Response Surface Exploration Research Articles

Response Surface Exploration for Elastic Modulus of Epoxy Concrete by Multi-Component Mixture Design

The concept of relating the physical characteristics of a multi-component composite with its component proportions involves the determination of a mathematical equation that adequately represents the response surface. The objective of this paper is to describe the shape of the response surface of elastic modulus of epoxy concrete over the simplex factor space (experimental region) with the help of the simplex-centroid design and the associated polynomial model and to determine the roles played by individual components. Epoxy concrete is a synthetic composite material specially developed for precision machine tool structures. The proposed model to which observations were fitted is the Scheffe's special cubic model for the simplex-centriod design. The adequacy of the fitted model was tested by both check points and F-test to ensure that it predicts a value at each point in the experimental region which is as close as possible to the true value of the response.

Dealing with emrqes in variables in response surface exploration

This paper is concerned with the analysis of data obtained from a designed experiment where the experimental design cannot be implemented exactly as planned, because errors in the levels of the variables cannot be avoided or measured. When the primary interest of the investigator lies In obtaining a satisfactory response surface model for the investigated relationship, the precision of the model estimates is essential for successful model building and accurate prediction of the response. An iterative procedure is proposed which estimates the effect of the variable in errors and obtains efficient weighted least squares estimates of the parameters of Interest.

Robust analysis of a response surface design

The paper discusses the outlier issue in experiments that are carefully planned for the purpose of response surface exploration. In such cases, a second order polynomial is fitted to the measurements in order to identify significant control variables, optimal settings of control variables, likely gains in the value of the response, etc. Outliers among the measurements cannot be avoided and will almost certainly have a highly confusing effect on the least-squares fit, leading to a wrong interpretation of the data. The paper discusses a practical example and uses it to exhibit two possible approaches. A combination of the least-squares technique with an expert knowledge of the design of the experiment can lead to valid interpretations by highlighting the trouble spots. The other possible approach uses robust fitting techniques which are much less easily fooled by outliers and automatically account for them. The second approach is better suited to casual users of response surface methodology.

UNCERTAINTY ANALYSIS OF THE WEPP SOIL EROSION MODEL

ABSTRACT Predictions from hydrologic and erosion models contain a large degree of uncertainty. The Modified Point Estimate Method (Harr, 1989) used in conjunction with a response surface exploration technique (Brooks, 1958) provides a simple, computationally efficient, and powerful tool for evaluating uncertainty of predictions by natural-resource models. The method allows analysis of models with a large number of input parameters which may be correlated and for which the exact input parameter distribution is unknown. The method was applied to the Water Erosion Prediction Project single rainfall-event erosion model. Sixty treatment combinations were selected to determine WEPP output uncertainties for a wide range of soil, crop, management, topographic, and storm conditions. The levels of the treatment combinations were randomly selected to span the entire factorial space of the 28 WEPP inputs, but with a finite number of treatment combinations. Five WEPP outputs were studied: peak runoff rate, average soil loss, average deposition, sediment yield, and sediment specific surface enrichment ratio. Maximum and average output uncertainties, given by the coefficient of variation, were determined for each output of the 60 treatments. Maximum coefficients of variation for peak runoff rate, soil loss, sediment yield, and sediment enrichment ratio were 196, 267, 323, and 47%, respectively. Average coefficients of variation for the same set of variables were 65, 99, 106, and 13%, respectively. Coefficient of variation was less for larger runoff and erosion events, which account for a large percentage of the total soil loss at a location over extended time periods. Significant, positive correlations existed between the coefficients of variation of peak runoff average soil loss, and average soil loss and sediment yield, indicating that the uncertainty in average soil loss and in sediment yield may be directly related to the uncertainty in peak runoff rate.

The Design of Experiments: Statistical Principles for Practical Application

Preface Part I. Overture: 1. Introduction 2. Elementary ideas of blocking: the randomised block design 3. Elementary ideas of treatment structure 4. General principles of linear models for the analysis of experimental data 5. Computers for analysing experimental data Part II. First Subject: 6. Replication 7. Blocking 8. Multiple blocking systems and cross-over designs 9. Randomisation 10. Covariance - extension of linear models 11. Model assumptions and more general models Part III. Second Subject: 12. Experimental objectives, treatments and treatment structures 13. Factorial structure and particular forms of effects 14. Split unit designs and repeated measurements 15. Incomplete bloxk size for factorial experiments 16. Some mathematical theory for comfounding and fractional replication 17. Quantitative factors and response functions 18. Response surface exploration Part IV. Coda: 19. Designing useful experiments References Index.

Some aspects of response surface theory, and the philosophy of statistical experimental design

This paper examines the problem of exploration of response surfaces. A discussion of the significant elements of design theory (and some fundamental issues) is presented in the context of the above problem. A sequential technique for fitting a polynomial surface of unknown degree is illustrated.

Chemometrics: A Textbook

1. Chemometrics and the Analytical Process. 2. Precision and Accuracy. 3. Evaluation of Precision and Accuracy. Comparison of Two Procedures. 4. Evaluation of Sources of Variation in Data. Analysis of Variance. 5. Calibration. 6. Reliability and Drift. 7. Sensitivity and Limit of Detection. 8. Selectivity and Specificity. 9. Information. 10. Costs. 11. The Time Constant. 12. Signals and Data. 13. Regression Methods. 14. Correlation Methods. 15. Signal Processing. 16. Response Surfaces and Models. 17. Exploration of Response Surfaces. 18. Optimization of Analytical Chemical Methods. 19. Optimization of Chromatographic Methods. 20. The Multivariate Approach. 21. Principal Components and Factor Analysis. 22. Clustering Techniques. 23. Supervised Pattern Recognition. 24. Decisions in the Analytical Laboratory. 25. Operations Research. 26. Decision Making. 27. Process Control. Appendix. Subject Index.

Automated exploration and exploitation of flow-injection response surfaces

Three-dimensional plots of instrumental responses vs. chemical concentrations or flow parameters have been 1 obtained in an automated manner on a computer-controlled flow-injection methods development system. Consideration of several alternative responses for flow-injection systems is shown to help characterize a chemistry more thoroughly and reveal the best experimental conditions. One may see the effects of individual experimental variables (reagent concentrations, pH, flow-rates, etc.), the interactions of these variables, instrumental factors and limitations of the surface exploration procedure employed. Chemical systems studied were the photometric determination of phosphate, palladium(II), iron(II) and persulfate. The propriety of automated response surface mapping is demonstrated and the efficacies of simplex and grid search approaches to response surface exploration are contrasted. Responses obtained include absorbance at peak maximum, relative standard deviation of maximum absorbance, time from injection to peak maximum and wavelength of maximum absorbance. Higher dimensional response surface representations of peak shape and absorbance spectra are also presented. The results show that the response chosen governs the general shape of the surface and the height at any point. This approach to automated characterization of chemical reactions in flow analysis is critically assessed.

Analyzing Optima in the Exploration of Multiple Response Surfaces

A method is presented for testing the equality of some or all (constrained or unconstrained) optima in a response surface analysis. An estimator of a common location of stationary points is obtained by a standard multivariate testing procedure and a confidence region associated with the common optimum is derived. The procedure is illustrated by the estimation of a common optimum in a multiple response experiment. The multivariate approach facilitates a more efficient estimation of the optimum than the usual univariate response surface analysis and provides additional tests of the response surface model.

Response surface exploration with spline functions: biased regressors

Nell'articolo si mostra come una superficie di risposta possa essere approssimata in un punto con una funzione spline polinomiale di primo ordine con k vincoli su ogni asse. Il vettore gradiente della superficie e poi stimato sui parametri del regressore spline in modo tale da minimizzare l'errore quadratico medio dello stimatore. Questo da luogo alla formulazione di un nuovo algoritmo per l'ottimizzazione delle superfici di risposta sia segmentate che non segmentate. Confronti numerici fra la procedura iterativa con spline con il metodo Speepest Ascent, con il metodo di Quasi-Newton e con il metodo delle direzioni coniugate mostrano che il metodo spline e spesso piu efficiente.

Sequential Exploration in Mixture Experiments

AbstractIn this paper sequential exploration of response surfaces for mixture experiments has been studied. The plan and the estimation and testing procedures for some of the well known models have been given.

Design of Experiments -- A Realistic Approach.

Review of Some Basic Statistical Concepts Some Intermediate Data Analysis Concepts A Scientific Approach to Experimentation Completely Randomized Design (CRD) Randomized Complete Block Design (RCBD) Nested (Hierarchical) and Nested Factorial Designs Split Plot Type Design Latin Square Type Designs 2n Factorial Experiments (Complete and Incomplete Blocks) Fractional Factorial Experiments for Two-Leveled Factors Three-Level Factorial Experiments Mixed Factorial Experiments and Other Incomplete Block Designs Response Surface Exploration Appendices

Kiss-Precise Sequential Rotatable Designs

SummaryA sequential procedure for the exploration of response surfaces is proposed. The procedure, which is for experiments with two factors, uses the kiss-precise configuration, i.e., the design points are on circles in mutual contact at each stage. Only three points need be added at each stage and the design points form a first-order rotatable design. A second-degree surface may be fitted when a near stationary region is reached.

A Theory of the Spatial Structure of Internal Trade In Underdeveloped Countries

A Theory of the Spatial Structure of Internal Trade In Underdeveloped Countries

Construction of Rotatable Designs Through Balanced Incomplete Block Designs

Rotatable designs were introduced by Box and Hunter (1954, 1957) for the exploration of response surfaces. They constructed these designs through geometrical configurations and obtained several second order designs. Afterwards, Gardiner and others (1959) obtained some third order designs through the same technique for two and three factors and a third order design for four factors. Bose and Draper (1959) obtained some second order designs by using a different method. Draper (1960 a) gave a method of construction of an infinite series of second order designs in three and more factors. Recently, Box and Behnken (1960 a) have obtained a class of second order rotatable designs from those of first order. Draper (1960 b) has obtained some third order rotatable designs in three dimensions and a third order rotatable design in four dimensions. Das (1961) has obtained such designs, both second and third orders up to 8 factors as fractional replicates of factorial designs. The method of construction of the designs presented in this paper is essentially based on that presented by Das (1961). After the manuscript of this paper was submitted for publication the authors' attention was drawn to the work of Box and Behnken (1960 b). They have obtained some second order designs by following a procedure which uses balanced incomplete block designs in the same manner as described below. They did not, however, extend the method to include other complementary sets of points which, as will be shown, allow one to obtain rotatable second and third order designs based on any balanced incomplete block design. In the present paper a method has been given by using the properties of balanced incomplete block designs through which second order rotatable designs with any number of factors, with a reasonably small number of points, can be obtained. By extending the method, third order rotatable designs, both sequential and nonsequential, up to 15 factors have been obtained with the help of doubly balanced incomplete block designs and complementary B.I.B. designs.

“Design and analysis of industrial experiments”*

SummaryDesign and analysis of industrial experiments This paper is based on a lecture on the “Design and Analysis of Industrial Experiments” given by Dr O. L. Davies on the 8th of May 1954 to the Industrial Section of the “Vereniging voor Statistiek”. Production, formulation, and testing are distinguished as three separate fields of chemical activity where experimental designs can be applied, and various numerical examples of such experiments are discussed in detail. They consist of a 23factorial design, a 24half replicate and a 25quarter replicate fractional factorial design, and a three‐way classification. In a final section the recent designs developed by Box for the exploration of response surfaces are briefly considered.

The Exploration and Exploitation of Response Surfaces: An Example of the Link between the Fitted Surface and the Basic Mechanism of the System

This is a sequel to an article which recently appeared in this journal [1] and had the same general title. The previous article described a number of applications of newly developed techniques [2] for the study of response surfaces. The present article shows how study of the form of the empirical surface can throw important light on the basic mechanism operating and can thus make possible developments in the fundamental theory of a process. This idea is illustrated in some detail with an example previously discussed only from the empirical standpoint. A theoretical surface, based on reaction kinetics is now derived, rate constants are estimated from the data and the theoretical surface is compared with the empirical surface previously obtained. It is then shown how the canonical variables of the empirical surface can relate to the basic physical laws controlling the system. In this connection the problem of suitable choice of metrics for the variables is discussed. In a final section some general remarks on the process of scientific investigation are appended.

Design and Analysis of Industrial Experiments.

Summary Design and analysis of industrial experiments This paper is based on a lecture on the “Design and Analysis of Industrial Experiments” given by Dr O. L. Davies on the 8th of May 1954 to the Industrial Section of the “Vereniging voor Statistiek”. Production, formulation, and testing are distinguished as three separate fields of chemical activity where experimental designs can be applied, and various numerical examples of such experiments are discussed in detail. They consist of a 23factorial design, a 24half replicate and a 25quarter replicate fractional factorial design, and a three-way classification. In a final section the recent designs developed by Box for the exploration of response surfaces are briefly considered.

The Exploration and Exploitation of Response Surfaces: Some General Considerations and Examples

The Exploration and Exploitation of Response Surfaces: Some General Considerations and Examples