# multifi_cokriging.py¶

Integrates the Multi-Fidelity Co-Kriging method described in [LeGratiet2013].

(Author: Remi Vauclin vauclin.remi@gmail.com)

This code was implemented using the package scikit-learn as basis. (Author: Vincent Dubourg, vincent.dubourg@gmail.com)

OpenMDAO adaptation. Regression and correlation functions were directly copied from scikit-learn package here to avoid scikit-learn dependency. (Author: Remi Lafage, remi.lafage@onera.fr)

ISAE/DMSM - ONERA/DCPS

class openmdao.surrogate_models.multifi_cokriging.MultiFiCoKriging(regr='constant', rho_regr='constant', normalize=True, theta=None, theta0=None, thetaL=None, thetaU=None)[source]

Bases: object

Integrate the Multi-Fidelity Co-Kriging method described in [LeGratiet2013].

Notes

Implementation is based on the Package Scikit-Learn (Author: Vincent Dubourg, vincent.dubourg@gmail.com) which translates the DACE Matlab toolbox, see [Rafec0a633dc4-NLNS2002].

References

Rafec0a633dc4-NLNS2002

H. B. Nielsen, S. N. Lophaven, and J. Sondergaard. DACE - A MATLAB Kriging Toolbox. (2002) http://www2.imm.dtu.dk/~hbn/dace/dace.pdf

Rafec0a633dc4-WBSWM1992

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/pss/1269548

Rafec0a633dc4-LeGratiet2013

L. Le Gratiet (2013). “Multi-fidelity Gaussian process regression for computer experiments.” PhD thesis, Universite Paris-Diderot-Paris VII.

Rafec0a633dc4-TBKH2011

Toal, D. J., Bressloff, N. W., Keane, A. J., & Holden, C. M. E. (2011). “The development of a hybridized particle swarm for kriging hyperparameter tuning.” Engineering optimization, 43(6), 675-699.

Examples

>>> from openmdao.surrogate_models.multifi_cokriging import MultiFiCoKriging
>>> import numpy as np
>>> # Xe: DOE for expensive code (nested in Xc)
>>> # Xc: DOE for cheap code
>>> # ye: expensive response
>>> # yc: cheap response
>>> Xe = np.array([[0],[0.4],[1]])
>>> Xc = np.vstack((np.array([[0.1],[0.2],[0.3],[0.5],[0.6],[0.7],[0.8],[0.9]]),Xe))
>>> ye = ((Xe*6-2)**2)*np.sin((Xe*6-2)*2)
>>> yc = 0.5*((Xc*6-2)**2)*np.sin((Xc*6-2)*2)+(Xc-0.5)*10. - 5
>>> model = MultiFiCoKriging(theta0=1, thetaL=1e-5, thetaU=50.)
>>> model.fit([Xc, Xe], [yc, ye])
>>> # Prediction on x=0.05
>>> np.abs(float(model.predict([0.05])[0])- ((0.05*6-2)**2)*np.sin((0.05*6-2)*2)) < 0.05
True


Attributes

 corr (Object) Correlation function to use, default is squared_exponential_correlation. normalize (bool, optional) When true, normalize X and Y so that the mean is at zero. regr (string or callable) A regression function returning an array of outputs of the linear regression functional basis for Universal Kriging purpose. regr is assumed to be the same for all levels of code. Default assumes a simple constant regression trend. Available built-in regression models are: ‘constant’, ‘linear’ rho_regr (string or callable or None) A regression function returning an array of outputs of the linear regression functional basis. Defines the regression function for the autoregressive parameter rho. rho_regr is assumed to be the same for all levels of code. Default assumes a simple constant regression trend. Available built-in regression models are: ‘constant’, ‘linear’ theta (double, array_like or list or None) Value of correlation parameters if they are known; no optimization is run. Default is None, so that optimization is run. if double: value is replicated for all features and all levels. if array_like: an array with shape (n_features, ) for isotropic calculation. It is replicated for all levels. if list: a list of nlevel arrays specifying value for each level theta0 (double, array_like or list or None) Starting point for the maximum likelihood estimation of the best set of parameters. Default is None and meaning use of the default 0.5*np.ones(n_features) if double: value is replicated for all features and all levels. if array_like: an array with shape (n_features, ) for isotropic calculation. It is replicated for all levels. if list: a list of nlevel arrays specifying value for each level thetaL (double, array_like or list or None) Lower bound on the autocorrelation parameters for maximum likelihood estimation. Default is None meaning use of the default 1e-5*np.ones(n_features). if double: value is replicated for all features and all levels. if array_like: An array with shape matching theta0’s. It is replicated for all levels of code. if list: a list of nlevel arrays specifying value for each level thetaU (double, array_like or list or None) Upper bound on the autocorrelation parameters for maximum likelihood estimation. Default is None meaning use of default value 50*np.ones(n_features). if double: value is replicated for all features and all levels. if array_like: An array with shape matching theta0’s. It is replicated for all levels of code. if list: a list of nlevel arrays specifying value for each level X_mean (float) Mean of the low fidelity training data for X. X_std (float) Standard deviation of the low fidelity training data for X. y_mean (float) Mean of the low fidelity training data for y. y_std (float) Standard deviation of the low fidelity training data for y.
__init__(self, regr='constant', rho_regr='constant', normalize=True, theta=None, theta0=None, thetaL=None, thetaU=None)[source]

Initialize all attributes.

Parameters
regrstring or callable, optional

A regression function returning an array of outputs of the linear regression functional basis for Universal Kriging purpose. regr is assumed to be the same for all levels of code. Default assumes a simple constant regression trend. Available built-in regression models are: ‘constant’, ‘linear’

rho_regrstring or callable, optional

A regression function returning an array of outputs of the linear regression functional basis. Defines the regression function for the autoregressive parameter rho. rho_regr is assumed to be the same for all levels of code. Default assumes a simple constant regression trend. Available built-in regression models are: ‘constant’, ‘linear’

thetadouble, array_like or list, optional

Value of correlation parameters if they are known; no optimization is run. Default is None, so that optimization is run. if double: value is replicated for all features and all levels. if array_like: an array with shape (n_features, ) for isotropic calculation. It is replicated for all levels. if list: a list of nlevel arrays specifying value for each level

theta0double, array_like or list, optional

Starting point for the maximum likelihood estimation of the best set of parameters. Default is None and meaning use of the default 0.5*np.ones(n_features) if double: value is replicated for all features and all levels. if array_like: an array with shape (n_features, ) for isotropic calculation. It is replicated for all levels. if list: a list of nlevel arrays specifying value for each level

thetaLdouble, array_like or list, optional

Lower bound on the autocorrelation parameters for maximum likelihood estimation. Default is None meaning use of the default 1e-5*np.ones(n_features). if double: value is replicated for all features and all levels. if array_like: An array with shape matching theta0’s. It is replicated for all levels of code. if list: a list of nlevel arrays specifying value for each level

thetaUdouble, array_like or list, optional

Upper bound on the autocorrelation parameters for maximum likelihood estimation. Default is None meaning use of default value 50*np.ones(n_features). if double: value is replicated for all features and all levels. if array_like: An array with shape matching theta0’s. It is replicated for all levels of code. if list: a list of nlevel arrays specifying value for each level

normalizebool, optional

When true, normalize X and Y so that the mean is at zero.

fit(self, X, y, initial_range=0.3, tol=1e-06)[source]

Implement the Multi-Fidelity co-kriging model fitting method.

Parameters
Xlist of double array_like elements

A list of arrays with the input at which observations were made, from lowest fidelity to highest fidelity. Designs must be nested with X[i] = np.vstack([…, X[i+1])

ylist of double array_like elements

A list of arrays with the observations of the scalar output to be predicted, from lowest fidelity to highest fidelity.

initial_rangefloat

Initial range for the optimizer.

tolfloat

Optimizer terminates when the tolerance tol is reached.

predict(self, X, eval_MSE=True)[source]

Perform the predictions of the kriging model on X.

Parameters
Xarray_like

An array with shape (n_eval, n_features) giving the point(s) at which the prediction(s) should be made.

eval_MSEboolean, optional

A boolean specifying whether the Mean Squared Error should be evaluated or not. Default assumes evalMSE is True.

Returns
array_like

An array with shape (n_eval, ) with the Best Linear Unbiased Prediction at X. If all_levels is set to True, an array with shape (n_eval, nlevel) giving the BLUP for all levels.

array_like, optional (if eval_MSE is True)

An array with shape (n_eval, ) with the Mean Squared Error at X. If all_levels is set to True, an array with shape (n_eval, nlevel) giving the MSE for all levels.

rlf(self, lvl, theta=None)[source]

Determine BLUP parameters and evaluate negative reduced likelihood function for 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
lvlInteger

Level of fidelity

thetaarray_like, optional

An array containing the autocorrelation parameters at which the Gaussian Process model parameters should be determined. Default uses the built-in autocorrelation parameters (ie theta = self.theta).

Returns
double

The value of the negative concentrated reduced likelihood function associated to the given autocorrelation parameters theta.

class openmdao.surrogate_models.multifi_cokriging.MultiFiCoKrigingSurrogate(regr='constant', rho_regr='constant', normalize=True, theta=None, theta0=None, thetaL=None, thetaU=None, tolerance=1e-06, initial_range=0.3)[source]

OpenMDAO adapter of multi-fidelity recursive cokriging method described in [LeGratiet2013].

See MultiFiCoKriging class.

Attributes

 initial_range (float) Initial range for the optimizer. model (MultiFiCoKriging) Contains MultiFiCoKriging surrogate. tolerance (float) Optimizer terminates when the tolerance tol is reached.
__init__(self, regr='constant', rho_regr='constant', normalize=True, theta=None, theta0=None, thetaL=None, thetaU=None, tolerance=1e-06, initial_range=0.3)[source]

Initialize all attributes.

Parameters
normalizebool, optional

When true, normalize X and Y so that the mean is at zero.

regrstring or callable, optional

A regression function returning an array of outputs of the linear regression functional basis for Universal Kriging purpose. regr is assumed to be the same for all levels of code. Default assumes a simple constant regression trend. Available built-in regression models are: ‘constant’, ‘linear’

rho_regrstring or callable, optional

A regression function returning an array of outputs of the linear regression functional basis. Defines the regression function for the autoregressive parameter rho. rho_regr is assumed to be the same for all levels of code. Default assumes a simple constant regression trend. Available built-in regression models are: ‘constant’, ‘linear’

thetadouble, array_like or list, optional

Value of correlation parameters if they are known; no optimization is run. Default is None, so that optimization is run. if double: value is replicated for all features and all levels. if array_like: an array with shape (n_features, ) for isotropic calculation. It is replicated for all levels. if list: a list of nlevel arrays specifying value for each level

theta0double, array_like or list, optional

Starting point for the maximum likelihood estimation of the best set of parameters. Default is None and meaning use of the default 0.5*np.ones(n_features) if double: value is replicated for all features and all levels. if array_like: an array with shape (n_features, ) for isotropic calculation. It is replicated for all levels. if list: a list of nlevel arrays specifying value for each level

thetaLdouble, array_like or list, optional

Lower bound on the autocorrelation parameters for maximum likelihood estimation. Default is None meaning use of the default 1e-5*np.ones(n_features). if double: value is replicated for all features and all levels. if array_like: An array with shape matching theta0’s. It is replicated for all levels of code. if list: a list of nlevel arrays specifying value for each level

thetaUdouble, array_like or list, optional

Upper bound on the autocorrelation parameters for maximum likelihood estimation. Default is None meaning use of default value 50*np.ones(n_features). if double: value is replicated for all features and all levels. if array_like: An array with shape matching theta0’s. It is replicated for all levels of code. if list: a list of nlevel arrays specifying value for each level

tolerancefloat

Optimizer terminates when the tolerance tol is reached.

initial_rangefloat

Initial range for the optimizer.

linearize(self, x)

Calculate the jacobian of the interpolant at the requested point.

Parameters
xarray-like

Point at which the surrogate Jacobian is evaluated.

predict(self, new_x)[source]

Calculate a predicted value of the response based on the current trained model.

Parameters
new_xarray_like

An array with shape (n_eval, n_features) giving the point(s) at which the prediction(s) should be made.

Returns
array_like

An array with shape (n_eval, ) with the Best Linear Unbiased Prediction at X. If all_levels is set to True, an array with shape (n_eval, nlevel) giving the BLUP for all levels.

array_like

An array with shape (n_eval, ) with the square root of the Mean Squared Error at X.

train(self, x, y)

Calculate a predicted value of the response based on the current trained model.

Parameters
xarray-like

Point(s) at which the surrogate is evaluated.

yarray-like

Model responses at given inputs.

train_multifi(self, X, Y)[source]

Train the surrogate model with the given set of inputs and outputs.

Parameters
Xarray_like

An array with shape (n_samples_X, n_features) with the input at which observations were made.

Yarray_like

An array with shape (n_samples_X, n_features) with the observations of the scalar output to be predicted.

vectorized_predict(self, x)

Calculate predicted values of the response based on the current trained model.

Parameters
xarray-like

Vectorized point(s) at which the surrogate is evaluated.

openmdao.surrogate_models.multifi_cokriging.constant_regression(x)[source]

Zero order polynomial (constant, p = 1) regression model.

x –> f(x) = 1

Parameters
xarray_like

Input data.

Returns
array_like

Constant regression output.

openmdao.surrogate_models.multifi_cokriging.l1_cross_distances(X, Y=None)[source]

Compute the nonzero componentwise L1 cross-distances between the vectors in X and Y.

Parameters
Xarray_like

An array with shape (n_samples_X, n_features)

Yarray_like

An array with shape (n_samples_Y, n_features)

Returns
array with shape (n_samples * (n_samples - 1) / 2, n_features)

The array of componentwise L1 cross-distances.

openmdao.surrogate_models.multifi_cokriging.linear_regression(x)[source]

First order polynomial (linear, p = n+1) regression model.

x –> f(x) = [ 1, x_1, …, x_n ].T

Parameters
xarray_like

Input data.

Returns
array_like

Linear regression output.

openmdao.surrogate_models.multifi_cokriging.squared_exponential_correlation(theta, d)[source]

Squared exponential correlation model (Radial Basis Function).

(Infinitely differentiable stochastic process, very smooth):

                                    n
theta, dx --> r(theta, dx) = exp(  sum  - theta_i * (dx_i)^2 )
i = 1

Parameters
thetaarray_like

An array with shape 1 (isotropic) or n (anisotropic) giving the autocorrelation parameter(s).

darray_like

An array with shape (n_eval, n_features) giving the componentwise distances between locations x and x’ at which the correlation model should be evaluated.

Returns
rarray_like

An array with shape (n_eval, ) containing the values of the autocorrelation model.