Empirical models, response surface

Theoretical Models of the Response Surface Mathematical models for response surfaces are divided into two categories those based on theory and those that are empirical. Theoretical models are derived from known chemical and physical relationships between the response and the factors. In spectrophotometry, for example, Beer s law is a theoretical model relating a substance s absorbance. A, to its concentration, Ca... 

Empirical Models of the Response Surface In many cases the underlying theoretical relationship between the response and its factors is unknown, making impossible a theoretical model of the response surface. A model can still be developed if we make some reasonable assumptions about the equation describing the response surface. For example, a response surface for two factors, A and B, might be represented by an equation that is first-order in both factors... 

The terms Po, Pa, Pt, Pat, Paa, and Pt,t, are adjustable parameters whose values are determined by using linear regression to fit the data to the equation. Such equations are empirical models of the response surface because they have no basis in a theoretical understanding of the relationship between the response and its factors. An empirical model may provide an excellent description of the response surface over a wide range of factor levels. It is more common, however, to find that an empirical model only applies to the range of factor levels for which data have been collected. 

To develop an empirical model for a response surface, it is necessary to collect the right data using an appropriate experimental design. Two popular experimental designs are considered in the following sections. 

Equation 14.9 gives the empirical model of the response surface for the data in Table 14.4 when the factors are in coded form. Convert the equation to its uncoded form. 

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

The following set of experiments provides practical examples of the optimization of experimental conditions. Examples include simplex optimization, factorial designs used to develop empirical models of response surfaces, and the fitting of experimental data to theoretical models of the response surface. 

The resulting data of the Box-Behnken design were used to formulate a statistically significant empirical model capable of relating the extent of sugar 3deld to the four factors. A commonly used empirical model for response surface analysis is a quadratic polynomial of the type... 

G. E. P. Box and N. R. Draper, Empirical Model Building and Response Surfaces, John Wiley Sons, New York, 1987. 

A complete list of the reaction conditions tested for this response surface design can be found in [76], The center point reaction condition was repeated six times. This was done to measure the variability of the reaction system. Also, the space velocity is kept constant, as it was the least important factor predicted by screening design, for all the reaction conditions. The purpose of this nested response surface design was to develop an empirical model in the form of Eqn (5) to relate the five reaction condition variables and the three catalyst composition variables to the observed catalytic performance. 

It is more commonly the case that all points on the response surface are not known instead, only a few values will have been obtained and the information will give an incomplete picture of the response surface, such as that shown in Figure 2.3. Common practice is to assume a functional relationship between the response and the factor (that is, to assume a model, either mechanistic or empirical) and find the values of the model parameters that fit the data. If a model of the form... 

Answering this question requires some knowledge of the response surface in the region around 85 C. We will assume that experiments have been carried out in this region for each process, and that the response surfaces shown in Figures 2.17 and 2.18 are good approximations of the true behavior of the system. Empirical models that adequately describe the dependence of yield on temperature in the region of factor space near 85 °C are... 

Assume the two inputs you chose in Problem 12.1 will exhibit factor interaction. If they would be expected to interact, what would be the mechanistic basis of that interaction What might be the approximate mathematical form of that interaction Would you use a mechanistic or empirical model to approximate the response surface Why Sketch the response surface predicted by this model. 

With the use of such an empirical model, the crushing strength can be optimized with respect to the relative amounts of excipients. The empirical model above can be represented as a surface in the 3-D space, where the axis are x, X2 and CR. Hence, this methodology is called response surface methodology. Good textbooks have appeared in the area of RSM [14,16]. 

G.E.P. Box, Draper N.R., Empirical model-building and response surfaces, John... 

A paper by Box, G.E.P., and Youle, P.V. (1955). The Exploration and Exploitation of Response Surfaces An Example of the Link Between the Fitted Surface and the Basic Mechanism of the System, Biometrics, 11, 287, explores the relationships between an empirical model and a fundamental mechanism. In their Section 9, they discuss some aspects of the process of scientific investigation. What do they perceive as the relationships among experiment, theory, and knowledge ... 

A researcher is therefore recommended to use the design of experiments or to achieve an optimum in an experimental way. A researcher who designs an experiment does not know beforehand where in the studied response surface the optimum is located and what the shape of the surface is. Therefore he uses two approaches to reach the optimum. By one approach, he approximates in the given experimental region his experimental data by an assumed empirical model, or fits the response surface to the degree of the needed polynomial accuracy. Based on such an analytical model, he performs analytical optimization. Reaching an optimum in this case is more efficient if the obtained analytical model is adequate. By another approach, the researcher does not form an analytical model, but he does his experiments iteratively by prior established rules until he reaches the optimum. 

Andersson developed a semi-empirical model for the charge distribution around the (V=0) bonds in V205, V6013, and V02.73 The surfaces of the lower oxides were treated, upon the basis of ESCA results discussed on p. 107, as being in an oxidized state, which is proposed to be the case under the usual conditions in (amm) oxidation reactions. The main result is that 02-in the form of (V=0) groups is responsible for the catalytic oxidation of hydrocarbons. 

Response Surface Methodology (RSM) is a statistical method which uses quantitative data from appropriately designed experiments to determine and simultaneously solve multi-variate equations (3). In this technique regression analysis is performed on the data to provide an equation or mathematical model. Mathematical models are empirically derived equations which best express the changes in measured response to the planned systematic... 

Sukigara, S., Gandhi, M., Ayutsede, J., Micklus, M., and Ko, F. "Regeneration of Bombyx mori silk by electrospinning. Part 2. Process optimization and empirical modeling using response surface methodology". Polymer 45(11), 3701-3708 (2004). 

A potential concern in the use of a two-level factorial design is the implicit assumption of linearity in the true response function. Perfect linearity is not necessary, as the purpose of a screening experiment is to identify effects and interactions that are potentially important, not to produce an accurate prediction equation or empirical model for the response. Even if the linear approximation is only very approximate, usually sufficient information will be generated to identify important effects. In fact, the two-factor interaction terms in equation (1) do model some curvature in the response function, as the interaction terms twist the plane generated by the main effects. However, because the factor levels in screening experiments are usually aggressively spaced, there can be situations where the curvature in the response surface will not be adequately modeled by the two-factor interaction... 

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