"""
Implements 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
"""
import numpy as np
from numpy import atleast_2d as array2d
from scipy import linalg
from scipy.optimize import minimize
from scipy.spatial.distance import squareform
from openmdao.surrogate_models.surrogate_model import MultiFiSurrogateModel
import logging
_logger = logging.getLogger()
MACHINE_EPSILON = np.finfo(np.double).eps # machine precision
NUGGET = 10. * MACHINE_EPSILON # nugget for robustness
INITIAL_RANGE_DEFAULT = 0.3 # initial range for optimizer
TOLERANCE_DEFAULT = 1e-6 # stopping criterion for MLE optimization
THETA0_DEFAULT = 0.5
THETAL_DEFAULT = 1e-5
THETAU_DEFAULT = 50
if hasattr(linalg, 'solve_triangular'):
# only in scipy since 0.9
solve_triangular = linalg.solve_triangular
else:
# slower, but works
def solve_triangular(x, y, lower=True):
"""Solve triangular."""
return linalg.solve(x, y)
[docs]
def constant_regression(x):
"""
Zero order polynomial (constant, p = 1) regression model.
x --> f(x) = 1
Parameters
----------
x : array_like
Input data.
Returns
-------
array_like
Constant regression output.
"""
x = np.asarray(x, dtype=float)
n_eval = x.shape[0]
f = np.ones([n_eval, 1])
return f
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def linear_regression(x):
"""
First order polynomial (linear, p = n+1) regression model.
x --> f(x) = [ 1, x_1, ..., x_n ].T
Parameters
----------
x : array_like
Input data.
Returns
-------
array_like
Linear regression output.
"""
x = np.asarray(x, dtype=float)
n_eval = x.shape[0]
f = np.hstack([np.ones([n_eval, 1]), x])
return f
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def squared_exponential_correlation(theta, d):
"""
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
----------
theta : array_like
An array with shape 1 (isotropic) or n (anisotropic) giving the
autocorrelation parameter(s).
d : array_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
-------
array_like
An array with shape (n_eval, ) containing the values of the
autocorrelation model.
"""
theta = np.asarray(theta, dtype=float)
d = np.asarray(d, dtype=float)
if d.ndim > 1:
n_features = d.shape[1]
else:
n_features = 1
if theta.size == 1:
return np.exp(-theta[0] * np.sum(d ** 2, axis=1))
elif theta.size != n_features:
raise ValueError("Length of theta must be 1 or %s" % n_features)
else:
return np.exp(-np.sum(theta.reshape(1, n_features) * d ** 2, axis=1))
[docs]
def l1_cross_distances(X, Y=None):
"""
Compute the nonzero componentwise L1 cross-distances between the vectors in X and Y.
Parameters
----------
X : array_like
An array with shape (n_samples_X, n_features).
Y : array_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.
"""
X = array2d(X)
if Y is None:
n_samples, n_features = X.shape
n_nonzero_cross_dist = n_samples * (n_samples - 1) // 2
D = np.zeros((n_nonzero_cross_dist, n_features))
ll_1 = 0
for k in range(n_samples - 1):
ll_0 = ll_1
ll_1 = ll_0 + n_samples - k - 1
D[ll_0:ll_1] = np.abs(X[k] - X[(k + 1):])
else:
Y = array2d(Y)
n_samples_X, n_features_X = X.shape
n_samples_Y, n_features_Y = Y.shape
if n_features_X != n_features_Y:
raise ValueError("X and Y must have the same dimensions.")
n_features = n_features_X
n_nonzero_cross_dist = n_samples_X * n_samples_Y
D = np.zeros((n_nonzero_cross_dist, n_features))
ll_1 = 0
for k in range(n_samples_X):
ll_0 = ll_1
ll_1 = ll_0 + n_samples_Y # - k - 1
D[ll_0:ll_1] = np.abs(X[k] - Y)
return D
[docs]
class MultiFiCoKriging(object):
"""
Implement the Multi-Fidelity Co-Kriging method described in [LeGratiet2013].
Parameters
----------
regr : str 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_regr : str 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'.
normalize : bool, optional
When true, normalize X and Y so that the mean is at zero.
theta : double, 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.
theta0 : double, 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.
thetaL : double, 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.
thetaU : double, 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.
Attributes
----------
corr : object
Correlation function to use, default is squared_exponential_correlation.
n_features : ndarray
Number of features for each fidelity level.
n_samples : ndarray
Number of samples for each fidelity level.
nlevel : int
Number of fidelity levels.
normalize : bool, optional
When true, normalize X and Y so that the mean is at zero.
regr : str 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 : str 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.
_nfev : int
Number of function evaluations.
Notes
-----
Implementation is based on the Package Scikit-Learn
(Author: Vincent Dubourg, vincent.dubourg@gmail.com) which translates
the DACE Matlab toolbox, see [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.
"""
_regression_types = {
'constant': constant_regression,
'linear': linear_regression
}
[docs]
def __init__(self, regr='constant', rho_regr='constant', normalize=True,
theta=None, theta0=None, thetaL=None, thetaU=None):
"""
Initialize all attributes.
"""
self.corr = squared_exponential_correlation
self.regr = regr
self.rho_regr = rho_regr
self.theta = theta
self.theta0 = theta0
self.thetaL = thetaL
self.thetaU = thetaU
self.normalize = normalize
self.X_mean = 0
self.X_std = 1
self.y_mean = 0
self.y_std = 1
self.n_features = None
self.n_samples = None
self.nlevel = None
self._nfev = 0
def _build_R(self, lvl, theta):
"""
Build the correlation matrix with given theta for the specified level.
Parameters
----------
lvl : int
Level of fidelity
theta : array_like
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
-------
ndarray
Correlation matrix.
"""
D = self.D[lvl]
n_samples = self.n_samples[lvl]
R = np.eye(n_samples) * (1. + NUGGET)
corr = squareform(self.corr(theta, D))
R = R + corr
return R
[docs]
def fit(self, X, y, initial_range=INITIAL_RANGE_DEFAULT, tol=TOLERANCE_DEFAULT):
"""
Implement the Multi-Fidelity co-kriging model fitting method.
Parameters
----------
X : list 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]).
y : list 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_range : float
Initial range for the optimizer.
tol : float
Optimizer terminates when the tolerance tol is reached.
"""
# Run input checks
# Transforms floats and arrays in lists to have a multifidelity
# structure
self._check_list_structure(X, y)
# Checks if all parameters are structured as required
self._check_params()
X = self.X
y = self.y
nlevel = self.nlevel
n_samples = self.n_samples
# initialize lists
self.beta = nlevel * [0]
self.beta_rho = nlevel * [None]
self.beta_regr = nlevel * [None]
self.C = nlevel * [0]
self.D = nlevel * [0]
self.F = nlevel * [0]
self.p = nlevel * [0]
self.q = nlevel * [0]
self.G = nlevel * [0]
self.sigma2 = nlevel * [0]
self._R_adj = nlevel * [None]
# Training data will be normalized using statistical quantities from the low fidelity set.
if self.normalize:
self.X_mean = X_mean = np.mean(X[0], axis=0)
self.X_std = X_std = np.std(X[0], axis=0)
self.y_mean = y_mean = np.mean(y[0], axis=0)
self.y_std = y_std = np.std(y[0], axis=0)
X_std[X_std == 0.] = 1.
y_std[y_std == 0.] = 1.
for lvl in range(nlevel):
if self.normalize:
X[lvl] = (X[lvl] - X_mean) / X_std
y[lvl] = (y[lvl] - y_mean) / y_std
# Calculate matrix of distances D between samples
self.D[lvl] = l1_cross_distances(X[lvl])
if (np.min(np.sum(self.D[lvl], axis=1)) == 0.):
raise ValueError("Multiple input features cannot have the same value.")
# Regression matrix and parameters
self.F[lvl] = self.regr(X[lvl])
self.p[lvl] = self.F[lvl].shape[1]
# Concatenate the autoregressive part for levels > 0
if lvl > 0:
F_rho = self.rho_regr(X[lvl])
self.q[lvl] = F_rho.shape[1]
self.F[lvl] = np.hstack((F_rho * np.dot((self.y[lvl - 1])[-n_samples[lvl]:],
np.ones((1, self.q[lvl]))), self.F[lvl]))
else:
self.q[lvl] = 0
n_samples_F_i = self.F[lvl].shape[0]
if n_samples_F_i != n_samples[lvl]:
raise ValueError("Number of rows in F and X do not match. Most "
"likely something is going wrong with the "
"regression model.")
if int(self.p[lvl] + self.q[lvl]) >= n_samples_F_i:
raise ValueError(f"Ordinary least squares problem is undetermined "
f"n_samples={n_samples[lvl]} must be greater than the regression"
f" model size p+q={self.p[lvl] + self.q[lvl]}.")
# Set attributes
self.X = X
self.y = y
self.rlf_value = np.zeros(nlevel)
for lvl in range(nlevel):
# Determine Gaussian Process model parameters
if self.theta[lvl] is None:
# Maximum Likelihood Estimation of the parameters
sol = self._max_rlf(lvl=lvl, initial_range=initial_range, tol=tol)
self.theta[lvl] = sol['theta']
self.rlf_value[lvl] = sol['rlf_value']
if np.isinf(self.rlf_value[lvl]):
raise ValueError("Bad parameter region. Try increasing upper bound")
else:
self.rlf_value[lvl] = self.rlf(lvl=lvl)
if np.isinf(self.rlf_value[lvl]):
raise ValueError("Bad point. Try increasing theta0.")
return
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def rlf(self, lvl, theta=None):
"""
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
----------
lvl : int
Level of fidelity.
theta : array_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.
"""
if theta is None:
# Use built-in autocorrelation parameters
theta = self.theta[lvl]
# Initialize output
rlf_value = 1e20
# Retrieve data
n_samples = self.n_samples[lvl]
y = self.y[lvl]
F = self.F[lvl]
p = self.p[lvl]
q = self.q[lvl]
R = self._build_R(lvl, theta)
try:
C = linalg.cholesky(R, lower=True)
except linalg.LinAlgError:
_logger.warning(('Cholesky decomposition of R at level %i failed' % lvl) +
' with theta=' + str(theta))
return rlf_value
# Get generalized least squares solution
Ft = solve_triangular(C, F, lower=True)
Yt = solve_triangular(C, y, lower=True)
try:
Q, G = linalg.qr(Ft, econ=True)
except TypeError: # qr() got an unexpected keyword argument 'econ'
# DeprecationWarning: qr econ argument will be removed after scipy
# 0.7. The economy transform will then be available through the
# mode='economic' argument.
Q, G = linalg.qr(Ft, mode='economic')
pass
# Universal Kriging
beta = solve_triangular(G, np.dot(Q.T, Yt))
err = Yt - np.dot(Ft, beta)
err2 = np.dot(err.T, err)[0, 0]
self._err = err
sigma2 = err2 / (n_samples - p - q)
detR = ((np.diag(C))**(2. / n_samples)).prod()
rlf_value = (n_samples - p - q) * np.log10(sigma2) \
+ n_samples * np.log10(detR)
self.beta_rho[lvl] = beta[:q]
self.beta_regr[lvl] = beta[q:]
self.beta[lvl] = beta
self.sigma2[lvl] = sigma2
self.C[lvl] = C
self.G[lvl] = G
return rlf_value
def _max_rlf(self, lvl, initial_range, tol):
"""
Estimate autocorrelation parameter theta as maximizer of the reduced likelihood function.
(Minimization of the negative reduced likelihood function is used for convenience.)
Parameters
----------
lvl : int
Level of fidelity
initial_range : float
Initial range of the optimizer
tol : float
Optimizer terminates when the tolerance tol is reached.
Returns
-------
array_like
The optimal hyperparameters.
double
The optimal negative reduced likelihood function value.
dict
res['theta']: optimal theta
res['rlf_value']: optimal value for likelihood
"""
# Initialize input
thetaL = self.thetaL[lvl]
thetaU = self.thetaU[lvl]
def rlf_transform(x):
return self.rlf(theta=10.**x, lvl=lvl)
# Use specified starting point as first guess
theta0 = self.theta0[lvl]
x0 = np.log10(theta0[0])
constraints = []
for i in range(theta0.size):
constraints.append({'type': 'ineq', 'fun': lambda log10t, i=i:
log10t[i] - np.log10(thetaL[0][i])})
constraints.append({'type': 'ineq', 'fun': lambda log10t, i=i:
np.log10(thetaU[0][i]) - log10t[i]})
constraints = tuple(constraints)
sol = minimize(rlf_transform, x0, method='COBYLA',
constraints=constraints,
options={'rhobeg': initial_range,
'tol': tol, 'disp': 0})
log10_optimal_x = sol['x']
optimal_rlf_value = sol['fun']
self._nfev += sol['nfev']
optimal_theta = 10. ** log10_optimal_x
res = {}
res['theta'] = optimal_theta
res['rlf_value'] = optimal_rlf_value
return res
[docs]
def predict(self, X, eval_MSE=True):
"""
Perform the predictions of the kriging model on 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 : bool, 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.
"""
X = array2d(X)
nlevel = self.nlevel
n_eval, n_features_X = X.shape
# Normalize
if self.normalize:
X = (X - self.X_mean) / self.X_std
# Calculate kriging mean and variance at level 0
mu = np.zeros((n_eval, nlevel))
f = self.regr(X)
f0 = self.regr(X)
dx = l1_cross_distances(X, Y=self.X[0])
# Get regression function and correlation
F = self.F[0]
C = self.C[0]
beta = self.beta[0]
Ft = solve_triangular(C, F, lower=True)
yt = solve_triangular(C, self.y[0], lower=True)
r_ = self.corr(self.theta[0], dx).reshape(n_eval, self.n_samples[0])
gamma = solve_triangular(C.T, yt - np.dot(Ft, beta), lower=False)
# Scaled predictor
mu[:, 0] = (np.dot(f, beta) + np.dot(r_, gamma)).ravel()
if eval_MSE:
MSE = np.zeros((n_eval, nlevel))
r_t = solve_triangular(C, r_.T, lower=True)
G = self.G[0]
u_ = solve_triangular(G.T, f.T - np.dot(Ft.T, r_t), lower=True)
MSE[:, 0] = self.sigma2[0] * \
(1 - (r_t**2).sum(axis=0) + (u_**2).sum(axis=0))
# Calculate recursively kriging mean and variance at level i
for i in range(1, nlevel):
C = self.C[i]
F = self.F[i]
g = self.rho_regr(X)
dx = l1_cross_distances(X, Y=self.X[i])
r_ = self.corr(self.theta[i], dx).reshape(
n_eval, self.n_samples[i])
f = np.vstack((g.T * mu[:, i - 1], f0.T))
Ft = solve_triangular(C, F, lower=True)
yt = solve_triangular(C, self.y[i], lower=True)
r_t = solve_triangular(C, r_.T, lower=True)
G = self.G[i]
beta = self.beta[i]
# scaled predictor
mu[:, i] = (np.dot(f.T, beta)
+ np.dot(r_t.T, yt - np.dot(Ft, beta))).ravel()
if eval_MSE:
Q_ = (np.dot((yt - np.dot(Ft, beta)).T,
yt - np.dot(Ft, beta)))[0, 0]
u_ = solve_triangular(G.T, f - np.dot(Ft.T, r_t), lower=True)
sigma2_rho = np.dot(g,
self.sigma2[
i] * linalg.inv(np.dot(G.T, G))[:self.q[i], :self.q[i]]
+ np.dot(beta[:self.q[i]], beta[:self.q[i]].T))
sigma2_rho = (sigma2_rho * g).sum(axis=1)
MSE[:, i] = sigma2_rho * MSE[:, i - 1] \
+ Q_ / (2 * (self.n_samples[i] - self.p[i] - self.q[i])) \
* (1 - (r_t**2).sum(axis=0)) \
+ self.sigma2[i] * (u_**2).sum(axis=0)
# scaled predictor
for i in range(nlevel): # Predictor
mu[:, i] = self.y_mean + self.y_std * mu[:, i]
if eval_MSE:
MSE[:, i] = self.y_std**2 * MSE[:, i]
if eval_MSE:
return mu[:, -1].reshape((n_eval, 1)), MSE[:, -1].reshape((n_eval, 1))
else:
return mu[:, -1].reshape((n_eval, 1))
def _check_list_structure(self, X, y):
"""
Transform floats and arrays in the training data lists to have a multifidelity structure.
Parameters
----------
X : list 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])
y : list 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.
"""
if type(X) is not list:
nlevel = 1
X = [X]
else:
nlevel = len(X)
if type(y) is not list:
y = [y]
if len(X) != len(y):
raise ValueError("X and y must have the same length.")
n_samples = np.zeros(nlevel, dtype=int)
n_features = np.zeros(nlevel, dtype=int)
n_samples_y = np.zeros(nlevel, dtype=int)
for i in range(nlevel):
n_samples[i], n_features[i] = X[i].shape
if i > 0 and n_features[i] != n_features[i - 1]:
raise ValueError("All X must have the same number of columns.")
y[i] = np.asarray(y[i]).ravel()[:, np.newaxis]
n_samples_y[i] = y[i].shape[0]
if n_samples[i] != n_samples_y[i]:
raise ValueError("X and y must have the same number of rows.")
self.n_features = n_features[0]
if type(self.theta) is not list:
self.theta = nlevel * [self.theta]
elif len(self.theta) != nlevel:
raise ValueError(f"theta must be a list of {nlevel} element(s).")
if type(self.theta0) is not list:
self.theta0 = nlevel * [self.theta0]
elif len(self.theta0) != nlevel:
raise ValueError(f"theta0 must be a list of {nlevel} element(s).")
if type(self.thetaL) is not list:
self.thetaL = nlevel * [self.thetaL]
elif len(self.thetaL) != nlevel:
raise ValueError(f"thetaL must be a list of {nlevel} element(s).")
if type(self.thetaU) is not list:
self.thetaU = nlevel * [self.thetaU]
elif len(self.thetaU) != nlevel:
raise ValueError(f"thetaU must be a list of {nlevel} element(s).")
self.nlevel = nlevel
self.X = X[:]
self.y = y[:]
self.n_samples = n_samples
return
def _check_params(self):
"""
Perform sanity checks on all parameters.
"""
# Check regression model
if not callable(self.regr):
if self.regr in self._regression_types:
self.regr = self._regression_types[self.regr]
else:
raise ValueError(f"regr should be one of {self._regression_types.keys()} or "
f"callable, {self.regr} was given.")
# Check rho regression model
if not callable(self.rho_regr):
if self.rho_regr in self._regression_types:
self.rho_regr = self._regression_types[self.rho_regr]
else:
raise ValueError(f"regr should be one of {self._regression_types.keys()} or "
f"callable, {self.rho_regr} was given.")
for i in range(self.nlevel):
# Check correlation parameters
if self.theta[i] is not None:
self.theta[i] = array2d(self.theta[i])
if np.any(self.theta[i] <= 0):
raise ValueError("theta must be strictly positive.")
if self.theta0[i] is not None:
self.theta0[i] = array2d(self.theta0[i])
if np.any(self.theta0[i] <= 0):
raise ValueError("theta0 must be strictly positive.")
else:
self.theta0[i] = array2d(self.n_features * [THETA0_DEFAULT])
lth = self.theta0[i].size
if self.thetaL[i] is not None:
self.thetaL[i] = array2d(self.thetaL[i])
if self.thetaL[i].size != lth:
raise ValueError("theta0 and thetaL must have the same length.")
else:
self.thetaL[i] = array2d(self.n_features * [THETAL_DEFAULT])
if self.thetaU[i] is not None:
self.thetaU[i] = array2d(self.thetaU[i])
if self.thetaU[i].size != lth:
raise ValueError("theta0 and thetaU must have the same length.")
else:
self.thetaU[i] = array2d(self.n_features * [THETAU_DEFAULT])
if np.any(self.thetaL[i] <= 0) or np.any(self.thetaU[i] < self.thetaL[i]):
raise ValueError("The bounds must satisfy O < thetaL <= thetaU.")
return
[docs]
class MultiFiCoKrigingSurrogate(MultiFiSurrogateModel):
"""
OpenMDAO adapter of multi-fidelity recursive cokriging method described in [LeGratiet2013].
See MultiFiCoKriging class.
Parameters
----------
**kwargs : keyword args
Some implementations of record_derivatives need additional args.
Attributes
----------
model : MultiFiCoKriging
Contains MultiFiCoKriging surrogate.
"""
[docs]
def __init__(self, **kwargs):
"""
Initialize all attributes.
"""
super().__init__(**kwargs)
self.model = None
def _declare_options(self):
"""
Declare options before kwargs are processed in the init method.
"""
opt = self.options
opt.declare('normalize', default=True, types=bool,
desc="When true, normalize X and Y so that the mean is at zero.")
opt.declare('regr', default='constant', types=(object, ),
desc="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 can be accessed by "
"setting this option to the strings 'constant' or 'linear'")
opt.declare('rho_regr', default='constant', types=(object, ),
desc="A regression function returning an array of outputs of the linear "
"regression functional basis. Defines the regression function for the "
"autoregressive parameter 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 can be accessed by setting this option to the "
"strings 'constant' or 'linear'")
opt.declare('theta', default=None, allow_none=True,
desc="Value of correlation parameters. If they are known, then no "
"optimization is run. Default is None, so that optimization is run. if double, "
"then value is replicated for all features and all levels. if array_like, "
"then an array with shape (n_features, ) for isotropic calculation. It is "
"replicated for all levels. if list, then a list of nlevel arrays specifying "
"value for each level")
opt.declare('theta0', default=None, allow_none=True,
desc="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")
opt.declare('thetaL', default=None, allow_none=True,
desc="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 theta0s. It is replicate "
"for all levels of code. "
"if list: a list of nlevel arrays specifying value for each level")
opt.declare('thetaU', default=None, allow_none=True,
desc="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")
opt.declare('tolerance', default=TOLERANCE_DEFAULT,
desc='Optimizer terminates when the tolerance tol is reached.')
opt.declare('initial_range', default=INITIAL_RANGE_DEFAULT,
desc='Initial range for the optimizer.')
[docs]
def predict(self, new_x):
"""
Calculate a predicted value of the response based on the current trained model.
Parameters
----------
new_x : array_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.
"""
Y_pred, MSE = self.model.predict([new_x])
return Y_pred, np.sqrt(np.abs(MSE))
[docs]
def train_multifi(self, X, Y):
"""
Train the surrogate model with the given set of inputs and outputs.
Parameters
----------
X : list of double array_like elements
A 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]).
Y : list of double array_like elements
A list of arrays with the observations of the scalar output to be predicted,
from highest fidelity to lowest fidelity.
"""
opt = self.options
if not self.model:
self.model = MultiFiCoKriging(regr=opt['regr'], rho_regr=opt['rho_regr'],
theta=opt['theta'], theta0=opt['theta0'],
thetaL=opt['thetaL'], thetaU=opt['thetaU'],
normalize=opt['normalize'])
X, Y = self._fit_adapter(X, Y)
self.model.fit(X, Y, tol=opt['tolerance'],
initial_range=opt['initial_range'])
def _fit_adapter(self, X, Y):
"""
Manage special case with one fidelity.
where can be called as [[xval1],[xval2]] instead of [[[xval1],[xval2]]]
we detect if shape(X[0]) is like (m,) instead of (m, n)
Parameters
----------
X : array_like
An array with shape (n_samples_X, n_features)
Y : array_like
An array with shape (n_samples_Y, n_features)
Returns
-------
list 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])
list 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.
"""
if len(np.shape(np.array(X[0]))) == 1:
X = [X]
Y = [Y]
X = [np.array(x) for x in reversed(X)]
Y = [np.array(y) for y in reversed(Y)]
return (X, Y)
