sklearn.gaussian_process
.GaussianProcess¶
Warning
DEPRECATED

class
sklearn.gaussian_process.
GaussianProcess
(*args, **kwargs)[source]¶ The legacy Gaussian Process model class.
Deprecated since version 0.18: This class will be removed in 0.20. Use the
GaussianProcessRegressor
instead.Read more in the User Guide.
Parameters: regr : string or callable, optional
A regression function returning an array of outputs of the linear regression functional basis. The number of observations n_samples should be greater than the size p of this basis. Default assumes a simple constant regression trend. Available builtin regression models are:
'constant', 'linear', 'quadratic'
corr : string or callable, optional
A stationary autocorrelation function returning the autocorrelation between two points x and x’. Default assumes a squaredexponential autocorrelation model. Builtin correlation models are:
'absolute_exponential', 'squared_exponential', 'generalized_exponential', 'cubic', 'linear'
beta0 : double array_like, optional
The regression weight vector to perform Ordinary Kriging (OK). Default assumes Universal Kriging (UK) so that the vector beta of regression weights is estimated using the maximum likelihood principle.
storage_mode : string, optional
A string specifying whether the Cholesky decomposition of the correlation matrix should be stored in the class (storage_mode = ‘full’) or not (storage_mode = ‘light’). Default assumes storage_mode = ‘full’, so that the Cholesky decomposition of the correlation matrix is stored. This might be a useful parameter when one is not interested in the MSE and only plan to estimate the BLUP, for which the correlation matrix is not required.
verbose : boolean, optional
A boolean specifying the verbose level. Default is verbose = False.
theta0 : double array_like, optional
An array with shape (n_features, ) or (1, ). The parameters in the autocorrelation model. If thetaL and thetaU are also specified, theta0 is considered as the starting point for the maximum likelihood estimation of the best set of parameters. Default assumes isotropic autocorrelation model with theta0 = 1e1.
thetaL : double array_like, optional
An array with shape matching theta0’s. Lower bound on the autocorrelation parameters for maximum likelihood estimation. Default is None, so that it skips maximum likelihood estimation and it uses theta0.
thetaU : double array_like, optional
An array with shape matching theta0’s. Upper bound on the autocorrelation parameters for maximum likelihood estimation. Default is None, so that it skips maximum likelihood estimation and it uses theta0.
normalize : boolean, optional
Input X and observations y are centered and reduced wrt means and standard deviations estimated from the n_samples observations provided. Default is normalize = True so that data is normalized to ease maximum likelihood estimation.
nugget : double or ndarray, optional
Introduce a nugget effect to allow smooth predictions from noisy data. If nugget is an ndarray, it must be the same length as the number of data points used for the fit. The nugget is added to the diagonal of the assumed training covariance; in this way it acts as a Tikhonov regularization in the problem. In the special case of the squared exponential correlation function, the nugget mathematically represents the variance of the input values. Default assumes a nugget close to machine precision for the sake of robustness (nugget = 10. * MACHINE_EPSILON).
optimizer : string, optional
A string specifying the optimization algorithm to be used. Default uses ‘fmin_cobyla’ algorithm from scipy.optimize. Available optimizers are:
'fmin_cobyla', 'Welch'
‘Welch’ optimizer is dued to Welch et al., see reference [WBSWM1992426428]. It consists in iterating over several onedimensional optimizations instead of running one single multidimensional optimization.
random_start : int, optional
The number of times the Maximum Likelihood Estimation should be performed from a random starting point. The first MLE always uses the specified starting point (theta0), the next starting points are picked at random according to an exponential distribution (loguniform on [thetaL, thetaU]). Default does not use random starting point (random_start = 1).
random_state : int, RandomState instance or None, optional (default=None)
The generator used to shuffle the sequence of coordinates of theta in the Welch optimizer. If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by np.random.
Attributes
theta_ (array) Specified theta OR the best set of autocorrelation parameters (the sought maximizer of the reduced likelihood function). reduced_likelihood_function_value_ (array) The optimal reduced likelihood function value. Notes
The presentation implementation is based on a translation of the DACE Matlab toolbox, see reference [NLNS2002426428].
References
[NLNS2002426428] (1, 2) H.B. Nielsen, S.N. Lophaven, H. B. Nielsen and J. Sondergaard. DACE  A MATLAB Kriging Toolbox. (2002) http://imedea.uibcsic.es/master/cambioglobal/Modulo_V_cod101615/Lab/lab_maps/krigging/DACEkrigingsoft/dace/dace.pdf [WBSWM1992426428] (1, 2) W.J. Welch, R.J. Buck, J. Sacks, H.P. Wynn, T.J. Mitchell, and M.D. Morris (1992). Screening, predicting, and computer experiments. Technometrics, 34(1) 15–25. http://www.jstor.org/stable/1269548 Examples
>>> import numpy as np >>> from sklearn.gaussian_process import GaussianProcess >>> X = np.array([[1., 3., 5., 6., 7., 8.]]).T >>> y = (X * np.sin(X)).ravel() >>> gp = GaussianProcess(theta0=0.1, thetaL=.001, thetaU=1.) >>> gp.fit(X, y) GaussianProcess(beta0=None... ...
Methods
fit
(X, y)The Gaussian Process model fitting method. get_params
([deep])Get parameters for this estimator. predict
(X[, eval_MSE, batch_size])This function evaluates the Gaussian Process model at x. reduced_likelihood_function
([theta])This function determines the BLUP parameters and evaluates the reduced likelihood function for the given autocorrelation parameters theta. score
(X, y[, sample_weight])Returns the coefficient of determination R^2 of the prediction. set_params
(**params)Set the parameters of this estimator. 
__init__
(*args, **kwargs)[source]¶ DEPRECATED: GaussianProcess was deprecated in version 0.18 and will be removed in 0.20. Use the GaussianProcessRegressor instead.

fit
(X, y)[source]¶ The Gaussian Process model fitting method.
Parameters: X : double array_like
An array with shape (n_samples, n_features) with the input at which observations were made.
y : double array_like
An array with shape (n_samples, ) or shape (n_samples, n_targets) with the observations of the output to be predicted.
Returns: gp : self
A fitted Gaussian Process model object awaiting data to perform predictions.

get_params
(deep=True)[source]¶ Get parameters for this estimator.
Parameters: deep : boolean, optional
If True, will return the parameters for this estimator and contained subobjects that are estimators.
Returns: params : mapping of string to any
Parameter names mapped to their values.

predict
(X, eval_MSE=False, batch_size=None)[source]¶ This function evaluates the Gaussian Process model at x.
Parameters: X : array_like
An array with shape (n_eval, n_features) giving the point(s) at which the prediction(s) should be made.
eval_MSE : boolean, optional
A boolean specifying whether the Mean Squared Error should be evaluated or not. Default assumes evalMSE = False and evaluates only the BLUP (mean prediction).
batch_size : integer, optional
An integer giving the maximum number of points that can be evaluated simultaneously (depending on the available memory). Default is None so that all given points are evaluated at the same time.
Returns: y : array_like, shape (n_samples, ) or (n_samples, n_targets)
An array with shape (n_eval, ) if the Gaussian Process was trained on an array of shape (n_samples, ) or an array with shape (n_eval, n_targets) if the Gaussian Process was trained on an array of shape (n_samples, n_targets) with the Best Linear Unbiased Prediction at x.
MSE : array_like, optional (if eval_MSE == True)
An array with shape (n_eval, ) or (n_eval, n_targets) as with y, with the Mean Squared Error at x.

reduced_likelihood_function
(theta=None)[source]¶ This function determines the BLUP parameters and evaluates the reduced likelihood function for the given autocorrelation parameters theta.
Maximizing this function wrt the autocorrelation parameters theta is equivalent to maximizing the likelihood of the assumed joint Gaussian distribution of the observations y evaluated onto the design of experiments X.
Parameters: theta : array_like, optional
An array containing the autocorrelation parameters at which the Gaussian Process model parameters should be determined. Default uses the builtin autocorrelation parameters (ie
theta = self.theta_
).Returns: reduced_likelihood_function_value : double
The value of the reduced likelihood function associated to the given autocorrelation parameters theta.
par : dict
A dictionary containing the requested Gaussian Process model parameters:
sigma2
is the Gaussian Process variance.beta
is the generalized leastsquares regression weights for Universal Kriging or given beta0 for Ordinary Kriging.gamma
is the Gaussian Process weights.C
is the Cholesky decomposition of the correlation matrix [R].Ft
is the solution of the linear equation system [R] x Ft = FG
is the QR decomposition of the matrix Ft.

score
(X, y, sample_weight=None)[source]¶ Returns the coefficient of determination R^2 of the prediction.
The coefficient R^2 is defined as (1  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: X : arraylike, shape = (n_samples, n_features)
Test samples.
y : arraylike, shape = (n_samples) or (n_samples, n_outputs)
True values for X.
sample_weight : arraylike, shape = [n_samples], optional
Sample weights.
Returns: score : float
R^2 of self.predict(X) wrt. y.
