Statistical Tests for Cross-Validation of Kriging Models

Kriging or Gaussian process models are popular metamodels (surrogate models or emulators) of simulation models; these metamodels give predictors for input combinations that are not simulated. To validate these metamodels for computationally expensive simulation models, the analysts often apply computationally efficient cross-validation. In this paper, we derive new statistical tests for so-called leave-one-out cross-validation. Graphically, we present these tests as scatterplots augmented with confidence intervals that use the estimated variances of the Kriging predictors. To estimate the true variances of these predictors, we might use bootstrapping. Like other statistical tests, our tests—with or without bootstrapping—have type I and type II error probabilities; to estimate these probabilities, we use Monte Carlo experiments. We also use such experiments to investigate statistical convergence. To illustrate the application of our tests, we use (i) an example with two inputs and (ii) the popular borehole example with eight inputs. Summary of Contribution: Simulation models are very popular in operations research (OR) and are also known as computer simulations or computer experiments. A popular topic is design and analysis of computer experiments. This paper focuses on Kriging methods and cross-validation methods applied to simulation models; these methods and models are often applied in OR. More specifically, the paper provides the following; (1) the basic variant of a new statistical test for leave-one–out cross-validation; (2) a bootstrap method for the estimation of the true variance of the Kriging predictor; and (3) Monte Carlo experiments for the evaluation of the consistency of the Kriging predictor, the convergence of the Studentized prediction error to the standard normal variable, and the convergence of the expected experimentwise type I error rate to the prespecified nominal value. The new statistical test is illustrated through examples, including the popular borehole model.

Optimal biased Kriging: Homeogram tapering and applications to geoid undulations in Korea

This article studies the Optimal Biased Kriging (OBK) approach which is an alternative geostatistical method that gives up the unbiasedness condition of Ordinary Kriging (OK) to gain an improved Mean Squared Prediction Error (MSPE). The system of equations for the optimal linear biased Kriging predictor is derived and itsMSPE is compared with that of Ordinary Kriging. A major impediment in implementing this system of equations and performing Kriging interpolation with massive datasets is the inversion of the spatial coherency matrix. This problem is investigated and a novel method, called “homeogram tapering”, which exploits spatial sorting techniques to create sparse matrices for efficient matrix inversion, is described. Finally, as an application, results from experiments performed on a geoid undulation dataset from Korea are presented. A precise geoid is usually the indispensable basis for meaningful hydrological studies over wide areas. These experiments use the theory presented here along with a relatively new spatial coherency measure, called the homeogram, also known as the non-centered covariance function.

Hierarchical Metamodeling of the Air Bending Process

Despite the tremendous effort of researchers and manufacturing engineers in improving the predictability of the air bending process, there is still a strong need for comprehensive and dependable prediction models. Currently, available modeling approaches all present some relevant limitations in practical applications. In this paper, we propose a new method, which represents an improvement over all existing modeling and prediction techniques. The proposed method can be used for accurate prediction of the main response variables of the air bending process: the angle α after springback and the bend deduction BD. The metamodeling method is based on the hierarchical fusion of different kinds of data: the deterministic low-fidelity response of numerical finite element method (FEM) simulations and the stochastic high fidelity response of experimental tests. The metamodel has been built over a very large database, unprecedented in the scientific literature on air bending, made of more than 500 numerical simulations and nearly 300 experimental tests. The fusion is achieved first by interpolating the FEM simulations with a kriging predictor; then, the hierarchical metamodel is built as a linear regression model of the experimental data, using the kriging predictor among the regressors. The accuracy of the method has been proved using a variant of the leave-one-out cross validation technique. The quality of the prediction yielded by the proposed method significantly over-performs the current prediction of the press brake on-line numerical control.

Kriging Metamodeling in Rotordynamics: Application for Predicting Critical Speeds and Vibrations of a Flexible Rotor

Rotating machinery produces vibrations depending upon the design of the rotor systems as well as any faults or uncertainties in the machine that can increase the vibrations of such systems. This study illustrates the effectiveness of using surrogate modeling based on kriging in order to predict the vibrational behavior (i.e., the critical speeds and the vibration amplitudes) of a complex flexible rotor in the presence of uncertainties. The basic idea of kriging is to predict unknown values of a function by using a small size set of known data. The kriging estimation is based on a weighted average of the known values of the function in the neighborhood of the point for which the value of the function has to be calculated. The crucial dependence of a kriging predictor versus the correlation functions and different orders will be illustrated. This paper also shows that reducing the number of samples required to have predictive models can be achieved by performing an initial understanding of the mechanical system of interest and by considering certain characteristics directly deriving from the physics of the problem studied.

Alternative Kriging-HDMR optimization method with expected improvement sampling strategy

Purpose The purpose of study is to overcome the error estimation of standard deviation derived from Expected improvement (EI) criterion. Compared with other popular methods, a quantitative model assessment and analysis tool, termed high-dimensional model representation (HDMR), is suggested to be integrated with an EI-assisted sampling strategy. Design/methodology/approach To predict standard deviation directly, Kriging is imported. Furthermore, to compensate for the underestimation of error in the Kriging predictor, a Pareto frontier (PF)-EI (PFEI) criterion is also suggested. Compared with other surrogate-assisted optimization methods, the distinctive characteristic of HDMR is to disclose the correlations among component functions. If only low correlation terms are considered, the number of function evaluations for HDMR grows only polynomially with the number of input variables and correlative terms. Findings To validate the suggested method, various nonlinear and high-dimensional mathematical functions are tested. The results show the suggested method is potential for solving complicated real engineering problems. Originality/value In this study, the authors hope to integrate superiorities of PFEI and HDMR to improve optimization performance.

Kriging-based optimization design for a new style shell with black box constraints

Complex engineering applications generally have the black box and computationally expensive characteristics. Surrogate-based optimization algorithms can effectively solve expensive black box optimization problems. This paper employs the kriging predictor to construct a surrogate model and uses an initial multistart optimization process to realize the global search on this kriging model. Based on a proposed trust region framework, a local search is carried out around the current promising solution. The whole optimization algorithm is implemented to solve a new style shell design of the autonomous underwater vehicle. Based on finite element analysis, buoyancy–weight ratio, maximum von Mises stress, and buckling critical load of the new style shell are calculated and stored as expensive sample values to construct the kriging model. Finally, the better design parameters of the new shell are obtained by this proposed optimization algorithm. In addition, compared with the traditional shell, the new shell shows the stronger stability and better buoyancy–weight ratio.