.. _mfk-ref-label: Multi-Fidelity Kriging (MFK) ================================ MFK is a multi-fidelity modeling method which uses an autoregressive model of order 1 (AR1). .. math :: y_\text{high}({\bf x})=\rho(x) \cdot y_\text{low}({\bf x}) + \delta({\bf x}) where :math:`\rho(x)` is a scaling/correlation factor (constant, linear or quadratic) and :math:`\delta(\cdot)` is a discrepancy function. The additive AR1 formulation was first introduced by Kennedy and O'Hagan [1]_. The implementation here follows the one proposed by Le Gratiet [2]_. It offers the advantage of being recursive, easily extended to :math:`n` levels of fidelity and offers better scaling for high numbers of samples. This method only uses nested sampling training points as described by Le Gratiet [2]_. References ---------- .. [1] Kennedy, M.C. and O'Hagan, A., Bayesian calibration of computer models. Journal of the Royal Statistical Society. 2001 .. [2] Le Gratiet, L., Multi-fidelity Gaussian process regression for computer experiments. PhD Thesis. 2013 Usage ----- .. code-block:: python import numpy as np import matplotlib.pyplot as plt from smt.applications.mfk import MFK, NestedLHS # low fidelity model def lf_function(x): import numpy as np return ( 0.5 * ((x * 6 - 2) ** 2) * np.sin((x * 6 - 2) * 2) + (x - 0.5) * 10.0 - 5 ) # high fidelity model def hf_function(x): import numpy as np return ((x * 6 - 2) ** 2) * np.sin((x * 6 - 2) * 2) # Problem set up xlimits = np.array([[0.0, 1.0]]) xdoes = NestedLHS(nlevel=2, xlimits=xlimits, random_state=0) xt_c, xt_e = xdoes(7) # Evaluate the HF and LF functions yt_e = hf_function(xt_e) yt_c = lf_function(xt_c) sm = MFK(theta0=xt_e.shape[1] * [1.0]) # low-fidelity dataset names being integers from 0 to level-1 sm.set_training_values(xt_c, yt_c, name=0) # high-fidelity dataset without name sm.set_training_values(xt_e, yt_e) # train the model sm.train() x = np.linspace(0, 1, 101, endpoint=True).reshape(-1, 1) # query the outputs y = sm.predict_values(x) _mse = sm.predict_variances(x) _derivs = sm.predict_derivatives(x, kx=0) plt.figure() plt.plot(x, hf_function(x), label="reference") plt.plot(x, y, linestyle="-.", label="mean_gp") plt.scatter(xt_e, yt_e, marker="o", color="k", label="HF doe") plt.scatter(xt_c, yt_c, marker="*", color="g", label="LF doe") plt.legend(loc=0) plt.ylim(-10, 17) plt.xlim(-0.1, 1.1) plt.xlabel(r"$x$") plt.ylabel(r"$y$") plt.show() :: ___________________________________________________________________________ MFK ___________________________________________________________________________ Problem size # training points. : 7 ___________________________________________________________________________ Training Training ... Training - done. Time (sec): 0.4344051 ___________________________________________________________________________ Evaluation # eval points. : 101 Predicting ... Predicting - done. Time (sec): 0.0002229 Prediction time/pt. (sec) : 0.0000022 ___________________________________________________________________________ Evaluation # eval points. : 101 Predicting ... Predicting - done. Time (sec): 0.0001872 Prediction time/pt. (sec) : 0.0000019 .. figure:: mfk_TestMFK_run_mfk_example.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 - ['pow_exp', 'abs_exp', 'squar_exp', 'act_exp', 'matern52', 'matern32'] - None - 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 - TNC - ['Cobyla', 'TNC'] - ['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) * - rho_regr - constant - ['constant', 'linear', 'quadratic'] - None - Regression function type for rho * - optim_var - False - [True, False] - ['bool'] - If True, the variance at HF samples is forced to zero * - propagate_uncertainty - True - [True, False] - ['bool'] - If True, the variance cotribution of lower fidelity levels are considered