multifi_cokriging.py¶
Integrates the MultiFidelity CoKriging method described in [LeGratiet2013].
(Author: Remi Vauclin vauclin.remi@gmail.com)
This code was implemented using the package scikitlearn as basis. (Author: Vincent Dubourg, vincent.dubourg@gmail.com)
OpenMDAO adaptation. Regression and correlation functions were directly copied from scikitlearn package here to avoid scikitlearn 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 MultiFidelity CoKriging method described in [LeGratiet2013].
Notes
Implementation is based on the Package ScikitLearn (Author: Vincent Dubourg, vincent.dubourg@gmail.com) which translates the DACE Matlab toolbox, see [Rafec0a633dc4NLNS2002].
References
 Rafec0a633dc4NLNS2002
H. B. Nielsen, S. N. Lophaven, and J. Sondergaard. DACE  A MATLAB Kriging Toolbox. (2002) http://www2.imm.dtu.dk/~hbn/dace/dace.pdf
 Rafec0a633dc4WBSWM1992
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
 Rafec0a633dc4LeGratiet2013
L. Le Gratiet (2013). “Multifidelity Gaussian process regression for computer experiments.” PhD thesis, Universite ParisDiderotParis VII.
 Rafec0a633dc4TBKH2011
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), 675699.
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*62)**2)*np.sin((Xe*62)*2) >>> yc = 0.5*((Xc*62)**2)*np.sin((Xc*62)*2)+(Xc0.5)*10.  5 >>> model = MultiFiCoKriging(theta0=1, thetaL=1e5, thetaU=50.) >>> model.fit([Xc, Xe], [yc, ye]) >>> # Prediction on x=0.05 >>> np.abs(float(model.predict([0.05])[0]) ((0.05*62)**2)*np.sin((0.05*62)*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 builtin 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 builtin 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 1e5*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 builtin 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 builtin 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 1e5*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=1e06)[source]¶ Implement the MultiFidelity cokriging 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 builtin 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=1e06, initial_range=0.3)[source]¶ Bases:
openmdao.surrogate_models.surrogate_model.MultiFiSurrogateModel
OpenMDAO adapter of multifidelity 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=1e06, 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 builtin 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 builtin 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 1e5*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
 xarraylike
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
 xarraylike
Point(s) at which the surrogate is evaluated.
 yarraylike
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
 xarraylike
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 crossdistances 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 crossdistances.

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.