Robustness of Kriging when interpolating in random simulation with heterogeneous variances: Some experiments

This paper investigates the use of Kriging in random simulation when the simulation output variances are not constant. Kriging gives a response surface or metamodel that can be used for interpolation. Because Ordinary Kriging assumes constant variances, this paper also applies Detrended Kriging to estimate a non-constant signal function, and then standardizes the residual noise through the heterogeneous variances estimated from replicated simulation runs. Numerical examples, however, suggest that Ordinary Kriging is a robust interpolation method.     

Application-driven sequential designs for simulation experiments: Kriging metamodelling

This paper proposes a novel method to select an experimental design for interpolation in simulation. Although the paper focuses on Kriging in deterministic simulation, the method also applies to other types of metamodels (besides Kriging), and to stochastic simulation. The paper focuses on simulations that require much computer time, so it is important to select a design with a small number of observations. The proposed method is therefore sequential. The novelty of the method is that it accounts for the specific input/output function of the particular simulation model at hand; that is, the method is application-driven or customized. This customization is achieved through cross-validation and jackknifing. The new method is tested through two academic applications, which demonstrate that the method indeed gives better results than either sequential designs based on an approximate Kriging prediction variance formula or designs with prefixed sample sizes.     

Classic Kriging <Emphasis Type="Italic">versus</Emphasis> Kriging with bootstrapping or conditional simulation: classic Kriging’s robust confidence intervals and optimization

Kriging is a popular method for estimating the global optimum of a simulated system. Kriging approximates the input/output function of the simulation model. Kriging also estimates the variances of the predictions of outputs for input combinations not yet simulated. These predictions and their variances are used by ‘efficient global optimization’ (EGO), to balance local and global search. This article focuses on two related questions: (1) How to select the next combination to be simulated when searching for the global optimum? (2) How to derive confidence intervals for outputs of input combinations not yet simulated? Classic Kriging simply plugs the estimated Kriging parameters into the formula for the predictor variance, so theoretically this variance is biased. This article concludes that practitioners may ignore this bias, because classic Kriging gives acceptable confidence intervals and estimates of the optimal input combination. This conclusion is based on bootstrapping and conditional simulation.     

Customized sequential designs for random simulation experiments: Kriging metamodeling and bootstrapping

This paper proposes a novel method to select an experimental design for interpolation in random simulation, especially discrete event simulation. (Though the paper focuses on Kriging, this design approach may also apply to other types of metamodels such as non-linear regression models and splines.) Assuming that simulation requires much computer time, it is important to select a design with a small number of observations (or simulation runs). The proposed method is therefore sequential. Its novelty is that it accounts for the specific input/output behavior (or response function) of the particular simulation at hand; i.e., the method is customized or application-driven. A tool for this customization is bootstrapping, which enables the estimation of the variances of predictions for inputs not yet simulated. The method is tested through two classic simulation models, namely the expected steady-state waiting time of the M/M/1 queuing model, and the mean costs of a terminating (s, S) inventory simulation. For these two simulation models the novel design indeed gives better results than a popular alternative design, namely Latin Hypercube Sampling (LHS) with a prefixed sample.     

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 for sensitivity analysis and optimization. It ends with topics for future research.     

Firepower concentration in cellular automaton combat models—an alternative to Lanchester

One of a new generation of combat models is examined to determine how its behaviour differs from older approaches based on first-order linear differential equations. This new methodology, which uses ‘cellular automaton’ or ‘agent-based’ models, has been around for a decade, prompting closer scrutiny. The method gives entities within a combat simulation the autonomy to react to circumstances in their local area. The reaction is determined by each entity's ‘personality’. It is found that the automata tend to either fight as a massed force, or form dispersed patterns of clusters within clusters. Such a pattern is known as a ‘fractal’. By adopting this pattern, a non-intuitive relationship between the kill probability of the automata and the force attrition rate develops. This provides a compelling example of how the result presented by earlier workers—that automaton models may evolve into fractal distributions—can have significance for operational researchers.     

Kriging空间分析法及其在城市大气污染中的应用

目前常用的大气污染物浓度插值方法没有考虑变量的空间分布特征 ,无法检验插值结果 ,不适用于复杂的城市大气环境的空间分析 .本文引用非参数地质统计学的指示克立格法 ,定义了大气环境研究领域的指示函数 ,据此分析了某市 SO2 的空间变异特征 ,估计了其 SO2 的空间分布 ,并给出了估计的精度 (估计误差 ) ,最后研究了监测点位置与估计误差空间分布之间的关系 .结果表明指示克立格法是城市大气污染空间分析、插值的有力工具 .     

Exchange rate uncertainty and employment: an algorithm describing ‘play’

The paper deals with the impacts of exchange rate uncertainty on the relationship between macroeconomic labour market variables. Under uncertainty, areas of weak reactions—so‐called ‘play’ areas—have to be considered at the macrolevel. The width of the play area is a positive function of the degree of uncertainty. When changes go beyond the play‐area suddenly strong reactions (‘spurts’) occur. These non‐linear dynamics are captured in a simplified linearized way. An algorithm describing linear play hysteresis is developed and implemented into a regression framework. As an empirical application, the exchange rate impacts on German employment are analysed considering play effects. Copyright © 2001 John Wiley & Sons, Ltd.     

Evaluating Strategy — Its Role Within the Context of Strategic Control

The paper uses a case study to illustrate a number of points in relation to strategy development and strategic control. A group decision support system — ‘Strategic Options Development and Analysis’ (SODA) — was used. The method focuses on the ‘linkages’ between strategic issues rather than on the issues themselves. The paper focuses upon the strategy review process. Three related elements of strategy' are considered as a part of the review: an evaluation of the extent to which the strategy has been implemented and is embedded in the organization; an evaluation of the assumptions underpinning the strategy; and an evaluation of the influence the strategy has on those at the ‘coal-face’.     

带缺失数据的偏正态众数回归模型的参数估计

针对现实生活中大量数据存在偏斜的情况,构建偏正态数据下的众数回归模型.又加之数据的缺失常有发生,采用插补方法处理缺失数据集,为比较插补效果,考虑对响应变量随机缺失情形进行统计推断研究.利用高斯牛顿迭代法给出众数回归模型参数的极大似然估计,比较该模型在均值插补,回归插补,众数插补三种插补条件下的插补效果.随机模拟和实例分...     

Monotonicity-preserving bootstrapped Kriging metamodels for expensive simulations

Kriging metamodels (also called Gaussian process or spatial correlation models) approximate the Input/Output functions implied by the underlying simulation models. Such metamodels serve sensitivity analysis, especially for computationally expensive simulations. In practice, simulation analysts often know that this Input/Output function is monotonic. To obtain a Kriging metamodel that preserves this characteristic, this article uses distribution-free bootstrapping assuming each input combination is simulated several times to obtain more reliable averaged outputs. Nevertheless, these averages still show sampling variation, so the Kriging metamodel does not need to be an exact interpolator; bootstrapping gives a noninterpolating Kriging metamodel. Bootstrapping may use standard Kriging software. The method is illustrated through the popular M/M/1 model with either the mean or the 90% quantile as output; these outputs are monotonic functions of the traffic rate. The empirical results demonstrate that monotonicity-preserving bootstrapped Kriging gives higher probability of covering the true outputs, without lengthening the confidence interval.     

Data mining feature selection for credit scoring models

The features used may have an important effect on the performance of credit scoring models. The process of choosing the best set of features for credit scoring models is usually unsystematic and dominated by somewhat arbitrary trial. This paper presents an empirical study of four machine learning feature selection methods. These methods provide an automatic data mining technique for reducing the feature space. The study illustrates how four feature selection methods—‘ReliefF’, ‘Correlation-based’, ‘Consistency-based’ and ‘Wrapper’ algorithms help to improve three aspects of the performance of scoring models: model simplicity, model speed and model accuracy. The experiments are conducted on real data sets using four classification algorithms—‘model tree (M5)’, ‘neural network (multi-layer perceptron with back-propagation)’, ‘logistic regression’, and ‘k-nearest-neighbours’.     

Validation of regression metamodels in simulation: Bootstrap approach

Simulation experiments are often analyzed through a linear regression model of their input/output data. Such an analysis yields a metamodel or response surface for the underlying simulation model. This metamodel can be validated through various statistics; this article studies (1) the coefficient of determination (R-square) for generalized least squares, and (2) a lack-of-fit F-statistic originally formulated by Rao [Biometrika 46 (1959) 49], who assumed multivariate normality. To derive the distributions of these two validation statistics, this paper shows how to apply bootstrapping—without assuming normality. To illustrate the performance of these bootstrapped validation statistics, the paper uses Monte Carlo experiments with simple models. For these models (i) R-square is a conservative statistic (rejecting a valid metamodel relatively rarely), so its power is low; (ii) Rao’s original statistic may reject a valid metamodel too often; (iii) bootstrapping Rao’s statistic gives only slightly conservative results, so its power is relatively high.     

Planning logistics operations in the oil industry

In this paper we apply stochastic programming modelling and solution techniques to planning problems for a consortium of oil companies. A multiperiod supply, transformation and distribution scheduling problem—the Depot and Refinery Optimization Problem (DROP)—is formulated for strategic or tactical level planning of the consortium's activities. This deterministic model is used as a basis for implementing a stochastic programming formulation with uncertainty in the product demands and spot supply costs (DROPS), whose solution process utilizes the deterministic equivalent linear programming problem. We employ our STOCHGEN general purpose stochastic problem generator to ‘recreate’ the decision (scenario) tree for the unfolding future as this deterministic equivalent. To project random demands for oil products at different spatial locations into the future and to generate random fluctuations in their future prices/costs a stochastic input data simulator is developed and calibrated to historical industry data. The models are written in the modelling language XPRESS-MP and solved by the XPRESS suite of linear programming solvers. From the viewpoint of implementation of large-scale stochastic programming models this study involves decisions in both space and time and careful revision of the original deterministic formulation. The first part of the paper treats the specification, generation and solution of the deterministic DROP model. The stochastic version of the model (DROPS) and its implementation are studied in detail in the second part and a number of related research questions and implications discussed.     

A signal-recovery system: asymptotic properties,and construction of an infinite-volume process

We consider a linear sequence of ‘nodes’, each of which can be in state 0 (‘off’) or 1 (‘on’). Signals from outside are sent to the rightmost node and travel instantaneously as far as possible to the left along nodes which are ‘on’. These nodes are immediately switched off, and become on again after a recovery time. The recovery times are independent exponentially distributed random variables. We present results for finite systems and use some of these results to construct an infinite-volume process (with signals ‘coming from infinity’), which has some peculiar properties. This construction is related to a question by Aldous and we hope that it sheds some light on, and stimulates further investigation of, that question.     

Can we have confidence in generic structures?

The system dynamics concept of ‘generic structure’ is dividable into three sub-types. This paper analyses the validity of these three, using both practical and theoretical perspectives. Firstly, a new set of measures is developed for generating validity—‘confidence’—amongst a group using generic structures in a practical modelling situation. It is concluded that different confidence criteria are implicitly employed; there is an argument for trading-off model precision and analytical quality for simplicity and ease of use and future research is needed to combine these ‘process’ and ‘content’ aspects of confidence. From a theoretical stance it is shown that with two of the sub-types a scientific notion of confidence is achievable whereas the third (‘archetypes’) involves merely metaphorical thinking. It is concluded that the theoretical status of archetypes requires further development, whilst ensuring that its benefits are retained.     

Multivariate versus univariate Kriging metamodels for multi-response simulation models

To analyze the input/output behavior of simulation models with multiple responses, we may apply either univariate or multivariate Kriging (Gaussian process) metamodels. In multivariate Kriging we face a major problem: the covariance matrix of all responses should remain positive-definite; we therefore use the recently proposed “nonseparable dependence” model. To evaluate the performance of univariate and multivariate Kriging, we perform several Monte Carlo experiments that simulate Gaussian processes. These Monte Carlo results suggest that the simpler univariate Kriging gives smaller mean square error.     

Conditional information criteria for selecting variables in linear mixed models

In this paper, we consider the problem of selecting the variables of the fixed effects in the linear mixed models where the random effects are present and the observation vectors have been obtained from many clusters. As the variable selection procedure, here we use the Akaike Information Criterion, AIC. In the context of the mixed linear models, two kinds of AIC have been proposed: marginal AIC and conditional AIC. In this paper, we derive three versions of conditional AIC depending upon different estimators of the regression coefficients and the random effects. Through the simulation studies, it is shown that the proposed conditional AIC’s are superior to the marginal and conditional AIC’s proposed in the literature in the sense of selecting the true model. Finally, the results are extended to the case when the random effects in all the clusters are of the same dimension but have a common unknown covariance matrix.     

Some new perspectives in best approximation and interpolation of random data

A new notion of universally optimal experimental design is introduced, relevant from the perspective of adaptive nonparametric estimation. It is demonstrated that both discrete and continuous Chebyshev designs are universally optimal in the problem of fitting properly weighted algebraic polynomials to random data. The result is a direct consequence of the well-known relation between Chebyshev’s polynomials and the trigonometric functions. Optimal interpolating designs in rational regression proved particularly elusive in the past. The question can be effectively handled using its connection to elliptic interpolation, in which the ordinary circular sinus, appearing in the classical trigonometric interpolation, is replaced by the Abel-Jacobi elliptic sinus sn(x, k) of a modulus k. First, it is demonstrated that — in a natural setting of equidistant design — the elliptic interpolant is never optimal in the so-called normal case k ∈ (?1, 1), except for the trigonometric case k = 0. However, the equidistant elliptic interpolation is always optimal in the imaginary case k ∈ i?. Through a relation between elliptic and rational functions, the result leads to a long sought optimal design, for properly weighted rational interpolants. Both the poles and nodes of the interpolants can be conveniently expressed in terms of classical Jacobi’s theta functions.