# Copyright (c) 2012-2014, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
import numpy as np
from .model import Model
from .parameterization.variational import VariationalPosterior
from .mapping import Mapping
from .. import likelihoods
from .. import kern
from ..inference.latent_function_inference import exact_gaussian_inference, expectation_propagation
from ..util.normalizer import Standardize
from paramz import ObsAr
import logging
import warnings
logger = logging.getLogger("GP")
[docs]class GP(Model):
"""
General purpose Gaussian process model
:param X: input observations
:param Y: output observations
:param kernel: a GPy kernel
:param likelihood: a GPy likelihood
:param inference_method: The :class:`~GPy.inference.latent_function_inference.LatentFunctionInference` inference method to use for this GP
:rtype: model object
:param Norm normalizer:
normalize the outputs Y.
Prediction will be un-normalized using this normalizer.
If normalizer is True, we will normalize using Standardize.
If normalizer is False, no normalization will be done.
.. Note:: Multiple independent outputs are allowed using columns of Y
"""
def __init__(self, X, Y, kernel, likelihood, mean_function=None, inference_method=None, name='gp', Y_metadata=None, normalizer=False):
super(GP, self).__init__(name)
assert X.ndim == 2
if isinstance(X, (ObsAr, VariationalPosterior)):
self.X = X.copy()
else: self.X = ObsAr(X)
assert Y.ndim == 2
logger.info("initializing Y")
if normalizer is True:
self.normalizer = Standardize()
elif normalizer is False:
self.normalizer = None
else:
self.normalizer = normalizer
if self.normalizer is not None:
self.normalizer.scale_by(Y)
self.Y_normalized = ObsAr(self.normalizer.normalize(Y))
self.Y = Y
elif isinstance(Y, np.ndarray):
self.Y = ObsAr(Y)
self.Y_normalized = self.Y
else:
self.Y = Y
self.Y_normalized = self.Y
if Y.shape[0] != self.num_data:
#There can be cases where we want inputs than outputs, for example if we have multiple latent
#function values
warnings.warn("There are more rows in your input data X, \
than in your output data Y, be VERY sure this is what you want")
_, self.output_dim = self.Y.shape
assert ((Y_metadata is None) or isinstance(Y_metadata, dict))
self.Y_metadata = Y_metadata
assert isinstance(kernel, kern.Kern)
#assert self.input_dim == kernel.input_dim
self.kern = kernel
assert isinstance(likelihood, likelihoods.Likelihood)
self.likelihood = likelihood
if self.kern._effective_input_dim != self.X.shape[1]:
warnings.warn("Your kernel has a different input dimension {} then the given X dimension {}. Be very sure this is what you want and you have not forgotten to set the right input dimenion in your kernel".format(self.kern._effective_input_dim, self.X.shape[1]))
#handle the mean function
self.mean_function = mean_function
if mean_function is not None:
assert isinstance(self.mean_function, Mapping)
assert mean_function.input_dim == self.input_dim
assert mean_function.output_dim == self.output_dim
self.link_parameter(mean_function)
#find a sensible inference method
logger.info("initializing inference method")
if inference_method is None:
if isinstance(likelihood, likelihoods.Gaussian) or isinstance(likelihood, likelihoods.MixedNoise):
inference_method = exact_gaussian_inference.ExactGaussianInference()
else:
inference_method = expectation_propagation.EP()
print("defaulting to " + str(inference_method) + " for latent function inference")
self.inference_method = inference_method
logger.info("adding kernel and likelihood as parameters")
self.link_parameter(self.kern)
self.link_parameter(self.likelihood)
self.posterior = None
[docs] def to_dict(self, save_data=True):
"""
Convert the object into a json serializable dictionary.
Note: It uses the private method _save_to_input_dict of the parent.
:param boolean save_data: if true, it adds the training data self.X and self.Y to the dictionary
:return dict: json serializable dictionary containing the needed information to instantiate the object
"""
input_dict = super(GP, self)._save_to_input_dict()
input_dict["class"] = "GPy.core.GP"
if not save_data:
input_dict["X"] = None
input_dict["Y"] = None
else:
try:
input_dict["X"] = self.X.values.tolist()
except:
input_dict["X"] = self.X.tolist()
try:
input_dict["Y"] = self.Y.values.tolist()
except:
input_dict["Y"] = self.Y.tolist()
input_dict["kernel"] = self.kern.to_dict()
input_dict["likelihood"] = self.likelihood.to_dict()
if self.mean_function is not None:
input_dict["mean_function"] = self.mean_function.to_dict()
input_dict["inference_method"] = self.inference_method.to_dict()
# TODO: We should create a Metadata class
if self.Y_metadata is not None:
# make Y_metadata serializable
input_dict["Y_metadata"] = {k: self.Y_metadata[k].tolist() for k in self.Y_metadata.keys()}
if self.normalizer is not None:
input_dict["normalizer"] = self.normalizer.to_dict()
return input_dict
@staticmethod
def _format_input_dict(input_dict, data=None):
import GPy
import numpy as np
if (input_dict['X'] is None) or (input_dict['Y'] is None):
assert(data is not None)
input_dict["X"], input_dict["Y"] = np.array(data[0]), np.array(data[1])
elif data is not None:
warnings.warn("WARNING: The model has been saved with X,Y! The original values are being overridden!")
input_dict["X"], input_dict["Y"] = np.array(data[0]), np.array(data[1])
else:
input_dict["X"], input_dict["Y"] = np.array(input_dict['X']), np.array(input_dict['Y'])
input_dict["kernel"] = GPy.kern.Kern.from_dict(input_dict["kernel"])
input_dict["likelihood"] = GPy.likelihoods.likelihood.Likelihood.from_dict(input_dict["likelihood"])
mean_function = input_dict.get("mean_function")
if mean_function is not None:
input_dict["mean_function"] = GPy.core.mapping.Mapping.from_dict(mean_function)
else:
input_dict["mean_function"] = mean_function
input_dict["inference_method"] = GPy.inference.latent_function_inference.LatentFunctionInference.from_dict(input_dict["inference_method"])
# converts Y_metadata from serializable to array. We should create a Metadata class
Y_metadata = input_dict.get("Y_metadata")
if isinstance(Y_metadata, dict):
input_dict["Y_metadata"] = {k: np.array(Y_metadata[k]) for k in Y_metadata.keys()}
else:
input_dict["Y_metadata"] = Y_metadata
normalizer = input_dict.get("normalizer")
if normalizer is not None:
input_dict["normalizer"] = GPy.util.normalizer._Norm.from_dict(normalizer)
else:
input_dict["normalizer"] = normalizer
return input_dict
@staticmethod
def _build_from_input_dict(input_dict, data=None):
input_dict = GP._format_input_dict(input_dict, data)
return GP(**input_dict)
[docs] def save_model(self, output_filename, compress=True, save_data=True):
self._save_model(output_filename, compress=True, save_data=True)
# The predictive variable to be used to predict using the posterior object's
# woodbury_vector and woodbury_inv is defined as predictive_variable
# as long as the posterior has the right woodbury entries.
# It is the input variable used for the covariance between
# X_star and the posterior of the GP.
# This is usually just a link to self.X (full GP) or self.Z (sparse GP).
# Make sure to name this variable and the predict functions will "just work"
# In maths the predictive variable is:
# K_{xx} - K_{xp}W_{pp}^{-1}K_{px}
# W_{pp} := \texttt{Woodbury inv}
# p := _predictive_variable
@property
def _predictive_variable(self):
return self.X
@property
def num_data(self):
return self.X.shape[0]
@property
def input_dim(self):
return self.X.shape[1]
[docs] def set_XY(self, X=None, Y=None):
"""
Set the input / output data of the model
This is useful if we wish to change our existing data but maintain the same model
:param X: input observations
:type X: np.ndarray
:param Y: output observations
:type Y: np.ndarray
"""
self.update_model(False)
if Y is not None:
if self.normalizer is not None:
self.normalizer.scale_by(Y)
self.Y_normalized = ObsAr(self.normalizer.normalize(Y))
self.Y = Y
else:
self.Y = ObsAr(Y)
self.Y_normalized = self.Y
if X is not None:
if self.X in self.parameters:
# LVM models
if isinstance(self.X, VariationalPosterior):
assert isinstance(X, type(self.X)), "The given X must have the same type as the X in the model!"
index = self.X._parent_index_
self.unlink_parameter(self.X)
self.X = X
self.link_parameter(self.X, index=index)
else:
index = self.X._parent_index_
self.unlink_parameter(self.X)
from ..core import Param
self.X = Param('latent mean', X)
self.link_parameter(self.X, index=index)
else:
self.X = ObsAr(X)
self.update_model(True)
[docs] def set_X(self,X):
"""
Set the input data of the model
:param X: input observations
:type X: np.ndarray
"""
self.set_XY(X=X)
[docs] def set_Y(self,Y):
"""
Set the output data of the model
:param X: output observations
:type X: np.ndarray
"""
self.set_XY(Y=Y)
[docs] def parameters_changed(self):
"""
Method that is called upon any changes to :class:`~GPy.core.parameterization.param.Param` variables within the model.
In particular in the GP class this method re-performs inference, recalculating the posterior and log marginal likelihood and gradients of the model
.. warning::
This method is not designed to be called manually, the framework is set up to automatically call this method upon changes to parameters, if you call
this method yourself, there may be unexpected consequences.
"""
self.posterior, self._log_marginal_likelihood, self.grad_dict = self.inference_method.inference(self.kern, self.X, self.likelihood, self.Y_normalized, self.mean_function, self.Y_metadata)
self.likelihood.update_gradients(self.grad_dict['dL_dthetaL'])
self.kern.update_gradients_full(self.grad_dict['dL_dK'], self.X)
if self.mean_function is not None:
self.mean_function.update_gradients(self.grad_dict['dL_dm'], self.X)
[docs] def log_likelihood(self):
"""
The log marginal likelihood of the model, :math:`p(\mathbf{y})`, this is the objective function of the model being optimised
"""
return self._log_marginal_likelihood
def _raw_predict(self, Xnew, full_cov=False, kern=None):
"""
For making predictions, does not account for normalization or likelihood
full_cov is a boolean which defines whether the full covariance matrix
of the prediction is computed. If full_cov is False (default), only the
diagonal of the covariance is returned.
.. math::
p(f*|X*, X, Y) = \int^{\inf}_{\inf} p(f*|f,X*)p(f|X,Y) df
= N(f*| K_{x*x}(K_{xx} + \Sigma)^{-1}Y, K_{x*x*} - K_{xx*}(K_{xx} + \Sigma)^{-1}K_{xx*}
\Sigma := \texttt{Likelihood.variance / Approximate likelihood covariance}
"""
mu, var = self.posterior._raw_predict(kern=self.kern if kern is None else kern, Xnew=Xnew, pred_var=self._predictive_variable, full_cov=full_cov)
if self.mean_function is not None:
mu += self.mean_function.f(Xnew)
return mu, var
[docs] def predict(self, Xnew, full_cov=False, Y_metadata=None, kern=None,
likelihood=None, include_likelihood=True):
"""
Predict the function(s) at the new point(s) Xnew. This includes the
likelihood variance added to the predicted underlying function
(usually referred to as f).
In order to predict without adding in the likelihood give
`include_likelihood=False`, or refer to self.predict_noiseless().
:param Xnew: The points at which to make a prediction
:type Xnew: np.ndarray (Nnew x self.input_dim)
:param full_cov: whether to return the full covariance matrix, or just
the diagonal
:type full_cov: bool
:param Y_metadata: metadata about the predicting point to pass to the
likelihood
:param kern: The kernel to use for prediction (defaults to the model
kern). this is useful for examining e.g. subprocesses.
:param include_likelihood: Whether or not to add likelihood noise to
the predicted underlying latent function f.
:type include_likelihood: bool
:returns: (mean, var):
mean: posterior mean, a Numpy array, Nnew x self.input_dim
var: posterior variance, a Numpy array, Nnew x 1 if full_cov=False,
Nnew x Nnew otherwise
If full_cov and self.input_dim > 1, the return shape of var is
Nnew x Nnew x self.input_dim. If self.input_dim == 1, the return
shape is Nnew x Nnew. This is to allow for different normalizations
of the output dimensions.
Note: If you want the predictive quantiles (e.g. 95% confidence
interval) use :py:func:`~GPy.core.gp.GP.predict_quantiles`.
"""
# Predict the latent function values
mean, var = self._raw_predict(Xnew, full_cov=full_cov, kern=kern)
if include_likelihood:
# now push through likelihood
if likelihood is None:
likelihood = self.likelihood
mean, var = likelihood.predictive_values(mean, var, full_cov,
Y_metadata=Y_metadata)
if self.normalizer is not None:
mean = self.normalizer.inverse_mean(mean)
# We need to create 3d array for the full covariance matrix with
# multiple outputs.
if full_cov & (mean.shape[1] > 1):
var = self.normalizer.inverse_covariance(var)
else:
var = self.normalizer.inverse_variance(var)
return mean, var
[docs] def predict_noiseless(self, Xnew, full_cov=False, Y_metadata=None, kern=None):
"""
Convenience function to predict the underlying function of the GP (often
referred to as f) without adding the likelihood variance on the
prediction function.
This is most likely what you want to use for your predictions.
:param Xnew: The points at which to make a prediction
:type Xnew: np.ndarray (Nnew x self.input_dim)
:param full_cov: whether to return the full covariance matrix, or just
the diagonal
:type full_cov: bool
:param Y_metadata: metadata about the predicting point to pass to the likelihood
:param kern: The kernel to use for prediction (defaults to the model
kern). this is useful for examining e.g. subprocesses.
:returns: (mean, var):
mean: posterior mean, a Numpy array, Nnew x self.input_dim
var: posterior variance, a Numpy array, Nnew x 1 if full_cov=False, Nnew x Nnew otherwise
If full_cov and self.input_dim > 1, the return shape of var is Nnew x Nnew x self.input_dim. If self.input_dim == 1, the return shape is Nnew x Nnew.
This is to allow for different normalizations of the output dimensions.
Note: If you want the predictive quantiles (e.g. 95% confidence interval) use :py:func:`~GPy.core.gp.GP.predict_quantiles`.
"""
return self.predict(Xnew, full_cov, Y_metadata, kern, None, False)
[docs] def predict_quantiles(self, X, quantiles=(2.5, 97.5), Y_metadata=None, kern=None, likelihood=None):
"""
Get the predictive quantiles around the prediction at X
:param X: The points at which to make a prediction
:type X: np.ndarray (Xnew x self.input_dim)
:param quantiles: tuple of quantiles, default is (2.5, 97.5) which is the 95% interval
:type quantiles: tuple
:param kern: optional kernel to use for prediction
:type predict_kw: dict
:returns: list of quantiles for each X and predictive quantiles for interval combination
:rtype: [np.ndarray (Xnew x self.output_dim), np.ndarray (Xnew x self.output_dim)]
"""
m, v = self._raw_predict(X, full_cov=False, kern=kern)
if likelihood is None:
likelihood = self.likelihood
quantiles = likelihood.predictive_quantiles(m, v, quantiles, Y_metadata=Y_metadata)
if self.normalizer is not None:
quantiles = [self.normalizer.inverse_mean(q) for q in quantiles]
return quantiles
[docs] def predictive_gradients(self, Xnew, kern=None):
"""
Compute the derivatives of the predicted latent function with respect
to X*
Given a set of points at which to predict X* (size [N*,Q]), compute the
derivatives of the mean and variance. Resulting arrays are sized:
dmu_dX* -- [N*, Q ,D], where D is the number of output in this GP
(usually one).
Note that this is not the same as computing the mean and variance of
the derivative of the function!
dv_dX* -- [N*, Q], (since all outputs have the same variance)
:param X: The points at which to get the predictive gradients
:type X: np.ndarray (Xnew x self.input_dim)
:returns: dmu_dX, dv_dX
:rtype: [np.ndarray (N*, Q ,D), np.ndarray (N*,Q) ]
"""
if kern is None:
kern = self.kern
mean_jac = np.empty((Xnew.shape[0], Xnew.shape[1], self.output_dim))
for i in range(self.output_dim):
mean_jac[:, :, i] = kern.gradients_X(
self.posterior.woodbury_vector[:, i:i+1].T, Xnew,
self._predictive_variable)
# Gradients wrt the diagonal part k_{xx}
dv_dX = kern.gradients_X_diag(np.ones(Xnew.shape[0]), Xnew)
# Grads wrt 'Schur' part K_{xf}K_{ff}^{-1}K_{fx}
if self.posterior.woodbury_inv.ndim == 3:
var_jac = np.empty(dv_dX.shape +
(self.posterior.woodbury_inv.shape[2],))
var_jac[:] = dv_dX[:, :, None]
for i in range(self.posterior.woodbury_inv.shape[2]):
alpha = -2.*np.dot(kern.K(Xnew, self._predictive_variable),
self.posterior.woodbury_inv[:, :, i])
var_jac[:, :, i] += kern.gradients_X(alpha, Xnew,
self._predictive_variable)
else:
var_jac = dv_dX
alpha = -2.*np.dot(kern.K(Xnew, self._predictive_variable),
self.posterior.woodbury_inv)
var_jac += kern.gradients_X(alpha, Xnew, self._predictive_variable)
if self.normalizer is not None:
mean_jac = self.normalizer.inverse_mean(mean_jac) \
- self.normalizer.inverse_mean(0.)
if self.output_dim > 1:
var_jac = self.normalizer.inverse_covariance(var_jac)
else:
var_jac = self.normalizer.inverse_variance(var_jac)
return mean_jac, var_jac
[docs] def predict_jacobian(self, Xnew, kern=None, full_cov=False):
"""
Compute the derivatives of the posterior of the GP.
Given a set of points at which to predict X* (size [N*,Q]), compute the
mean and variance of the derivative. Resulting arrays are sized:
dL_dX* -- [N*, Q ,D], where D is the number of output in this GP (usually one).
Note that this is the mean and variance of the derivative,
not the derivative of the mean and variance! (See predictive_gradients for that)
dv_dX* -- [N*, Q], (since all outputs have the same variance)
If there is missing data, it is not implemented for now, but
there will be one output variance per output dimension.
:param X: The points at which to get the predictive gradients.
:type X: np.ndarray (Xnew x self.input_dim)
:param kern: The kernel to compute the jacobian for.
:param boolean full_cov: whether to return the cross-covariance terms between
the N* Jacobian vectors
:returns: dmu_dX, dv_dX
:rtype: [np.ndarray (N*, Q ,D), np.ndarray (N*,Q,(D)) ]
"""
if kern is None:
kern = self.kern
mean_jac = np.empty((Xnew.shape[0],Xnew.shape[1],self.output_dim))
for i in range(self.output_dim):
mean_jac[:,:,i] = kern.gradients_X(self.posterior.woodbury_vector[:,i:i+1].T, Xnew, self._predictive_variable)
dK_dXnew_full = np.empty((self._predictive_variable.shape[0], Xnew.shape[0], Xnew.shape[1]))
one = np.ones((1,1))
for i in range(self._predictive_variable.shape[0]):
dK_dXnew_full[i] = kern.gradients_X(one, Xnew, self._predictive_variable[[i]])
if full_cov:
dK2_dXdX = kern.gradients_XX(one, Xnew)
else:
dK2_dXdX = kern.gradients_XX_diag(one, Xnew)
#dK2_dXdX = np.zeros((Xnew.shape[0], Xnew.shape[1], Xnew.shape[1]))
#for i in range(Xnew.shape[0]):
# dK2_dXdX[i:i+1,:,:] = kern.gradients_XX(one, Xnew[i:i+1,:])
def compute_cov_inner(wi):
if full_cov:
var_jac = dK2_dXdX - np.einsum('qnm,msr->nsqr', dK_dXnew_full.T.dot(wi), dK_dXnew_full) # n,s = Xnew.shape[0], m = pred_var.shape[0]
else:
var_jac = dK2_dXdX - np.einsum('qnm,mnr->nqr', dK_dXnew_full.T.dot(wi), dK_dXnew_full)
return var_jac
if self.posterior.woodbury_inv.ndim == 3: # Missing data:
if full_cov:
var_jac = np.empty((Xnew.shape[0],Xnew.shape[0],Xnew.shape[1],Xnew.shape[1],self.output_dim))
for d in range(self.posterior.woodbury_inv.shape[2]):
var_jac[:, :, :, :, d] = compute_cov_inner(self.posterior.woodbury_inv[:, :, d])
else:
var_jac = np.empty((Xnew.shape[0],Xnew.shape[1],Xnew.shape[1],self.output_dim))
for d in range(self.posterior.woodbury_inv.shape[2]):
var_jac[:, :, :, d] = compute_cov_inner(self.posterior.woodbury_inv[:, :, d])
else:
var_jac = compute_cov_inner(self.posterior.woodbury_inv)
return mean_jac, var_jac
[docs] def predict_wishart_embedding(self, Xnew, kern=None, mean=True, covariance=True):
"""
Predict the wishart embedding G of the GP. This is the density of the
input of the GP defined by the probabilistic function mapping f.
G = J_mean.T*J_mean + output_dim*J_cov.
:param array-like Xnew: The points at which to evaluate the magnification.
:param :py:class:`~GPy.kern.Kern` kern: The kernel to use for the magnification.
Supplying only a part of the learning kernel gives insights into the density
of the specific kernel part of the input function. E.g. one can see how dense the
linear part of a kernel is compared to the non-linear part etc.
"""
if kern is None:
kern = self.kern
mu_jac, var_jac = self.predict_jacobian(Xnew, kern, full_cov=False)
mumuT = np.einsum('iqd,ipd->iqp', mu_jac, mu_jac)
Sigma = np.zeros(mumuT.shape)
if var_jac.ndim == 4: # Missing data
Sigma = var_jac.sum(-1)
else:
Sigma = self.output_dim*var_jac
G = 0.
if mean:
G += mumuT
if covariance:
G += Sigma
return G
[docs] def predict_wishard_embedding(self, Xnew, kern=None, mean=True, covariance=True):
warnings.warn("Wrong naming, use predict_wishart_embedding instead. Will be removed in future versions!", DeprecationWarning)
return self.predict_wishart_embedding(Xnew, kern, mean, covariance)
[docs] def predict_magnification(self, Xnew, kern=None, mean=True, covariance=True, dimensions=None):
"""
Predict the magnification factor as
sqrt(det(G))
for each point N in Xnew.
:param bool mean: whether to include the mean of the wishart embedding.
:param bool covariance: whether to include the covariance of the wishart embedding.
:param array-like dimensions: which dimensions of the input space to use [defaults to self.get_most_significant_input_dimensions()[:2]]
"""
G = self.predict_wishart_embedding(Xnew, kern, mean, covariance)
if dimensions is None:
dimensions = self.get_most_significant_input_dimensions()[:2]
G = G[:, dimensions][:,:,dimensions]
from ..util.linalg import jitchol
mag = np.empty(Xnew.shape[0])
for n in range(Xnew.shape[0]):
try:
mag[n] = np.sqrt(np.exp(2*np.sum(np.log(np.diag(jitchol(G[n, :, :]))))))
except:
mag[n] = np.sqrt(np.linalg.det(G[n, :, :]))
return mag
[docs] def posterior_samples_f(self,X, size=10, **predict_kwargs):
"""
Samples the posterior GP at the points X.
:param X: The points at which to take the samples.
:type X: np.ndarray (Nnew x self.input_dim)
:param size: the number of a posteriori samples.
:type size: int.
:returns: set of simulations
:rtype: np.ndarray (Nnew x D x samples)
"""
predict_kwargs["full_cov"] = True # Always use the full covariance for posterior samples.
m, v = self._raw_predict(X, **predict_kwargs)
if self.normalizer is not None:
m, v = self.normalizer.inverse_mean(m), self.normalizer.inverse_variance(v)
def sim_one_dim(m, v):
return np.random.multivariate_normal(m, v, size).T
if self.output_dim == 1:
return sim_one_dim(m.flatten(), v)[:, np.newaxis, :]
else:
fsim = np.empty((X.shape[0], self.output_dim, size))
for d in range(self.output_dim):
if v.ndim == 3:
fsim[:, d, :] = sim_one_dim(m[:, d], v[:, :, d])
else:
fsim[:, d, :] = sim_one_dim(m[:, d], v)
return fsim
[docs] def posterior_samples(self, X, size=10, Y_metadata=None, likelihood=None, **predict_kwargs):
"""
Samples the posterior GP at the points X.
:param X: the points at which to take the samples.
:type X: np.ndarray (Nnew x self.input_dim.)
:param size: the number of a posteriori samples.
:type size: int.
:param noise_model: for mixed noise likelihood, the noise model to use in the samples.
:type noise_model: integer.
:returns: Ysim: set of simulations,
:rtype: np.ndarray (D x N x samples) (if D==1 we flatten out the first dimension)
"""
fsim = self.posterior_samples_f(X, size, **predict_kwargs)
if likelihood is None:
likelihood = self.likelihood
if fsim.ndim == 3:
for d in range(fsim.shape[1]):
fsim[:, d] = likelihood.samples(fsim[:, d], Y_metadata=Y_metadata)
else:
fsim = likelihood.samples(fsim, Y_metadata=Y_metadata)
return fsim
[docs] def optimize(self, optimizer=None, start=None, messages=False, max_iters=1000, ipython_notebook=True, clear_after_finish=False, **kwargs):
"""
Optimize the model using self.log_likelihood and self.log_likelihood_gradient, as well as self.priors.
kwargs are passed to the optimizer. They can be:
:param max_iters: maximum number of function evaluations
:type max_iters: int
:param messages: whether to display during optimisation
:type messages: bool
:param optimizer: which optimizer to use (defaults to self.preferred optimizer), a range of optimisers can be found in :module:`~GPy.inference.optimization`, they include 'scg', 'lbfgs', 'tnc'.
:type optimizer: string
:param bool ipython_notebook: whether to use ipython notebook widgets or not.
:param bool clear_after_finish: if in ipython notebook, we can clear the widgets after optimization.
"""
self.inference_method.on_optimization_start()
try:
ret = super(GP, self).optimize(optimizer, start, messages, max_iters, ipython_notebook, clear_after_finish, **kwargs)
except KeyboardInterrupt:
print("KeyboardInterrupt caught, calling on_optimization_end() to round things up")
self.inference_method.on_optimization_end()
raise
return ret
[docs] def infer_newX(self, Y_new, optimize=True):
"""
Infer X for the new observed data *Y_new*.
:param Y_new: the new observed data for inference
:type Y_new: numpy.ndarray
:param optimize: whether to optimize the location of new X (True by default)
:type optimize: boolean
:return: a tuple containing the posterior estimation of X and the model that optimize X
:rtype: (:class:`~GPy.core.parameterization.variational.VariationalPosterior` and numpy.ndarray, :class:`~GPy.core.model.Model`)
"""
from ..inference.latent_function_inference.inferenceX import infer_newX
return infer_newX(self, Y_new, optimize=optimize)
[docs] def log_predictive_density(self, x_test, y_test, Y_metadata=None):
"""
Calculation of the log predictive density
.. math:
p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*})
:param x_test: test locations (x_{*})
:type x_test: (Nx1) array
:param y_test: test observations (y_{*})
:type y_test: (Nx1) array
:param Y_metadata: metadata associated with the test points
"""
mu_star, var_star = self._raw_predict(x_test)
return self.likelihood.log_predictive_density(y_test, mu_star, var_star, Y_metadata=Y_metadata)
[docs] def log_predictive_density_sampling(self, x_test, y_test, Y_metadata=None, num_samples=1000):
"""
Calculation of the log predictive density by sampling
.. math:
p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*})
:param x_test: test locations (x_{*})
:type x_test: (Nx1) array
:param y_test: test observations (y_{*})
:type y_test: (Nx1) array
:param Y_metadata: metadata associated with the test points
:param num_samples: number of samples to use in monte carlo integration
:type num_samples: int
"""
mu_star, var_star = self._raw_predict(x_test)
return self.likelihood.log_predictive_density_sampling(y_test, mu_star, var_star, Y_metadata=Y_metadata, num_samples=num_samples)
def _raw_posterior_covariance_between_points(self, X1, X2):
"""
Computes the posterior covariance between points. Does not account for
normalization or likelihood
:param X1: some input observations
:param X2: other input observations
:returns:
cov: raw posterior covariance: k(X1,X2) - k(X1,X) G^{-1} K(X,X2)
"""
return self.posterior.covariance_between_points(self.kern, self.X, X1, X2)
[docs] def posterior_covariance_between_points(self, X1, X2, Y_metadata=None,
likelihood=None,
include_likelihood=True):
"""
Computes the posterior covariance between points. Includes likelihood
variance as well as normalization so that evaluation at (x,x) is consistent
with model.predict
:param X1: some input observations
:param X2: other input observations
:param Y_metadata: metadata about the predicting point to pass to the
likelihood
:param include_likelihood: Whether or not to add likelihood noise to
the predicted underlying latent function f.
:type include_likelihood: bool
:returns:
cov: posterior covariance, a Numpy array, Nnew x Nnew if
self.output_dim == 1, and Nnew x Nnew x self.output_dim otherwise.
"""
cov = self._raw_posterior_covariance_between_points(X1, X2)
if include_likelihood:
# Predict latent mean and push through likelihood
mean, _ = self._raw_predict(X1, full_cov=True)
if likelihood is None:
likelihood = self.likelihood
_, cov = likelihood.predictive_values(mean, cov, full_cov=True,
Y_metadata=Y_metadata)
if self.normalizer is not None:
if self.output_dim > 1:
cov = self.normalizer.inverse_covariance(cov)
else:
cov = self.normalizer.inverse_variance(cov)
return cov