Conditions of Slope-Rotatability for Third-Order Polynomial Regression Models

In experimental design for response surface analysis, it is sometimes of interest to estimate the difference of responses at two points. If differences at points close together are involved, the design that reliably estimates the slope of the response surface is important. In particular, Hader and Park (1978 Hader , R. J. , Park , S. H. ( 1978 ). Slope-rotatable central composite designs . Technometrics 20 : 413 – 417 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) suggested the concept of slope-rotatability and studied slope rotatable central composite designs. Until now, many response surface designs including central composite designs have been suggested for fitting second order response surface models. However, we often need to fit third-order polynomial regression models. In this article, we suggest extended central composite designs (ECCDs) to fit third-order models and find the necessary and sufficient conditions for slope-rotatability over all directions in the third-order polynomial models.     

Experimental design for response surface gradient estimation

In this paper, a new experimental design and gradient estimation procedure is presented for Phase I response surface optimization. The design is motivated by basic principles of differential calculus, which imply that if a point in Rⁿ has been reached by exactly minimizing a function along a given direction, then the gradient of the function at that point must be orthogonal to the search direction followed. While exact line search is not required for the new design to be effective, this principle implies that the dimension of the gradient estimation procedure may often be reduced from n to n-1 variables, and the experimenter is able to concentrate experimental effort within the most productive region around the center of the design. The new design and gradient estimation procedures are presented, and bias and variance properties are derived. The effectiveness of the new design is shown to depend on the experimenter's ability to terminate line search within a near-stationary region of the line search function A simple heuristic is presented which indicates whether the new design should be used at a given experimental region.     

Characterizing projected designs: repeat and mirror-image runs

Two-level designs are useful to examine a large number of factors in an efficient manner. It is typically anticipated that only a few factors will be identified as important ones. The results can then be reanalyzed using a projection of the original design, projected into the space of the factors that matter. An interesting question is how many intrinsically different type of projections are possible from an initial given design. We examine this question here for the Plackett and Burman screening series with N= 12, 20 and 24 runs and projected dimensions k≤5. As a characterization criterion, we look at the number of repeat and mirror-image runs in the projections. The idea can be applied toany two-level design projected into fewer dimensions.     

Response surface model diagnosis in two-way tables

Graphical model diagnosis procedures for two-way tables have already been proposed by Mandel, as well as by Bradu and Gabriel who used Gabriel's biplot as a main tool.

In this paper, a graphical model diagnosis extending Mandel's approach is developed. Response surface models are obtained whereby the response is expressed in terms of ‘truth-connected’ latent variates and, ultimately, in terms of originally measured external variates.

Meaningful models are obtained in some cases, accurate smoothing and interpolation algorithms in others.

As a by–product, Euclidean maps, which represent a twodimensional scaling (for rows or columns) also displaying ,ANOVA features, are obtained. To an extent, these maps can be viewed as a substitute for a model in the event of partial failure of the modelling operation.     

On choosing between fixed and random block effects in some no-interaction models

The purpose of this article is to strengthen the understanding of the relationship between a fixed-blocks and random-blocks analysis in models that do not include interactions between treatments and blocks. Treating the block effects as random has been recommended in the literature for balanced incomplete block designs (BIBD) because it results in smaller variances of treatment contrasts. This reduction in variance is large if the block-to-block variation relative to the total variation is small. However, this analysis is also more complicated because it results in a subjective interpretation of results if the block variance component is non-positive. The probability of a non-positive variance component is large precisely in those situations where a random-blocks analysis is useful – that is, when the block-to-block variation, relative to the total variation, is small. In contrast, the analysis in which the block effects are fixed is computationally simpler and less subjective. The loss in power for some BIBD with a fixed effects analysis is trivial. In such cases, we recommend treating the block effects as fixed. For response surface experiments designed in blocks, however, an opposite recommendation is made. When block effects are fixed, the variance of the estimated response surface is not uniquely estimated, and in practice this variance is obtained by ignoring the block effect. It is argued that a more reasonable approach is to treat the block effects to be random than to ignore it.     

COMBINING MULTIPLE MARKERS FOR MULTI‐CATEGORY CLASSIFICATION: AN ROC SURFACE APPROACH

The Receiver Operating Characteristic (ROC) curve and the Area Under the ROC Curve (AUC) are effective statistical tools for evaluating the accuracy of diagnostic tests for binary‐class medical data. However, many real‐world biomedical problems involve more than two categories. The Volume Under the ROC Surface (VUS) and Hypervolume Under the ROC Manifold (HUM) measures are extensions for the AUC under three‐class and multiple‐class models. Inference methods for such measures have been proposed recently. We develop a method of constructing a linear combination of markers for which the VUS or HUM of the combined markers is maximized. Asymptotic validity of the estimator is justified by extending the results for maximum rank correlation estimation that are well known in econometrics. A bootstrap resampling method is then applied to estimate the sampling variability. Simulations and examples are provided to demonstrate our methods.     

SAW滤波器等效电路模型参数的精确计算法

基于声表面波滤波器中叉指换能器（ＩＤＴ）上电荷分布特性，结合优化设计的叉指加权系数及加权模式，推导出了ＳＡＷ滤波器等效电路模型参数的精确计算方法。其计算表达式仅由加权系数，叉指间相互位置及电荷分布决定。模拟结果表明此算法精确、简便有效     

圆弧车刀廓形的作图设计

从作图设计的观点出发，通过对成形车刀前南、后角、刀倾角及主偏角变化的分析，阐明了圆弧车刀廓形的作图设计方法．     

整体波场延拓法基准面校正

提出了一种新的近地表问题解决方法——整体波场延拓法基准面校正(WEDD)。借助地震理论中的互换原理和观测面沉降概念,在基准面校正过程中将炮点和检波点的处理统一起来,并利用炮点-全偏移距域双平方根算子实现波场的单域整体延拓,最后提取目标深度的波场值完成参考面的重新定义。模型试算结果表明了WEDD方法的有效性。     

Estimation of Acute Toxicity by Fitting a Dose-Time-Response Surface

In acute toxicity testing, organisms are continuously exposed to progressively increasing concentrations of a chemical and deaths of test organisms are recorded at several selected times. The results of the test are traditionally summarized by a dose-response curve, and the time course of effect is usually ignored for lack of a suitable model. A model which integrates the combined effects of dose and exposure duration on response is derived from the biological mechanisms of aquatic toxicity, and a statistically efficient approach for estimating acute toxicity by fitting the proposed model is developed in this paper. The proposed procedure has been computerized as software and a typical data set is used to illustrate the theory and procedure. The new statistical technique is also tested by a data base of a variety of chemical and fish species.