GEKPLS ====== GEKPLS is a gradient-enhaced kriging with partial least squares approach. Gradient-enhaced kriging (GEK) is an extention of kriging which supports gradient information [1]_. GEK is usually more accurate than kriging, however, it is not computationally efficient when the number of inputs, the number of sampling points, or both, are high. This is mainly due to the size of the corresponding correlation matrix that increases proportionally with both the number of inputs and the number of sampling points. To address these issues, GEKPLS exploits the gradient information with a slight increase of the size of the correlation matrix and reduces the number of hyperparameters. The key idea of GEKPLS consists in generating a set of approximating points around each sampling points using the first order Taylor approximation method. Then, the PLS method is applied several times, each time on a different number of sampling points with the associated sampling points. Each PLS provides a set of coefficients that gives the contribution of each variable nearby the associated sampling point to the output. Finally, an average of all PLS coefficients is computed to get the global influence to the output. Denoting these coefficients by :math:`\left(w_1^{(k)},\dots,w_{nx}^{(k)}\right)`, the GEKPLS Gaussian kernel function is given by: .. math :: k\left({\bf x^{(i)}},{\bf x^{(j)}}\right)=\sigma\prod\limits_{l=1}^{nx} \prod\limits_{k=1}^h\exp\left(-\theta_k\left(w_l^{(k)}x_l^{(i)}-w_l^{(k)}x_l^{(j)}\right)^{2}\right) This approach reduces the number of hyperparameters (reduced dimension) from :math:`nx` to :math:`h` with :math:`nx>>h`. As previously mentioned, PLS is applied several times with respect to each sampling point, which provides the influence of each input variable around that point. The idea here is to add only m approximating points :math:`(m \in [1, nx])` around each sampling point. Only the :math:`m` highest coefficients given by the first principal component are considered, which usually contains the most useful information. More details of such approach are given in [2]_. .. [1] Forrester, I. J. and Sobester, A. and Keane, A. J., Engineering Design via Surrogate Modeling: A Practical Guide. Wiley, 2008 (Chapter 7). .. [2] Bouhlel, M. A., & Martins, J. R. (2019). Gradient-enhanced kriging for high-dimensional problems. Engineering with Computers, 35(1), 157-173. Usage ----- .. code-block:: python import numpy as np import matplotlib.pyplot as plt from smt.surrogate_models import GEKPLS, DesignSpace from smt.problems import Sphere from smt.sampling_methods import LHS # Construction of the DOE fun = Sphere(ndim=2) sampling = LHS(xlimits=fun.xlimits, criterion="m") xt = sampling(20) yt = fun(xt) # Compute the gradient for i in range(2): yd = fun(xt, kx=i) yt = np.concatenate((yt, yd), axis=1) design_space = DesignSpace(fun.xlimits) # Build the GEKPLS model n_comp = 2 sm = GEKPLS( design_space=design_space, theta0=[1e-2] * n_comp, extra_points=1, print_prediction=False, n_comp=n_comp, ) sm.set_training_values(xt, yt[:, 0]) for i in range(2): sm.set_training_derivatives(xt, yt[:, 1 + i].reshape((yt.shape[0], 1)), i) sm.train() # Test the model X = np.arange(fun.xlimits[0, 0], fun.xlimits[0, 1], 0.25) Y = np.arange(fun.xlimits[1, 0], fun.xlimits[1, 1], 0.25) X, Y = np.meshgrid(X, Y) Z = np.zeros((X.shape[0], X.shape[1])) for i in range(X.shape[0]): for j in range(X.shape[1]): Z[i, j] = sm.predict_values( np.hstack((X[i, j], Y[i, j])).reshape((1, 2)) ) fig = plt.figure() ax = fig.add_subplot(projection="3d") ax.plot_surface(X, Y, Z) plt.show() :: ___________________________________________________________________________ GEKPLS ___________________________________________________________________________ Problem size # training points. : 20 ___________________________________________________________________________ Training Training ... Training - done. Time (sec): 0.0415080 .. figure:: gekpls_Test_test_gekpls.png :scale: 80 % :align: center Options ------- .. list-table:: List of options :header-rows: 1 :widths: 15, 10, 20, 20, 30 :stub-columns: 0 * - Option - Default - Acceptable values - Acceptable types - Description * - print_global - True - None - ['bool'] - Global print toggle. If False, all printing is suppressed * - print_training - True - None - ['bool'] - Whether to print training information * - print_prediction - True - None - ['bool'] - Whether to print prediction information * - print_problem - True - None - ['bool'] - Whether to print problem information * - print_solver - True - None - ['bool'] - Whether to print solver information * - poly - constant - ['constant', 'linear', 'quadratic'] - ['str'] - Regression function type * - corr - squar_exp - ['abs_exp', 'squar_exp'] - ['str'] - Correlation function type * - pow_exp_power - 1.9 - None - ['float'] - Power for the pow_exp kernel function (valid values in (0.0, 2.0]), This option is set automatically when corr option is squar, abs, or matern. * - categorical_kernel - MixIntKernelType.CONT_RELAX - [, , , , ] - None - The kernel to use for categorical inputs. Only for non continuous Kriging * - hierarchical_kernel - MixHrcKernelType.ALG_KERNEL - [, ] - None - The kernel to use for mixed hierarchical inputs. Only for non continuous Kriging * - nugget - 2.220446049250313e-14 - None - ['float'] - a jitter for numerical stability * - theta0 - [0.01] - None - ['list', 'ndarray'] - Initial hyperparameters * - theta_bounds - [1e-06, 20.0] - None - ['list', 'ndarray'] - bounds for hyperparameters * - hyper_opt - Cobyla - ['Cobyla'] - ['str'] - Optimiser for hyperparameters optimisation * - eval_noise - False - [True, False] - ['bool'] - noise evaluation flag * - noise0 - [0.0] - None - ['list', 'ndarray'] - Initial noise hyperparameters * - noise_bounds - [2.220446049250313e-14, 10000000000.0] - None - ['list', 'ndarray'] - bounds for noise hyperparameters * - use_het_noise - False - [True, False] - ['bool'] - heteroscedastic noise evaluation flag * - n_start - 10 - None - ['int'] - number of optimizer runs (multistart method) * - xlimits - None - None - ['list', 'ndarray'] - definition of a design space of float (continuous) variables: array-like of size nx x 2 (lower, upper bounds) * - design_space - None - None - ['BaseDesignSpace', 'list', 'ndarray'] - definition of the (hierarchical) design space: use `smt.utils.design_space.DesignSpace` as the main API. Also accepts list of float variable bounds * - random_state - 41 - None - ['NoneType', 'int', 'RandomState'] - Numpy RandomState object or seed number which controls random draws for internal optim (set by default to get reproductibility) * - n_comp - 2 - None - ['int'] - Number of principal components * - eval_n_comp - False - [True, False] - ['bool'] - n_comp evaluation flag * - eval_comp_treshold - 1.0 - None - ['float'] - n_comp evaluation treshold for Wold's R criterion * - cat_kernel_comps - None - None - ['list'] - Number of components for PLS categorical kernel * - delta_x - 0.0001 - None - ['int', 'float'] - Step used in the FOTA * - extra_points - 0 - None - ['int'] - Number of extra points per training point