solving a system of linear
equations induced by each kernel functor, one per response dimension, in
a generalization of Equation (3.4) of [muyskens2021muygps]_. For each
batch element :math:`\\mathbf{x}_i` we compute
.. math::
\\widehat{Y}_{NN} (\\mathbf{x}_i \\mid X_{N_i})_{:,j} =
K^{(j)}_\\theta (\\mathbf{x}_i, X_{N_i})
(K^{(j)}_\\theta (X_{N_i}, X_{N_i}) + \\varepsilon_j I_k)^{-1}
Y(X_{N_i})_{:,j}.
Here :math:`X_{N_i}` is the set of nearest neighbors of
:math:`\\mathbf{x}_i` in the training data, :math:`K^{(j)}_\\theta` is
the kernel functor associated with the jth internal model, corresponding
to the jth response dimension, :math:`\\varepsilon_j I_k` is a diagonal
homoscedastic noise matrix whose diagonal is the value of the
`self.models[j].eps` hyperparameter, and :math:`Y(X_{N_i})_{:,j}` is the
`(batch_count,)` vector of the jth responses of the nearest neighbors
given by a slice of the `batch_nn_targets` argument.
Args:
pairwise_diffs:
A tensor of shape
`(batch_count, nn_count, nn_count, feature_count)` containing
the `(nn_count, nn_count, feature_count)`-shaped pairwise
nearest neighbor difference tensors corresponding to each of the
batch elements.
crosswise_diffs:
A matrix of shape `(batch_count, nn_count, feature_count)` whose
rows list the difference between each feature of each batch
element element and its nearest neighbors.
batch_nn_targets:
A tensor of shape `(batch_count, nn_count, response_count)`
listing the vector-valued responses for the nearest neighbors
of each batch element.
Returns:
A matrix of shape `(batch_count, response_count)` whose rows are
the predicted response for each of the given indices.
"""
batch_count, nn_count, response_count = batch_nn_targets.shape
responses = mm.zeros((batch_count, response_count))
for i, model in enumerate(self.models):
K = model.kernel(pairwise_diffs)
Kcross = model.kernel(crosswise_diffs)
responses = mm.assign(
responses,
model.posterior_mean(
K,
Kcross,
batch_nn_targets[:, :, i].reshape(batch_count, nn_count, 1),
).reshape(batch_count),
slice(None),
i,
)
return responses
[docs] def posterior_variance(
self,
pairwise_diffs: mm.ndarray,
crosswise_diffs: mm.ndarray,
) -> mm.ndarray:
"""
Performs simultaneous posterior variance inference on provided
difference tensors.
Return the local posterior variances of each prediction, corresponding
to the diagonal elements of a covariance matrix. For each batch element
:math:`\\mathbf{x}_i`, we compute
.. math::
Var(\\widehat{Y}_{NN} (\\mathbf{x}_i \\mid X_{N_i}))_j =
K^{(j)}_\\theta (\\mathbf{x}_i, \\mathbf{x}_i) -
K^{(j)}_\\theta (\\mathbf{x}_i, X_{N_i})
(K^{(j)}_\\theta (X_{N_i}, X_{N_i}) + \\varepsilon I_k)^{-1}
K^{(j)}_\\theta (X_{N_i}, \\mathbf{x}_i).
Args:
pairwise_diffs:
A tensor of shape
`(batch_count, nn_count, nn_count, feature_count)` containing
the `(nn_count, nn_count, feature_count)`-shaped pairwise
nearest neighbor difference tensors corresponding to each of the
batch elements.
crosswise_diffs:
A matrix of shape `(batch_count, nn_count, feature_count)` whose
rows list the difference between each feature of each batch
element element and its nearest neighbors.
Returns:
A vector of shape `(batch_count, response_count)` consisting of the
diagonal elements of the posterior variance for each model.
"""
batch_count, _, _ = crosswise_diffs.shape
response_count = len(self.models)
diagonal_variance = mm.zeros((batch_count, response_count))
for i, model in enumerate(self.models):
K = model.kernel(pairwise_diffs)
Kcross = model.kernel(crosswise_diffs)
ss = self.sigma_sq()[i]
diagonal_variance = mm.assign(
diagonal_variance,
model.posterior_variance(K, Kcross).reshape(batch_count) * ss,
slice(None),
i,
)
return diagonal_variance
[docs] def fast_coefficients(
self,
pairwise_diffs_fast: mm.ndarray,
train_nn_targets_fast: mm.ndarray,
) -> mm.ndarray:
"""
Produces coefficient tensor for fast posterior mean inference given in
Equation (8) of [dunton2022fast]_.
To form the tensor, we compute
.. math::
\\mathbf{C}_{N^*}(i, :, j) =
(K_{\\hat{\\theta_j}} (X_{N^*}, X_{N^*}) +
\\varepsilon I_k)^{-1} Y(X_{N^*}).
Here :math:`X_{N^*}` is the union of the nearest neighbor of the ith
test point and the `nn_count - 1` nearest neighbors of this nearest
neighbor, :math:`K_{\\hat{\\theta_j}}` is the trained kernel functor
corresponding the jth response and specified by `self.models`,
:math:`\\varepsilon I_k` is a diagonal homoscedastic noise matrix whose
diagonal is the value of the `self.eps` hyperparameter,
and :math:`Y(X_{N^*})` is the `(train_count, response_count)`
matrix of responses corresponding to the training features indexed
by $N^*$.
Args:
pairwise_diffs:
A tensor of shape
`(train_count, nn_count, nn_count, feature_count)` containing
the `(nn_count, nn_count, feature_count)`-shaped pairwise
nearest neighbor difference tensors corresponding to each of the
batch elements.
batch_nn_targets:
A tensor of shape `(train_count, nn_count, response_count)`
listing the vector-valued responses for the nearest neighbors
of each batch element.
Returns:
A tensor of shape `(batch_count, nn_count, response_count)`
whose entries comprise the precomputed coefficients for fast
posterior mean inference.
"""
train_count, nn_count, response_count = train_nn_targets_fast.shape
coeffs_tensor = mm.zeros((train_count, nn_count, response_count))
for i, model in enumerate(self.models):
K = model.kernel(pairwise_diffs_fast)
perturb_fn = select_perturb_fn(model.eps)
mm.assign(
coeffs_tensor,
model.fast_coefficients(
perturb_fn(K, model.eps()),
train_nn_targets_fast[:, :, i],
),
slice(None),
slice(None),
i,
)
return coeffs_tensor
[docs] def fast_posterior_mean(
self,
crosswise_diffs: mm.ndarray,
coeffs_tensor: mm.ndarray,
) -> mm.ndarray:
"""
Performs fast posterior mean inference using provided crosswise
differences and precomputed coefficient matrix.
Returns the predicted response in the form of a posterior
mean for each element of the batch of observations, as computed in
Equation (9) of [dunton2022fast]_. For each test point
:math:`\\mathbf{z}`, we compute
.. math::
\\widehat{Y} (\\mathbf{z} \\mid X) =
K_\\theta (\\mathbf{z}, X_{N^*}) \\mathbf{C}_{N^*}.
Here :math:`X_{N^*}` is the union of the nearest neighbor of the queried
test point :math:`\\mathbf{z}` and the nearest neighbors of that
training point, :math:`K_\\theta` is the kernel functor specified by
`self.kernel`, and :math:`\\mathbf{C}_{N^*}` is the matrix of
precomputed coefficients given in Equation (8) of [dunton2022fast]_.
Args:
crosswise_diffs:
A matrix of shape `(batch_count, nn_count, feature_count)` whose
rows list the difference between each feature of each batch
element element and its nearest neighbors.
coeffs_tensor:
A tensor of shape `(batch_count, nn_count, response_count)`
providing the precomputed coefficients.
Returns:
A matrix of shape `(batch_count, response_count)` whose rows are
the predicted response for each of the given indices.
"""
Kcross = mm.zeros(coeffs_tensor.shape)
for i, model in enumerate(self.models):
mm.assign(
Kcross,
model.kernel(crosswise_diffs),
slice(None),
slice(None),
i,
)
return _mmuygps_fast_posterior_mean(Kcross, coeffs_tensor)
[docs] def apply_new_noise(self, new_noise):
"""
Updates the heteroscedastic noise parameters of a MultivariateMuyGPs
model.
Args:
new_noise:
A matrix of shape
`(test_count, nn_count, nn_count, response_count)` containing
the measurement noise corresponding to the nearest neighbors
of each test point and each response.
Returns:
A MultivariateMuyGPs model with updated heteroscedastic noise
parameters.
"""
ret = deepcopy(self)
for i, model in enumerate(ret.models):
model.eps = HeteroscedasticNoise(new_noise[:, :, :, i], "fixed")
model._mean_fn = PosteriorMean(model.eps)
model._var_fn = PosteriorVariance(model.eps, model.sigma_sq)
return ret