# sklearn.gaussian_process.GaussianProcessRegressor¶

class sklearn.gaussian_process.GaussianProcessRegressor(kernel=None, *, alpha=1e-10, optimizer='fmin_l_bfgs_b', n_restarts_optimizer=0, normalize_y=False, copy_X_train=True, random_state=None)[source]

Gaussian process regression (GPR).

The implementation is based on Algorithm 2.1 of Gaussian Processes for Machine Learning (GPML) by Rasmussen and Williams.

In addition to standard scikit-learn estimator API, GaussianProcessRegressor:

• allows prediction without prior fitting (based on the GP prior)

• provides an additional method sample_y(X), which evaluates samples drawn from the GPR (prior or posterior) at given inputs

• exposes a method log_marginal_likelihood(theta), which can be used externally for other ways of selecting hyperparameters, e.g., via Markov chain Monte Carlo.

Read more in the User Guide.

New in version 0.18.

Parameters
kernelkernel instance, default=None

The kernel specifying the covariance function of the GP. If None is passed, the kernel ConstantKernel(1.0, constant_value_bounds="fixed" * RBF(1.0, length_scale_bounds="fixed") is used as default. Note that the kernel hyperparameters are optimized during fitting unless the bounds are marked as “fixed”.

alphafloat or ndarray of shape (n_samples,), default=1e-10

Value added to the diagonal of the kernel matrix during fitting. This can prevent a potential numerical issue during fitting, by ensuring that the calculated values form a positive definite matrix. It can also be interpreted as the variance of additional Gaussian measurement noise on the training observations. Note that this is different from using a WhiteKernel. If an array is passed, it must have the same number of entries as the data used for fitting and is used as datapoint-dependent noise level. Allowing to specify the noise level directly as a parameter is mainly for convenience and for consistency with Ridge.

optimizer“fmin_l_bfgs_b” or callable, default=”fmin_l_bfgs_b”

Can either be one of the internally supported optimizers for optimizing the kernel’s parameters, specified by a string, or an externally defined optimizer passed as a callable. If a callable is passed, it must have the signature:

def optimizer(obj_func, initial_theta, bounds):
# * 'obj_func' is the objective function to be minimized, which
#   takes the hyperparameters theta as parameter and an
#   optional flag eval_gradient, which determines if the
# * 'initial_theta': the initial value for theta, which can be
#   used by local optimizers
# * 'bounds': the bounds on the values of theta
....
# Returned are the best found hyperparameters theta and
# the corresponding value of the target function.
return theta_opt, func_min


Per default, the ‘L-BGFS-B’ algorithm from scipy.optimize.minimize is used. If None is passed, the kernel’s parameters are kept fixed. Available internal optimizers are:

'fmin_l_bfgs_b'

n_restarts_optimizerint, default=0

The number of restarts of the optimizer for finding the kernel’s parameters which maximize the log-marginal likelihood. The first run of the optimizer is performed from the kernel’s initial parameters, the remaining ones (if any) from thetas sampled log-uniform randomly from the space of allowed theta-values. If greater than 0, all bounds must be finite. Note that n_restarts_optimizer == 0 implies that one run is performed.

normalize_ybool, default=False

Whether the target values y are normalized, the mean and variance of the target values are set equal to 0 and 1 respectively. This is recommended for cases where zero-mean, unit-variance priors are used. Note that, in this implementation, the normalisation is reversed before the GP predictions are reported.

Changed in version 0.23.

copy_X_trainbool, default=True

If True, a persistent copy of the training data is stored in the object. Otherwise, just a reference to the training data is stored, which might cause predictions to change if the data is modified externally.

random_stateint, RandomState instance or None, default=None

Determines random number generation used to initialize the centers. Pass an int for reproducible results across multiple function calls. See :term: Glossary <random_state>.

Attributes
X_train_array-like of shape (n_samples, n_features) or list of object

Feature vectors or other representations of training data (also required for prediction).

y_train_array-like of shape (n_samples,) or (n_samples, n_targets)

Target values in training data (also required for prediction)

kernel_kernel instance

The kernel used for prediction. The structure of the kernel is the same as the one passed as parameter but with optimized hyperparameters

L_array-like of shape (n_samples, n_samples)

Lower-triangular Cholesky decomposition of the kernel in X_train_

alpha_array-like of shape (n_samples,)

Dual coefficients of training data points in kernel space

log_marginal_likelihood_value_float

The log-marginal-likelihood of self.kernel_.theta

Examples

>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import DotProduct, WhiteKernel
>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
>>> kernel = DotProduct() + WhiteKernel()
>>> gpr = GaussianProcessRegressor(kernel=kernel,
...         random_state=0).fit(X, y)
>>> gpr.score(X, y)
0.3680...
>>> gpr.predict(X[:2,:], return_std=True)
(array([653.0..., 592.1...]), array([316.6..., 316.6...]))


Methods

 fit(X, y) Fit Gaussian process regression model. get_params([deep]) Get parameters for this estimator. log_marginal_likelihood([theta, …]) Returns log-marginal likelihood of theta for training data. predict(X[, return_std, return_cov]) Predict using the Gaussian process regression model sample_y(X[, n_samples, random_state]) Draw samples from Gaussian process and evaluate at X. score(X, y[, sample_weight]) Return the coefficient of determination $$R^2$$ of the prediction. set_params(**params) Set the parameters of this estimator.
fit(X, y)[source]

Fit Gaussian process regression model.

Parameters
Xarray-like of shape (n_samples, n_features) or list of object

Feature vectors or other representations of training data.

yarray-like of shape (n_samples,) or (n_samples, n_targets)

Target values

Returns
selfreturns an instance of self.
get_params(deep=True)[source]

Get parameters for this estimator.

Parameters
deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns
paramsdict

Parameter names mapped to their values.

log_marginal_likelihood(theta=None, eval_gradient=False, clone_kernel=True)[source]

Returns log-marginal likelihood of theta for training data.

Parameters
thetaarray-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.

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_kernelbool, 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_likelihoodfloat

Log-marginal likelihood of theta for training data.

Gradient of the log-marginal likelihood with respect to the kernel hyperparameters at position theta. Only returned when eval_gradient is True.

predict(X, return_std=False, return_cov=False)[source]

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, also its standard deviation (return_std=True) or covariance (return_cov=True). Note that at most one of the two can be requested.

Parameters
Xarray-like of shape (n_samples, n_features) or list of object

Query points where the GP is evaluated.

return_stdbool, default=False

If True, the standard-deviation of the predictive distribution at the query points is returned along with the mean.

return_covbool, default=False

If True, the covariance of the joint predictive distribution at the query points is returned along with the mean.

Returns
y_meanndarray of shape (n_samples, [n_output_dims])

Mean of predictive distribution a query points.

y_stdndarray of shape (n_samples,), optional

Standard deviation of predictive distribution at query points. Only returned when return_std is True.

y_covndarray of shape (n_samples, n_samples), optional

Covariance of joint predictive distribution a query points. Only returned when return_cov is True.

sample_y(X, n_samples=1, random_state=0)[source]

Draw samples from Gaussian process and evaluate at X.

Parameters
Xarray-like of shape (n_samples, n_features) or list of object

Query points where the GP is evaluated.

n_samplesint, default=1

The number of samples drawn from the Gaussian process

random_stateint, 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_samplesndarray of shape (n_samples_X, [n_output_dims], n_samples)

Values of n_samples samples drawn from Gaussian process and evaluated at query points.

score(X, y, sample_weight=None)[source]

Return the coefficient of determination $$R^2$$ of the prediction.

The coefficient $$R^2$$ is defined as $$(1 - \frac{u}{v})$$, where $$u$$ is the residual sum of squares ((y_true - y_pred) ** 2).sum() and $$v$$ is the total sum of squares ((y_true - y_true.mean()) ** 2).sum(). The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a $$R^2$$ score of 0.0.

Parameters
Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.

yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.

sample_weightarray-like of shape (n_samples,), default=None

Sample weights.

Returns
scorefloat

$$R^2$$ of self.predict(X) wrt. y.

Notes

The $$R^2$$ score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params)[source]

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters
**paramsdict

Estimator parameters.

Returns
selfestimator instance

Estimator instance.