skopt.learning.GaussianProcessRegressor

class skopt.learning.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, noise=None)

GaussianProcessRegressor that allows noise tunability.

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.

Parameters

kernelkernel object

The kernel specifying the covariance function of the GP. If None is passed, the kernel “1.0 * RBF(1.0)” is used as default. Note that the kernel’s hyperparameters are optimized during fitting.

alphafloat or array-like, optional (default: 1e-10)

Value added to the diagonal of the kernel matrix during fitting. Larger values correspond to increased noise level in the observations and reduce potential numerical issue during fitting. 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. Note that this is equivalent to adding a WhiteKernel with c=alpha. Allowing to specify the noise level directly as a parameter is mainly for convenience and for consistency with Ridge.

optimizerstring or callable, optional (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 maximized, which
    #   takes the hyperparameters theta as parameter and an
    #   optional flag eval_gradient, which determines if the
    #   gradient is returned additionally to the function value
    # * '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 ‘fmin_l_bfgs_b’ algorithm from scipy.optimize is used. If None is passed, the kernel’s parameters are kept fixed. Available internal optimizers are:

'fmin_l_bfgs_b'

n_restarts_optimizerint, optional (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_yboolean, optional (default: False)

Whether the target values y are normalized, i.e., the mean of the observed target values become zero. This parameter should be set to True if the target values’ mean is expected to differ considerable from zero. When enabled, the normalization effectively modifies the GP’s prior based on the data, which contradicts the likelihood principle; normalization is thus disabled per default.

copy_X_trainbool, optional (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_stateinteger or numpy.RandomState, optional

The generator used to initialize the centers. If an integer is given, it fixes the seed. Defaults to the global numpy random number generator.

noisestring, “gaussian”, optional

If set to “gaussian”, then it is assumed that y is a noisy estimate of f(x) where the noise is gaussian.

Attributes

X_train_array-like, shape = (n_samples, n_features)

Feature values in training data (also required for prediction)

y_train_array-like, shape = (n_samples, [n_output_dims])

Target values in training data (also required for prediction)

kernel_ kernel object

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, shape = (n_samples, n_samples)

Lower-triangular Cholesky decomposition of the kernel in X_train_

alpha_array-like, 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

noise_float

Estimate of the gaussian noise. Useful only when noise is set to “gaussian”.

Methods

fit(X, y)	Fit Gaussian process regression model.
get_params([deep])	Get parameters for this estimator.
log_marginal_likelihood([theta, ...])	Return log-marginal likelihood of theta for training data.
predict(X[, return_std, return_cov, ...])	Predict output for X.
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 of the prediction.
set_params(**params)	Set the parameters of this estimator.

__init__(kernel=None, alpha=1e-10, optimizer='fmin_l_bfgs_b', n_restarts_optimizer=0, normalize_y=False, copy_X_train=True, random_state=None, noise=None)

fit(X, y)

Fit Gaussian process regression model.

Parameters

Xarray-like, shape = (n_samples, n_features)

Training data

yarray-like, shape = (n_samples, [n_output_dims])

Target values

Returns

self

Returns an instance of self.

get_params(deep=True)

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)

Return 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.

eval_gradientbool, 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_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.

log_likelihood_gradientndarray 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.

predict(X, return_std=False, return_cov=False, return_mean_grad=False, return_std_grad=False)

Predict output for X.

In addition to the mean of the predictive distribution, also its standard deviation (return_std=True) or covariance (return_cov=True), the gradient of the mean and the standard-deviation with respect to X can be optionally provided.

Parameters

Xarray-like, shape = (n_samples, n_features)

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.

return_mean_gradbool, default: False

Whether or not to return the gradient of the mean. Only valid when X is a single point.

return_std_gradbool, default: False

Whether or not to return the gradient of the std. Only valid when X is a single point.

Returns

y_meanarray, shape = (n_samples, [n_output_dims])

Mean of predictive distribution a query points

y_stdarray, shape = (n_samples,), optional

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

y_covarray, shape = (n_samples, n_samples), optional

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

y_mean_gradshape = (n_samples, n_features)

The gradient of the predicted mean

y_std_gradshape = (n_samples, n_features)

The gradient of the predicted std.

sample_y(X, n_samples=1, random_state=0)

Draw samples from Gaussian process and evaluate at X.

Parameters

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

Query points where the GP is evaluated.

n_samplesint, default=1

Number of samples drawn from the Gaussian process per query point.

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 Glossary.

Returns

y_samplesndarray 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.

score(X, y, sample_weight=None)

Return the coefficient of determination of the prediction.

The coefficient of determination \(R^2\) 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)

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.

Examples using skopt.learning.GaussianProcessRegressor

Use different base estimators for optimization