Source code for openmdao.surrogate_models.multifi_cokriging

"""
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
"""
from six.moves import range

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=np.float) n_eval = x.shape[0] f = np.ones([n_eval, 1]) return f
[docs]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=np.float) n_eval = x.shape[0] f = np.hstack([np.ones([n_eval, 1]), x]) return f
[docs]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 ------- r : array_like An array with shape (n_eval, ) containing the values of the autocorrelation model. """ theta = np.asarray(theta, dtype=np.float) d = np.asarray(d, dtype=np.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): """ Integrate the Multi-Fidelity Co-Kriging method described in [LeGratiet2013]. Attributes ---------- corr : Object Correlation function to use, default is squared_exponential_correlation. n_features : ndarry Number of features for each fidelity level. n_samples : ndarry 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 : 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. _nfev : int Number of function evaluations. _parent_name : str or None Absolute pathname of metamodel component that owns this surrogate. 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 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, parent_name=''): """ Initialize all attributes. Parameters ---------- regr : string 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 : string 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' 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 normalize : bool, optional When true, normalize X and Y so that the mean is at zero. parent_name : str Absolute pathname of metamodel component that owns this surrogate. """ 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._parent_name = parent_name 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 : Integer 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 Correlatioin 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.): self._raise("Multiple input features cannot have the same value.", exc_type=ValueError) # 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]: self._raise("Number of rows in F and X do not match. Most " "likely something is going wrong with the " "regression model.", exc_type=ValueError) if int(self.p[lvl] + self.q[lvl]) >= n_samples_F_i: self._raise("Ordinary least squares problem is undetermined " "n_samples=%d must be greater than the regression" " model size p+q=%d." % (n_samples[lvl], self.p[lvl] + self.q[lvl]), exc_type=ValueError) # 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]): self._raise("Bad parameter region. Try increasing upper bound", exc_type=ValueError) else: self.rlf_value[lvl] = self.rlf(lvl=lvl) if np.isinf(self.rlf_value[lvl]): self._raise("Bad point. Try increasing theta0.", exc_type=ValueError) return
[docs] 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 : Integer 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 : integer 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 : boolean, 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): self._raise("X and y must have the same length.", exc_type=ValueError) 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]: self._raise("All X must have the same number of columns.", exc_type=ValueError) 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]: self._raise("X and y must have the same number of rows.", exc_type=ValueError) self.n_features = n_features[0] if type(self.theta) is not list: self.theta = nlevel * [self.theta] elif len(self.theta) != nlevel: self._raise("theta must be a list of %d element(s)." % nlevel, exc_type=ValueError) if type(self.theta0) is not list: self.theta0 = nlevel * [self.theta0] elif len(self.theta0) != nlevel: self._raise("theta0 must be a list of %d element(s)." % nlevel, exc_type=ValueError) if type(self.thetaL) is not list: self.thetaL = nlevel * [self.thetaL] elif len(self.thetaL) != nlevel: self._raise("thetaL must be a list of %d element(s)." % nlevel, exc_type=ValueError) if type(self.thetaU) is not list: self.thetaU = nlevel * [self.thetaU] elif len(self.thetaU) != nlevel: self._raise("thetaU must be a list of %d element(s)." % nlevel, exc_type=ValueError) 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: self._raise("regr should be one of %s or callable, %s was given." % (self._regression_types.keys(), self.regr), exc_type=ValueError) # 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: self._raise("rho_regr should be one of %s or callable, %s was given." % (self._regression_types.keys(), self.rho_regr), exc_type=ValueError) 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): self._raise("theta must be strictly positive.", exc_type=ValueError) if self.theta0[i] is not None: self.theta0[i] = array2d(self.theta0[i]) if np.any(self.theta0[i] <= 0): self._raise("theta0 must be strictly positive.", exc_type=ValueError) 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: self._raise("theta0 and thetaL must have the same length.", exc_type=ValueError) 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: self._raise("theta0 and thetaU must have the same length.", exc_type=ValueError) 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]): self._raise("The bounds must satisfy O < thetaL <= thetaU.", exc_type=ValueError) return def _raise(self, msg, exc_type=RuntimeError): """ Raise the given exception type, with parent's name prepended to the message. Parameters ---------- msg : str The error message. exc_type : class The type of the exception to be raised. """ if self._parent_name is None: full_msg = msg else: full_msg = '{}: {}'.format(self._parent_name, msg) raise exc_type(full_msg)
[docs]class MultiFiCoKrigingSurrogate(MultiFiSurrogateModel): """ OpenMDAO adapter of multi-fidelity recursive cokriging method described in [LeGratiet2013]. See MultiFiCoKriging class. Attributes ---------- model : MultiFiCoKriging Contains MultiFiCoKriging surrogate. """
[docs] def __init__(self, **kwargs): """ Initialize all attributes. Parameters ---------- **kwargs : keyword args Some implementations of record_derivatives need additional args. """ super(MultiFiCoKrigingSurrogate, self).__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 : array_like An array with shape (n_samples_X, n_features) with the input at which observations were made. Y : array_like An array with shape (n_samples_X, n_features) with the observations of the scalar output to be predicted. """ 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'], parent_name=self._parent_name) 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)