Rs中响应曲面模型的稳健设计

本文运用再生核Hhilbert空间方法研究R^s中响应曲面模型的稳健设计问题。我们假设模型偏差包括由多元Hermite多项式高阶项产生的效应，如果偏差大，设计点应该布于布点附近，否则，设计点应该适当散开，所求设计是正交不变的，而且关于模型偏差是稳健的。     

Optimal design for smoothing splines

In the common nonparametric regression model we consider the problem of constructing optimal designs, if the unknown curve is estimated by a smoothing spline. A special basis for the space of natural splines is introduced and the local minimax property for these splines is used to derive two optimality criteria for the construction of optimal designs. The first criterion determines the design for a most precise estimation of the coefficients in the spline representation and corresponds to D-optimality, while the second criterion is the G-optimality criterion and corresponds to an accurate prediction of the curve. Several properties of the optimal designs are derived. In general, D- and G-optimal designs are not equivalent. Optimal designs are determined numerically and compared with the uniform design.     

Spatial Regression Models,Response Surfaces,and Process Optimization

Abstract

Spatial regression models are developed as a complementary alternative to second-order polynomial response surfaces in the context of process optimization. These models provide estimates of design variable effects and smooth, data-faithful approximations to the unknown response function over the design space. The predicted response surfaces are driven by the covariance structures of the models. Several structures, isotropic and anisotropic, are considered and connections with thin plate splines are reviewed. Estimation of covariance parameters is achieved via maximum likelihood and residual maximum likelihood. A feature of the spatial regression approach is the visually appealing graphical summaries that are produced. These allow rapid and intuitive identification of process windows on the design space for which the response achieves target performance. Relevant design issues are briefly discussed and spatial designs, such as the packing designs available in Gosset, are suggested as a suitable design complement. The spatial regression models also perform well with no global design, for example with data obtained from series of designs on the same space of design variables. The approach is illustrated with an example involving the optimization of components in a DNA amplification assay. A Monte Carlo comparison of the spatial models with both thin plate splines and second-order polynomial response surfaces for a scenario motivated by the example is also given. This shows superior performance of the spatial models to the second-order polynomials with respect to both prediction over the complete design space and for cross-validation prediction error in the region of the optimum. An anisotropic spatial regression model performs best for a high noise case and both this model and the thin plate spline for a low noise case. Spatial regression is recommended for construction of response surfaces in all process optimization applications.     

Optimal designs with respect to Elfving's partial minimax criterion in polynomial regression

For the polynomial regression model on the interval [a, b] the optimal design problem with respect to Elfving's minimax criterion is considered. It is shown that the minimax problem is related to the problem of determining optimal designs for the estimation of the individual parameters. Sufficient conditions are given guaranteeing that an optimal design for an individual parameter in the polynomial regression is also minimax optimal for a subset of the parameters. The results are applied to polynomial regression on symmetric intervals [–b, b] (b1) and on nonnegative or nonpositive intervals where the conditions reduce to very simple inequalities, involving the degree of the underlying regression and the index of the maximum of the absolute coefficients of the Chebyshev polynomial of the first kind on the given interval. In the most cases the minimax optimal design can be found explicitly.Research supported in part by the Deutsche Forschungsgemeinschaft.Research supported in part by NSF Grant DMS 9101730.     

A local minimax characterization for computing multiple nonsmooth saddle critical points

This paper is concerned with characterizations of nonsmooth saddle critical points for numerical algorithm design. Most characterizations for nonsmooth saddle critical points in the literature focus on existence issue and are converted to solve global minimax problems. Thus they are not helpful for numerical algorithm design. Inspired by the results on computational theory and methods for finding multiple smooth saddle critical points in [14, 15, 19, 21, 23], a local minimax characterization for multiple nonsmooth saddle critical points in either a Hilbert space or a reflexive Banach space is established in this paper to provide a mathematical justification for numerical algorithm design. A local minimax algorithm for computing multiple nonsmooth saddle critical points is presented by its flow chart. Dedicated to Terry Rockafellar on his 70th birthday     

D-Optimal Designs for Trigonometric Regression Models on a Partial Circle

In the common trigonometric regression model we investigate the D-optimal design problem, where the design space is a partial circle. It is demonstrated that the structure of the optimal design depends only on the length of the design space and that the support points (and weights) are analytic functions of this parameter. By means of a Taylor expansion we provide a recursive algorithm such that the D-optimal designs for Fourier regression models on a partial circle can be determined in all cases. In the linear and quadratic case the D-optimal design can be determined explicitly.     

错误指定多元线性回归模型的Bayes估计

本文在错误指定下给出了多元线性模型的最优线性 Bayes估计 ,在矩阵损失下讨论了其相对于最小二乘法估计的优良性 ,且获得 Bayes估计的容许性和极小极大性     

Minimax and maximin efficient designs for estimating the location-shift parameter of parallel models with dual responses

Minimax designs and maximin efficient designs for estimating the location-shift parameter of a parallel linear model with correlated dual responses over a symmetric compact design region are derived. A comparison of the behavior of efficiencies between the minimax and maximin efficient designs relative to locally optimal designs is also provided. Both minimax or maximin efficient designs have advantage in terms of estimating efficiencies in different situations.     

Convex Optimal Designs for Compound Polynomial Extrapolation

The extrapolation design problem for polynomial regression model on the design space [–1,1] is considered when the degree of the underlying polynomial model is with uncertainty. We investigate compound optimal extrapolation designs with two specific polynomial models, that is those with degrees |m, 2m}. We prove that to extrapolate at a point z, |z| > 1, the optimal convex combination of the two optimal extrapolation designs | _m ^* (z), _2m ^* (z)} for each model separately is a compound optimal extrapolation design to extrapolate at z. The results are applied to find the compound optimal discriminating designs for the two polynomial models with degree |m, 2m}, i.e., discriminating models by estimating the highest coefficient in each model. Finally, the relations between the compound optimal extrapolation design problem and certain nonlinear extremal problems for polynomials are worked out. It is shown that the solution of the compound optimal extrapolation design problem can be obtained by maximizing a (weighted) sum of two squared polynomials with degree m and 2m evaluated at the point z, |z| > 1, subject to the restriction that the sup-norm of the sum of squared polynomials is bounded.     

Minimax second-order designs for difference between estimated responses in extrapolation region

Minimization of the variance of the difference between estimated response at two response at two points maximized over all pairs of points in the extrapolation region is taken as the criterion for selecting designs. Optimal designs under the criterion are derived for second-order models.     

Minimax lower bound for kink location estimators in a nonparametric regression model with long-range dependence

In this paper, a lower bound is determined in the minimax sense for change point estimators of the first derivative of a regression function in the fractional white noise model. Similar minimax results presented previously in the area focus on change points in the derivatives of a regression function in the white noise model or consider estimation of the regression function in the presence of correlated errors.     

Bias of the structural quasi-score estimator of a measurement error model under misspecification of the regressor distribution

In a structural measurement error model the structural quasi-score (SQS) estimator is based on the distribution of the latent regressor variable. If this distribution is misspecified, the SQS estimator is (asymptotically) biased. Two types of misspecification are considered. Both assume that the statistician erroneously adopts a normal distribution as his model for the regressor distribution. In the first type of misspecification, the true model consists of a mixture of normal distributions which cluster around a single normal distribution, in the second type, the true distribution is a normal distribution admixed with a second normal distribution of low weight. In both cases of misspecification, the bias, of course, tends to zero when the size of misspecification tends to zero. However, in the first case the bias goes to zero in a flat way so that small deviations from the true model lead to a negligible bias, whereas in the second case the bias is noticeable even for small deviations from the true model.     

Robust designs for models with possible bias and correlated errors

This paper studies the model-robust design problem for general models with an unknown bias or contamination and the correlated errors. The true response function is assumed to be from a reproducing kernel Hilbert space and the errors are fitted by the qth order moving average process MA（q）, especially the MA（1） errors and the MA（2） errors. In both situations, design criteria are derived in terms of the average expected quadratic loss for the least squares estimation by using a minimax method. A case is studied and the orthogonality of the criteria is proved for this special response. The robustness of the design criteria is discussed through several numerical examples.     

Error estimation properties of Gaussian process models in stochastic simulations

The theoretical relationship between the prediction variance of a Gaussian process model (GPM) and its mean square prediction error is well known. This relationship has been studied for the case when deterministic simulations are used in GPM, with application to design of computer experiments and metamodeling optimization. This article analyzes the error estimation of Gaussian process models when the simulated data observations contain measurement noise. In particular, this work focuses on the correlation between the GPM prediction variance and the distribution of prediction errors over multiple experimental designs, as a function of location in the input space. The results show that the error estimation properties of a Gaussian process model using stochastic simulations are preserved when the signal-to-noise ratio in the data is larger than 10, regardless of the number of training points used in the metamodel. Also, this article concludes that the distribution of prediction errors approaches a normal distribution with a variance equal to the GPM prediction variance, even in the presence of significant bias in the GPM predictions.     

Minimax second-order designs over cuboidal regions for the difference between two estimated responses

Minimization of the variance of the difference between estimated responses at two points, maximized over all pairs of points in the factor space, is taken as the design criterion. Optimal designs under this criterion are derived, via a combination of algebraic and numerical techniques, for the full second-order regression model over cuboidal regions. Use of a convexity argument and a surrogate objective function significantly reduces the computational burden.     

矩阵损失下一类相依回归模型中的线性容许估计和Minimax估计

本文研究设计矩人有相同值域的相依回归模型,在矩阵损失下我们给出了回归系数的线性估计是线性容许的充要条件,它们推广了已有的结果,我们也在矩阵上给出了某个回归模型的回归系数的唯一的Ｍｉｎｉｍａｘ估计,它说明此时其它模型的信息不起作用     

Excess of locally D-optimal designs and homothetic transformations

The paper is devoted to the study of homothety’s influence on the number of optimal design support points under fixed values of a regression model’s parameters. The Ayen–Peters two-dimensional nonlinear in parameters model used in analytical chemistry is considered. It is shown that the number of optimal design support points must be greater than or equal to the number of parameters depending on certain conditions. The optimal designs with the minimal number of support points are constructed explicitly. Some numerical methods for constructing designs with greater number of points (we suggest to call them excess designs) are used.     

A New Design Criterion When Heteroscedasticity is Ignored

This paper examines the construction of optimal designs when one assumes a homoscedastic linear model, but the underlying model is heteroscedastic. A criterion that takes this type of misspecification into account is formulated and an equivalence theorem is given. We also provide explicit optimal designs for single-factor and multi-factor experiments under various heteroscedastic assumptions and discuss the relationship between the D-optimal design sought here and the conventional D-optimal design.     

Robustness of Bayesian D-optimal design for the logistic mixed model against misspecification of autocorrelation

In medicine and health sciences mixed effects models are often used to study time-structured data. Optimal designs for such studies have been shown useful to improve the precision of the estimators of the parameters. However, optimal designs for such studies are often derived under the assumption of a zero autocorrelation between the errors, especially for binary data. Ignoring or misspecifying the autocorrelation in the design stage can result in loss of efficiency. This paper addresses robustness of Bayesian D-optimal designs for the logistic mixed effects model for longitudinal data with a linear or quadratic time effect against incorrect specification of the autocorrelation. To find the Bayesian D-optimal allocations of time points for different values of the autocorrelation, under different priors for the fixed effects and different covariance structures of the random effects, a scalar function of the approximate variance–covariance matrix of the fixed effects is optimized. Two approximations are compared; one based on a first order penalized quasi likelihood (PQL1) and one based on an extended version of the generalized estimating equations (GEE). The results show that Bayesian D-optimal allocations of time points are robust against misspecification of the autocorrelation and are approximately equally spaced. Moreover, PQL1 and extended GEE give essentially the same Bayesian D-optimal allocation of time points for a given subject-to-measurement cost ratio. Furthermore, Bayesian optimal designs are hardly affected either by the choice of a covariance structure or by the choice of a prior distribution.     

On the determination of the worst sampling error in a communication system for pulse-amplitude-modulated signals. I: The general model

We present a mathematical model of a communication system perturbed by statistical sampling errors (timing jitter). The aim is to find an ‘optimal’ impulse response for the system, the optimization problem actually being a minimax problem. that is we put the model into a game-theoretical framework. The basic game turns out to be a statistical game similar to those arising from estimation problems in statistics. Earlier results concerning least-favourable sampling error distributions published by Krabs and Vogel are supplemented by estimations of the number of support points of the least-favourable distribution. Furthermore, we state the existence of the saddle points of our game, which formerly has only been proved for some special cases. In the first part we treat the general situation, where one strategy set for the game—the set of all feasible impulse responses—forms a vector space with infinite dimension. In the second part we discuss the problem in the case that the impulse responses are restricted to a finite-dimensional subspace of the whole infinite-dimensional subspace.