670 lines
27 KiB
Python
670 lines
27 KiB
Python
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"""Gaussian processes regression."""
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# Authors: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
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# Modified by: Pete Green <p.l.green@liverpool.ac.uk>
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# License: BSD 3 clause
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import warnings
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from numbers import Integral, Real
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from operator import itemgetter
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import numpy as np
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import scipy.optimize
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from scipy.linalg import cho_solve, cholesky, solve_triangular
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from ..base import BaseEstimator, MultiOutputMixin, RegressorMixin, _fit_context, clone
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from ..preprocessing._data import _handle_zeros_in_scale
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from ..utils import check_random_state
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from ..utils._param_validation import Interval, StrOptions
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from ..utils.optimize import _check_optimize_result
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from .kernels import RBF, Kernel
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from .kernels import ConstantKernel as C
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GPR_CHOLESKY_LOWER = True
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class GaussianProcessRegressor(MultiOutputMixin, RegressorMixin, BaseEstimator):
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"""Gaussian process regression (GPR).
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The implementation is based on Algorithm 2.1 of [RW2006]_.
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In addition to standard scikit-learn estimator API,
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:class:`GaussianProcessRegressor`:
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* allows prediction without prior fitting (based on the GP prior)
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* provides an additional method `sample_y(X)`, which evaluates samples
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drawn from the GPR (prior or posterior) at given inputs
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* exposes a method `log_marginal_likelihood(theta)`, which can be used
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externally for other ways of selecting hyperparameters, e.g., via
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Markov chain Monte Carlo.
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To learn the difference between a point-estimate approach vs. a more
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Bayesian modelling approach, refer to the example entitled
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:ref:`sphx_glr_auto_examples_gaussian_process_plot_compare_gpr_krr.py`.
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Read more in the :ref:`User Guide <gaussian_process>`.
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.. versionadded:: 0.18
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Parameters
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----------
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kernel : kernel instance, default=None
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The kernel specifying the covariance function of the GP. If None is
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passed, the kernel ``ConstantKernel(1.0, constant_value_bounds="fixed")
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* RBF(1.0, length_scale_bounds="fixed")`` is used as default. Note that
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the kernel hyperparameters are optimized during fitting unless the
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bounds are marked as "fixed".
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alpha : float or ndarray of shape (n_samples,), default=1e-10
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Value added to the diagonal of the kernel matrix during fitting.
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This can prevent a potential numerical issue during fitting, by
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ensuring that the calculated values form a positive definite matrix.
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It can also be interpreted as the variance of additional Gaussian
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measurement noise on the training observations. Note that this is
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different from using a `WhiteKernel`. If an array is passed, it must
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have the same number of entries as the data used for fitting and is
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used as datapoint-dependent noise level. Allowing to specify the
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noise level directly as a parameter is mainly for convenience and
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for consistency with :class:`~sklearn.linear_model.Ridge`.
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optimizer : "fmin_l_bfgs_b", callable or None, default="fmin_l_bfgs_b"
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Can either be one of the internally supported optimizers for optimizing
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the kernel's parameters, specified by a string, or an externally
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defined optimizer passed as a callable. If a callable is passed, it
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must have the signature::
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def optimizer(obj_func, initial_theta, bounds):
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# * 'obj_func': the objective function to be minimized, which
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# takes the hyperparameters theta as a parameter and an
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# optional flag eval_gradient, which determines if the
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# gradient is returned additionally to the function value
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# * 'initial_theta': the initial value for theta, which can be
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# used by local optimizers
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# * 'bounds': the bounds on the values of theta
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....
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# Returned are the best found hyperparameters theta and
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# the corresponding value of the target function.
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return theta_opt, func_min
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Per default, the L-BFGS-B algorithm from `scipy.optimize.minimize`
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is used. If None is passed, the kernel's parameters are kept fixed.
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Available internal optimizers are: `{'fmin_l_bfgs_b'}`.
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n_restarts_optimizer : int, default=0
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The number of restarts of the optimizer for finding the kernel's
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parameters which maximize the log-marginal likelihood. The first run
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of the optimizer is performed from the kernel's initial parameters,
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the remaining ones (if any) from thetas sampled log-uniform randomly
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from the space of allowed theta-values. If greater than 0, all bounds
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must be finite. Note that `n_restarts_optimizer == 0` implies that one
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run is performed.
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normalize_y : bool, default=False
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Whether or not to normalize the target values `y` by removing the mean
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and scaling to unit-variance. This is recommended for cases where
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zero-mean, unit-variance priors are used. Note that, in this
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implementation, the normalisation is reversed before the GP predictions
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are reported.
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.. versionchanged:: 0.23
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copy_X_train : bool, default=True
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If True, a persistent copy of the training data is stored in the
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object. Otherwise, just a reference to the training data is stored,
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which might cause predictions to change if the data is modified
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externally.
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n_targets : int, default=None
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The number of dimensions of the target values. Used to decide the number
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of outputs when sampling from the prior distributions (i.e. calling
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:meth:`sample_y` before :meth:`fit`). This parameter is ignored once
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:meth:`fit` has been called.
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.. versionadded:: 1.3
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random_state : int, RandomState instance or None, default=None
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Determines random number generation used to initialize the centers.
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Pass an int for reproducible results across multiple function calls.
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See :term:`Glossary <random_state>`.
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Attributes
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----------
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X_train_ : array-like of shape (n_samples, n_features) or list of object
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Feature vectors or other representations of training data (also
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required for prediction).
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y_train_ : array-like of shape (n_samples,) or (n_samples, n_targets)
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Target values in training data (also required for prediction).
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kernel_ : kernel instance
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The kernel used for prediction. The structure of the kernel is the
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same as the one passed as parameter but with optimized hyperparameters.
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L_ : array-like of shape (n_samples, n_samples)
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Lower-triangular Cholesky decomposition of the kernel in ``X_train_``.
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alpha_ : array-like of shape (n_samples,)
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Dual coefficients of training data points in kernel space.
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log_marginal_likelihood_value_ : float
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The log-marginal-likelihood of ``self.kernel_.theta``.
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n_features_in_ : int
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Number of features seen during :term:`fit`.
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.. versionadded:: 0.24
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feature_names_in_ : ndarray of shape (`n_features_in_`,)
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Names of features seen during :term:`fit`. Defined only when `X`
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has feature names that are all strings.
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.. versionadded:: 1.0
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See Also
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--------
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GaussianProcessClassifier : Gaussian process classification (GPC)
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based on Laplace approximation.
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References
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----------
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.. [RW2006] `Carl E. Rasmussen and Christopher K.I. Williams,
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"Gaussian Processes for Machine Learning",
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MIT Press 2006 <https://www.gaussianprocess.org/gpml/chapters/RW.pdf>`_
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Examples
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--------
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>>> from sklearn.datasets import make_friedman2
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>>> from sklearn.gaussian_process import GaussianProcessRegressor
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>>> from sklearn.gaussian_process.kernels import DotProduct, WhiteKernel
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>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
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>>> kernel = DotProduct() + WhiteKernel()
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>>> gpr = GaussianProcessRegressor(kernel=kernel,
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... random_state=0).fit(X, y)
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>>> gpr.score(X, y)
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0.3680...
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>>> gpr.predict(X[:2,:], return_std=True)
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(array([653.0..., 592.1...]), array([316.6..., 316.6...]))
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"""
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_parameter_constraints: dict = {
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"kernel": [None, Kernel],
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"alpha": [Interval(Real, 0, None, closed="left"), np.ndarray],
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"optimizer": [StrOptions({"fmin_l_bfgs_b"}), callable, None],
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"n_restarts_optimizer": [Interval(Integral, 0, None, closed="left")],
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"normalize_y": ["boolean"],
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"copy_X_train": ["boolean"],
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"n_targets": [Interval(Integral, 1, None, closed="left"), None],
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"random_state": ["random_state"],
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}
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def __init__(
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self,
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kernel=None,
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*,
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alpha=1e-10,
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optimizer="fmin_l_bfgs_b",
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n_restarts_optimizer=0,
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normalize_y=False,
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copy_X_train=True,
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n_targets=None,
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random_state=None,
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):
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self.kernel = kernel
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self.alpha = alpha
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self.optimizer = optimizer
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self.n_restarts_optimizer = n_restarts_optimizer
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self.normalize_y = normalize_y
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self.copy_X_train = copy_X_train
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self.n_targets = n_targets
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self.random_state = random_state
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@_fit_context(prefer_skip_nested_validation=True)
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def fit(self, X, y):
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"""Fit Gaussian process regression model.
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Parameters
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----------
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X : array-like of shape (n_samples, n_features) or list of object
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Feature vectors or other representations of training data.
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y : array-like of shape (n_samples,) or (n_samples, n_targets)
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Target values.
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Returns
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-------
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self : object
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GaussianProcessRegressor class instance.
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"""
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if self.kernel is None: # Use an RBF kernel as default
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self.kernel_ = C(1.0, constant_value_bounds="fixed") * RBF(
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1.0, length_scale_bounds="fixed"
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)
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else:
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self.kernel_ = clone(self.kernel)
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self._rng = check_random_state(self.random_state)
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if self.kernel_.requires_vector_input:
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dtype, ensure_2d = "numeric", True
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else:
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dtype, ensure_2d = None, False
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X, y = self._validate_data(
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X,
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y,
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multi_output=True,
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y_numeric=True,
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ensure_2d=ensure_2d,
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dtype=dtype,
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)
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n_targets_seen = y.shape[1] if y.ndim > 1 else 1
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if self.n_targets is not None and n_targets_seen != self.n_targets:
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raise ValueError(
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"The number of targets seen in `y` is different from the parameter "
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f"`n_targets`. Got {n_targets_seen} != {self.n_targets}."
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)
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# Normalize target value
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if self.normalize_y:
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self._y_train_mean = np.mean(y, axis=0)
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self._y_train_std = _handle_zeros_in_scale(np.std(y, axis=0), copy=False)
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# Remove mean and make unit variance
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y = (y - self._y_train_mean) / self._y_train_std
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else:
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shape_y_stats = (y.shape[1],) if y.ndim == 2 else 1
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self._y_train_mean = np.zeros(shape=shape_y_stats)
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self._y_train_std = np.ones(shape=shape_y_stats)
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if np.iterable(self.alpha) and self.alpha.shape[0] != y.shape[0]:
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if self.alpha.shape[0] == 1:
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self.alpha = self.alpha[0]
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else:
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raise ValueError(
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"alpha must be a scalar or an array with same number of "
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f"entries as y. ({self.alpha.shape[0]} != {y.shape[0]})"
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)
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self.X_train_ = np.copy(X) if self.copy_X_train else X
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self.y_train_ = np.copy(y) if self.copy_X_train else y
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if self.optimizer is not None and self.kernel_.n_dims > 0:
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# Choose hyperparameters based on maximizing the log-marginal
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# likelihood (potentially starting from several initial values)
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def obj_func(theta, eval_gradient=True):
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if eval_gradient:
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lml, grad = self.log_marginal_likelihood(
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theta, eval_gradient=True, clone_kernel=False
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)
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return -lml, -grad
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else:
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return -self.log_marginal_likelihood(theta, clone_kernel=False)
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# First optimize starting from theta specified in kernel
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optima = [
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(
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self._constrained_optimization(
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obj_func, self.kernel_.theta, self.kernel_.bounds
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)
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)
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]
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# Additional runs are performed from log-uniform chosen initial
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# theta
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if self.n_restarts_optimizer > 0:
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if not np.isfinite(self.kernel_.bounds).all():
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raise ValueError(
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"Multiple optimizer restarts (n_restarts_optimizer>0) "
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"requires that all bounds are finite."
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)
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bounds = self.kernel_.bounds
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for iteration in range(self.n_restarts_optimizer):
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theta_initial = self._rng.uniform(bounds[:, 0], bounds[:, 1])
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optima.append(
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self._constrained_optimization(obj_func, theta_initial, bounds)
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)
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# Select result from run with minimal (negative) log-marginal
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# likelihood
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lml_values = list(map(itemgetter(1), optima))
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self.kernel_.theta = optima[np.argmin(lml_values)][0]
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self.kernel_._check_bounds_params()
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self.log_marginal_likelihood_value_ = -np.min(lml_values)
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else:
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self.log_marginal_likelihood_value_ = self.log_marginal_likelihood(
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self.kernel_.theta, clone_kernel=False
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)
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# Precompute quantities required for predictions which are independent
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# of actual query points
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# Alg. 2.1, page 19, line 2 -> L = cholesky(K + sigma^2 I)
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K = self.kernel_(self.X_train_)
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K[np.diag_indices_from(K)] += self.alpha
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try:
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self.L_ = cholesky(K, lower=GPR_CHOLESKY_LOWER, check_finite=False)
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except np.linalg.LinAlgError as exc:
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exc.args = (
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(
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f"The kernel, {self.kernel_}, is not returning a positive "
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"definite matrix. Try gradually increasing the 'alpha' "
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"parameter of your GaussianProcessRegressor estimator."
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),
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) + exc.args
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|
raise
|
||
|
|
# Alg 2.1, page 19, line 3 -> alpha = L^T \ (L \ y)
|
||
|
|
self.alpha_ = cho_solve(
|
||
|
|
(self.L_, GPR_CHOLESKY_LOWER),
|
||
|
|
self.y_train_,
|
||
|
|
check_finite=False,
|
||
|
|
)
|
||
|
|
return self
|
||
|
|
|
||
|
|
def predict(self, X, return_std=False, return_cov=False):
|
||
|
|
"""Predict using the Gaussian process regression model.
|
||
|
|
|
||
|
|
We can also predict based on an unfitted model by using the GP prior.
|
||
|
|
In addition to the mean of the predictive distribution, optionally also
|
||
|
|
returns its standard deviation (`return_std=True`) or covariance
|
||
|
|
(`return_cov=True`). Note that at most one of the two can be requested.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : array-like of shape (n_samples, n_features) or list of object
|
||
|
|
Query points where the GP is evaluated.
|
||
|
|
|
||
|
|
return_std : bool, default=False
|
||
|
|
If True, the standard-deviation of the predictive distribution at
|
||
|
|
the query points is returned along with the mean.
|
||
|
|
|
||
|
|
return_cov : bool, default=False
|
||
|
|
If True, the covariance of the joint predictive distribution at
|
||
|
|
the query points is returned along with the mean.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
y_mean : ndarray of shape (n_samples,) or (n_samples, n_targets)
|
||
|
|
Mean of predictive distribution at query points.
|
||
|
|
|
||
|
|
y_std : ndarray of shape (n_samples,) or (n_samples, n_targets), optional
|
||
|
|
Standard deviation of predictive distribution at query points.
|
||
|
|
Only returned when `return_std` is True.
|
||
|
|
|
||
|
|
y_cov : ndarray of shape (n_samples, n_samples) or \
|
||
|
|
(n_samples, n_samples, n_targets), optional
|
||
|
|
Covariance of joint predictive distribution at query points.
|
||
|
|
Only returned when `return_cov` is True.
|
||
|
|
"""
|
||
|
|
if return_std and return_cov:
|
||
|
|
raise RuntimeError(
|
||
|
|
"At most one of return_std or return_cov can be requested."
|
||
|
|
)
|
||
|
|
|
||
|
|
if self.kernel is None or self.kernel.requires_vector_input:
|
||
|
|
dtype, ensure_2d = "numeric", True
|
||
|
|
else:
|
||
|
|
dtype, ensure_2d = None, False
|
||
|
|
|
||
|
|
X = self._validate_data(X, ensure_2d=ensure_2d, dtype=dtype, reset=False)
|
||
|
|
|
||
|
|
if not hasattr(self, "X_train_"): # Unfitted;predict based on GP prior
|
||
|
|
if self.kernel is None:
|
||
|
|
kernel = C(1.0, constant_value_bounds="fixed") * RBF(
|
||
|
|
1.0, length_scale_bounds="fixed"
|
||
|
|
)
|
||
|
|
else:
|
||
|
|
kernel = self.kernel
|
||
|
|
|
||
|
|
n_targets = self.n_targets if self.n_targets is not None else 1
|
||
|
|
y_mean = np.zeros(shape=(X.shape[0], n_targets)).squeeze()
|
||
|
|
|
||
|
|
if return_cov:
|
||
|
|
y_cov = kernel(X)
|
||
|
|
if n_targets > 1:
|
||
|
|
y_cov = np.repeat(
|
||
|
|
np.expand_dims(y_cov, -1), repeats=n_targets, axis=-1
|
||
|
|
)
|
||
|
|
return y_mean, y_cov
|
||
|
|
elif return_std:
|
||
|
|
y_var = kernel.diag(X)
|
||
|
|
if n_targets > 1:
|
||
|
|
y_var = np.repeat(
|
||
|
|
np.expand_dims(y_var, -1), repeats=n_targets, axis=-1
|
||
|
|
)
|
||
|
|
return y_mean, np.sqrt(y_var)
|
||
|
|
else:
|
||
|
|
return y_mean
|
||
|
|
else: # Predict based on GP posterior
|
||
|
|
# Alg 2.1, page 19, line 4 -> f*_bar = K(X_test, X_train) . alpha
|
||
|
|
K_trans = self.kernel_(X, self.X_train_)
|
||
|
|
y_mean = K_trans @ self.alpha_
|
||
|
|
|
||
|
|
# undo normalisation
|
||
|
|
y_mean = self._y_train_std * y_mean + self._y_train_mean
|
||
|
|
|
||
|
|
# if y_mean has shape (n_samples, 1), reshape to (n_samples,)
|
||
|
|
if y_mean.ndim > 1 and y_mean.shape[1] == 1:
|
||
|
|
y_mean = np.squeeze(y_mean, axis=1)
|
||
|
|
|
||
|
|
# Alg 2.1, page 19, line 5 -> v = L \ K(X_test, X_train)^T
|
||
|
|
V = solve_triangular(
|
||
|
|
self.L_, K_trans.T, lower=GPR_CHOLESKY_LOWER, check_finite=False
|
||
|
|
)
|
||
|
|
|
||
|
|
if return_cov:
|
||
|
|
# Alg 2.1, page 19, line 6 -> K(X_test, X_test) - v^T. v
|
||
|
|
y_cov = self.kernel_(X) - V.T @ V
|
||
|
|
|
||
|
|
# undo normalisation
|
||
|
|
y_cov = np.outer(y_cov, self._y_train_std**2).reshape(*y_cov.shape, -1)
|
||
|
|
# if y_cov has shape (n_samples, n_samples, 1), reshape to
|
||
|
|
# (n_samples, n_samples)
|
||
|
|
if y_cov.shape[2] == 1:
|
||
|
|
y_cov = np.squeeze(y_cov, axis=2)
|
||
|
|
|
||
|
|
return y_mean, y_cov
|
||
|
|
elif return_std:
|
||
|
|
# Compute variance of predictive distribution
|
||
|
|
# Use einsum to avoid explicitly forming the large matrix
|
||
|
|
# V^T @ V just to extract its diagonal afterward.
|
||
|
|
y_var = self.kernel_.diag(X).copy()
|
||
|
|
y_var -= np.einsum("ij,ji->i", V.T, V)
|
||
|
|
|
||
|
|
# Check if any of the variances is negative because of
|
||
|
|
# numerical issues. If yes: set the variance to 0.
|
||
|
|
y_var_negative = y_var < 0
|
||
|
|
if np.any(y_var_negative):
|
||
|
|
warnings.warn(
|
||
|
|
"Predicted variances smaller than 0. "
|
||
|
|
"Setting those variances to 0."
|
||
|
|
)
|
||
|
|
y_var[y_var_negative] = 0.0
|
||
|
|
|
||
|
|
# undo normalisation
|
||
|
|
y_var = np.outer(y_var, self._y_train_std**2).reshape(*y_var.shape, -1)
|
||
|
|
|
||
|
|
# if y_var has shape (n_samples, 1), reshape to (n_samples,)
|
||
|
|
if y_var.shape[1] == 1:
|
||
|
|
y_var = np.squeeze(y_var, axis=1)
|
||
|
|
|
||
|
|
return y_mean, np.sqrt(y_var)
|
||
|
|
else:
|
||
|
|
return y_mean
|
||
|
|
|
||
|
|
def sample_y(self, X, n_samples=1, random_state=0):
|
||
|
|
"""Draw samples from Gaussian process and evaluate at X.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : array-like of shape (n_samples_X, n_features) or list of object
|
||
|
|
Query points where the GP is evaluated.
|
||
|
|
|
||
|
|
n_samples : int, default=1
|
||
|
|
Number of samples drawn from the Gaussian process per query point.
|
||
|
|
|
||
|
|
random_state : int, RandomState instance or None, default=0
|
||
|
|
Determines random number generation to randomly draw samples.
|
||
|
|
Pass an int for reproducible results across multiple function
|
||
|
|
calls.
|
||
|
|
See :term:`Glossary <random_state>`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
y_samples : ndarray of shape (n_samples_X, n_samples), or \
|
||
|
|
(n_samples_X, n_targets, n_samples)
|
||
|
|
Values of n_samples samples drawn from Gaussian process and
|
||
|
|
evaluated at query points.
|
||
|
|
"""
|
||
|
|
rng = check_random_state(random_state)
|
||
|
|
|
||
|
|
y_mean, y_cov = self.predict(X, return_cov=True)
|
||
|
|
if y_mean.ndim == 1:
|
||
|
|
y_samples = rng.multivariate_normal(y_mean, y_cov, n_samples).T
|
||
|
|
else:
|
||
|
|
y_samples = [
|
||
|
|
rng.multivariate_normal(
|
||
|
|
y_mean[:, target], y_cov[..., target], n_samples
|
||
|
|
).T[:, np.newaxis]
|
||
|
|
for target in range(y_mean.shape[1])
|
||
|
|
]
|
||
|
|
y_samples = np.hstack(y_samples)
|
||
|
|
return y_samples
|
||
|
|
|
||
|
|
def log_marginal_likelihood(
|
||
|
|
self, theta=None, eval_gradient=False, clone_kernel=True
|
||
|
|
):
|
||
|
|
"""Return log-marginal likelihood of theta for training data.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
theta : array-like of shape (n_kernel_params,) default=None
|
||
|
|
Kernel hyperparameters for which the log-marginal likelihood is
|
||
|
|
evaluated. If None, the precomputed log_marginal_likelihood
|
||
|
|
of ``self.kernel_.theta`` is returned.
|
||
|
|
|
||
|
|
eval_gradient : bool, default=False
|
||
|
|
If True, the gradient of the log-marginal likelihood with respect
|
||
|
|
to the kernel hyperparameters at position theta is returned
|
||
|
|
additionally. If True, theta must not be None.
|
||
|
|
|
||
|
|
clone_kernel : bool, default=True
|
||
|
|
If True, the kernel attribute is copied. If False, the kernel
|
||
|
|
attribute is modified, but may result in a performance improvement.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
log_likelihood : float
|
||
|
|
Log-marginal likelihood of theta for training data.
|
||
|
|
|
||
|
|
log_likelihood_gradient : ndarray of shape (n_kernel_params,), optional
|
||
|
|
Gradient of the log-marginal likelihood with respect to the kernel
|
||
|
|
hyperparameters at position theta.
|
||
|
|
Only returned when eval_gradient is True.
|
||
|
|
"""
|
||
|
|
if theta is None:
|
||
|
|
if eval_gradient:
|
||
|
|
raise ValueError("Gradient can only be evaluated for theta!=None")
|
||
|
|
return self.log_marginal_likelihood_value_
|
||
|
|
|
||
|
|
if clone_kernel:
|
||
|
|
kernel = self.kernel_.clone_with_theta(theta)
|
||
|
|
else:
|
||
|
|
kernel = self.kernel_
|
||
|
|
kernel.theta = theta
|
||
|
|
|
||
|
|
if eval_gradient:
|
||
|
|
K, K_gradient = kernel(self.X_train_, eval_gradient=True)
|
||
|
|
else:
|
||
|
|
K = kernel(self.X_train_)
|
||
|
|
|
||
|
|
# Alg. 2.1, page 19, line 2 -> L = cholesky(K + sigma^2 I)
|
||
|
|
K[np.diag_indices_from(K)] += self.alpha
|
||
|
|
try:
|
||
|
|
L = cholesky(K, lower=GPR_CHOLESKY_LOWER, check_finite=False)
|
||
|
|
except np.linalg.LinAlgError:
|
||
|
|
return (-np.inf, np.zeros_like(theta)) if eval_gradient else -np.inf
|
||
|
|
|
||
|
|
# Support multi-dimensional output of self.y_train_
|
||
|
|
y_train = self.y_train_
|
||
|
|
if y_train.ndim == 1:
|
||
|
|
y_train = y_train[:, np.newaxis]
|
||
|
|
|
||
|
|
# Alg 2.1, page 19, line 3 -> alpha = L^T \ (L \ y)
|
||
|
|
alpha = cho_solve((L, GPR_CHOLESKY_LOWER), y_train, check_finite=False)
|
||
|
|
|
||
|
|
# Alg 2.1, page 19, line 7
|
||
|
|
# -0.5 . y^T . alpha - sum(log(diag(L))) - n_samples / 2 log(2*pi)
|
||
|
|
# y is originally thought to be a (1, n_samples) row vector. However,
|
||
|
|
# in multioutputs, y is of shape (n_samples, 2) and we need to compute
|
||
|
|
# y^T . alpha for each output, independently using einsum. Thus, it
|
||
|
|
# is equivalent to:
|
||
|
|
# for output_idx in range(n_outputs):
|
||
|
|
# log_likelihood_dims[output_idx] = (
|
||
|
|
# y_train[:, [output_idx]] @ alpha[:, [output_idx]]
|
||
|
|
# )
|
||
|
|
log_likelihood_dims = -0.5 * np.einsum("ik,ik->k", y_train, alpha)
|
||
|
|
log_likelihood_dims -= np.log(np.diag(L)).sum()
|
||
|
|
log_likelihood_dims -= K.shape[0] / 2 * np.log(2 * np.pi)
|
||
|
|
# the log likehood is sum-up across the outputs
|
||
|
|
log_likelihood = log_likelihood_dims.sum(axis=-1)
|
||
|
|
|
||
|
|
if eval_gradient:
|
||
|
|
# Eq. 5.9, p. 114, and footnote 5 in p. 114
|
||
|
|
# 0.5 * trace((alpha . alpha^T - K^-1) . K_gradient)
|
||
|
|
# alpha is supposed to be a vector of (n_samples,) elements. With
|
||
|
|
# multioutputs, alpha is a matrix of size (n_samples, n_outputs).
|
||
|
|
# Therefore, we want to construct a matrix of
|
||
|
|
# (n_samples, n_samples, n_outputs) equivalent to
|
||
|
|
# for output_idx in range(n_outputs):
|
||
|
|
# output_alpha = alpha[:, [output_idx]]
|
||
|
|
# inner_term[..., output_idx] = output_alpha @ output_alpha.T
|
||
|
|
inner_term = np.einsum("ik,jk->ijk", alpha, alpha)
|
||
|
|
# compute K^-1 of shape (n_samples, n_samples)
|
||
|
|
K_inv = cho_solve(
|
||
|
|
(L, GPR_CHOLESKY_LOWER), np.eye(K.shape[0]), check_finite=False
|
||
|
|
)
|
||
|
|
# create a new axis to use broadcasting between inner_term and
|
||
|
|
# K_inv
|
||
|
|
inner_term -= K_inv[..., np.newaxis]
|
||
|
|
# Since we are interested about the trace of
|
||
|
|
# inner_term @ K_gradient, we don't explicitly compute the
|
||
|
|
# matrix-by-matrix operation and instead use an einsum. Therefore
|
||
|
|
# it is equivalent to:
|
||
|
|
# for param_idx in range(n_kernel_params):
|
||
|
|
# for output_idx in range(n_output):
|
||
|
|
# log_likehood_gradient_dims[param_idx, output_idx] = (
|
||
|
|
# inner_term[..., output_idx] @
|
||
|
|
# K_gradient[..., param_idx]
|
||
|
|
# )
|
||
|
|
log_likelihood_gradient_dims = 0.5 * np.einsum(
|
||
|
|
"ijl,jik->kl", inner_term, K_gradient
|
||
|
|
)
|
||
|
|
# the log likehood gradient is the sum-up across the outputs
|
||
|
|
log_likelihood_gradient = log_likelihood_gradient_dims.sum(axis=-1)
|
||
|
|
|
||
|
|
if eval_gradient:
|
||
|
|
return log_likelihood, log_likelihood_gradient
|
||
|
|
else:
|
||
|
|
return log_likelihood
|
||
|
|
|
||
|
|
def _constrained_optimization(self, obj_func, initial_theta, bounds):
|
||
|
|
if self.optimizer == "fmin_l_bfgs_b":
|
||
|
|
opt_res = scipy.optimize.minimize(
|
||
|
|
obj_func,
|
||
|
|
initial_theta,
|
||
|
|
method="L-BFGS-B",
|
||
|
|
jac=True,
|
||
|
|
bounds=bounds,
|
||
|
|
)
|
||
|
|
_check_optimize_result("lbfgs", opt_res)
|
||
|
|
theta_opt, func_min = opt_res.x, opt_res.fun
|
||
|
|
elif callable(self.optimizer):
|
||
|
|
theta_opt, func_min = self.optimizer(obj_func, initial_theta, bounds=bounds)
|
||
|
|
else:
|
||
|
|
raise ValueError(f"Unknown optimizer {self.optimizer}.")
|
||
|
|
|
||
|
|
return theta_opt, func_min
|
||
|
|
|
||
|
|
def _more_tags(self):
|
||
|
|
return {"requires_fit": False}
|