ooDACE toolbox: a flexible object-oriented Kriging implementation

When analyzing data from computationally expensive simulation codes, surrogate modeling methods are firmly established as facilitators for design space exploration, sensitivity analysis, visualization and optimization. Kriging is a popular surrogate modeling technique used for the Design and Analysis of Computer Experiments (DACE). Hence, the past decade Kriging has been the subject of extensive research and many extensions have been proposed, e.g., co-Kriging, stochastic Kriging, blind Kriging, etc. However, few Kriging implementations are publicly available and tailored towards scientists and engineers. Furthermore, no Kriging toolbox exists that unifies several Kriging flavors. This paper addresses this need by presenting an efficient object-oriented Kriging implementation and several Kriging extensions, providing a flexible and easily extendable framework to test and implement new Kriging flavors while reusing as much code as possible

Network Kriging

Network service providers and customers are often concerned with aggregate performance measures that span multiple network paths. Unfortunately, forming such network-wide measures can be difficult, due to the issues of scale involved. In particular, the number of paths grows too rapidly with the number of endpoints to make exhaustive measurement practical. As a result, it is of interest to explore the feasibility of methods that dramatically reduce the number of paths measured in such situations while maintaining acceptable accuracy. We cast the problem as one of statistical prediction--in the spirit of the so-called `kriging' problem in spatial statistics--and show that end-to-end network properties may be accurately predicted in many cases using a surprisingly small set of carefully chosen paths. More precisely, we formulate a general framework for the prediction problem, propose a class of linear predictors for standard quantities of interest (e.g., averages, totals, differences) and show that linear algebraic methods of subset selection may be used to effectively choose which paths to measure. We characterize the performance of the resulting methods, both analytically and numerically. The success of our methods derives from the low effective rank of routing matrices as encountered in practice, which appears to be a new observation in its own right with potentially broad implications on network measurement generally.Comment: 16 pages, 9 figures, single-space

Kriging Metamodeling in Simulation: A Review

This article reviews Kriging (also called spatial correlation modeling). It presents the basic Kriging assumptions and formulas contrasting Kriging and classic linear regression metamodels. Furthermore, it extends Kriging to random simulation, and discusses bootstrapping to estimate the variance of the Kriging predictor. Besides classic one-shot statistical designs such as Latin Hypercube Sampling, it reviews sequentialized and customized designs. It ends with topics for future research.Kriging;Metamodel;Response Surface;Interpolation;Design

On Prediction Properties of Kriging: Uniform Error Bounds and Robustness

Kriging based on Gaussian random fields is widely used in reconstructing unknown functions. The kriging method has pointwise predictive distributions which are computationally simple. However, in many applications one would like to predict for a range of untried points simultaneously. In this work we obtain some error bounds for the (simple) kriging predictor under the uniform metric. It works for a scattered set of input points in an arbitrary dimension, and also covers the case where the covariance function of the Gaussian process is misspecified. These results lead to a better understanding of the rate of convergence of kriging under the Gaussian or the Mat\'ern correlation functions, the relationship between space-filling designs and kriging models, and the robustness of the Mat\'ern correlation functions

The Correct Kriging Variance Estimated by Bootstrapping

The classic Kriging variance formula is widely used in geostatistics and in the design and analysis of computer experiments.This paper proves that this formula is wrong.Furthermore, it shows that the formula underestimates the Kriging variance in expectation.The paper develops parametric bootstrapping to estimate the Kriging variance.The new method is tested on several artificial examples and a real-life case study.These results demonstrate that the classic formula underestimates the true Kriging variance.Kriging;Kriging variance;bootstrapping;design and analysis of computer experiments (DACE);Monte Carlo;global optimization;black-box optimization

Kriging Interpolating Cosmic Velocity Field

[abridged] Volume-weighted statistics of large scale peculiar velocity is preferred by peculiar velocity cosmology, since it is free of uncertainties of galaxy density bias entangled in mass-weighted statistics. However, measuring the volume-weighted velocity statistics from galaxy (halo/simulation particle) velocity data is challenging. For the first time, we apply the Kriging interpolation to obtain the volume-weighted velocity field. Kriging is a minimum variance estimator. It predicts the most likely velocity for each place based on the velocity at other places. We test the performance of Kriging quantified by the E-mode velocity power spectrum from simulations. Dependences on the variogram prior used in Kriging, the number $n_k$ of the nearby particles to interpolate and the density $n_P$ of the observed sample are investigated. First, we find that Kriging induces $1\%$ and $3\%$ systematics at $k\sim 0.1h{\rm Mpc}^{-1}$ when $n_P\sim 6\times 10^{-2} ({\rm Mpc}/h)^{-3}$ and $n_P\sim 6\times 10^{-3} ({\rm Mpc}/h)^{-3}$, respectively. The deviation increases for decreasing $n_P$ and increasing $k$. When $n_P\lesssim 6\times 10^{-4} ({\rm Mpc}/h)^{-3}$, a smoothing effect dominates small scales, causing significant underestimation of the velocity power spectrum. Second, increasing $n_k$ helps to recover small scale power. However, for $n_P\lesssim 6\times 10^{-4} ({\rm Mpc}/h)^{-3}$ cases, the recovery is limited. Finally, Kriging is more sensitive to the variogram prior for lower sample density. The most straightforward application of Kriging on the cosmic velocity field does not show obvious advantages over the nearest-particle method (Zheng et al. 2013) and could not be directly applied to cosmology so far. However, whether potential improvements may be achieved by more delicate versions of Kriging is worth further investigation.Comment: 11 pages, 5 figures, published in PR

Efficient simulation-driven design optimization of antennas using co-kriging

We present an efficient technique for design optimization of antenna structures. Our approach exploits coarse-discretization electromagnetic (EM) simulations of the antenna of interest that are used to create its fast initial model (a surrogate) through kriging. During the design process, the predictions obtained by optimizing the surrogate are verified using high-fidelity EM simulations, and this high-fidelity data is used to enhance the surrogate through co-kriging technique that accommodates all EM simulation data into one surrogate model. The co-kriging-based optimization algorithm is simple, elegant and is capable of yielding a satisfactory design at a low cost equivalent to a few high-fidelity EM simulations of the antenna structure. To our knowledge, this is a first application of co-kriging to antenna design. An application example is provided