# multifi_cokriging.py

# 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

classopenmdao.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].

- Parameters

regrstr or callable, optionalA 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_regrstr or callable, optionalA 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’.

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

thetadouble, array_like or list, optionalValue 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, optionalStarting 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, optionalLower 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, optionalUpper 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.

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

- 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

- 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

- LeGratiet2013
L. Le Gratiet (2013). “Multi-fidelity Gaussian process regression for computer experiments.” PhD thesis, Universite Paris-Diderot-Paris VII.

- 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

corrobjectCorrelation function to use, default is squared_exponential_correlation.

n_featuresndarrayNumber of features for each fidelity level.

n_samplesndarrayNumber of samples for each fidelity level.

nlevelintNumber of fidelity levels.

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

regrstr or callableA 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_regrstr or callable or NoneA 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 or NoneValue 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 or NoneStarting 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 or NoneLower 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 or NoneUpper 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_meanfloatMean of the low fidelity training data for X.

X_stdfloatStandard deviation of the low fidelity training data for X.

y_meanfloatMean of the low fidelity training data for y.

y_stdfloatStandard deviation of the low fidelity training data for y.

_nfevintNumber of function evaluations.

- __init__(
regr='constant',rho_regr='constant',normalize=True,theta=None,theta0=None,thetaL=None,thetaU=None)[source]Initialize all attributes.

- fit(
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 elementsA 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 elementsA list of arrays with the observations of the scalar output to be predicted, from lowest fidelity to highest fidelity.

initial_rangefloatInitial range for the optimizer.

tolfloatOptimizer terminates when the tolerance tol is reached.

- predict(
X,eval_MSE=True)[source]Perform the predictions of the kriging model on X.

- Parameters

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

eval_MSEbool, optionalA 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(
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

lvlintLevel of fidelity.

thetaarray_like, optionalAn 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.

classopenmdao.surrogate_models.multifi_cokriging.MultiFiCoKrigingSurrogate(**kwargs)[source]Bases:

`openmdao.surrogate_models.surrogate_model.MultiFiSurrogateModel`

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

See MultiFiCoKriging class.

- Parameters

**kwargskeyword argsSome implementations of record_derivatives need additional args.

- Attributes

modelMultiFiCoKrigingContains MultiFiCoKriging surrogate.

- __init__(
**kwargs)[source]Initialize all attributes.

- linearize(
x)Calculate the jacobian of the interpolant at the requested point.

- Parameters

xarray-likePoint at which the surrogate Jacobian is evaluated.

- predict(
new_x)[source]Calculate a predicted value of the response based on the current trained model.

- Parameters

new_xarray_likeAn 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(
x,y)Train the surrogate model with the given set of inputs and outputs.

- Parameters

xarray-likePoint(s) at which the surrogate is evaluated.

yarray-likeModel responses at given inputs.

- train_multifi(
X,Y)[source]Train the surrogate model with the given set of inputs and outputs.

- Parameters

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

Ylist of double array_like elementsA list of arrays with the observations of the scalar output to be predicted, from highest fidelity to lowest fidelity.

- vectorized_predict(
x)Calculate predicted values of the response based on the current trained model.

- Parameters

xarray-likeVectorized 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_likeInput 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_likeAn array with shape (n_samples_X, n_features).

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

darray_likeAn 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

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