Model validation, response surface

Iman RL, Helton JC, Campbell JE. An approach to sensitivity analysis of computer models Part II—Ranking of input variables, response surface validation, distribution effect and technique synopsis. / Quality Technol 1981 13 232-40. 

In reference 88, response surfaces from optimization were used to obtain an initial idea about the method robustness and about the interval of the factors to be examined in a later robustness test. In the latter, regression analysis was applied and a full quadratic model was fitted to the data for each response. The method was considered robust concerning its quantitative aspect, since no statistically significant coefficients occurred. However, for qualitative responses, e.g., resolution, significant factors were found and the results were further used to calculate system suitability values. In reference 89, first a second-order polynomial model was fitted to the data and validated. Then response surfaces were drawn for... 

Furthermore, optimal design theory assumes that the model is true within the region defined by the candidate design points, since the designs are optimal in terms of minimizing variance as opposed to bias due to lack-of-fit of the model. In reality, the response surface model is only assumed to be a locally adequate polynomial approximation to the truth it is not assumed to be the truth. Consequently, the experimental design chosen should reflect doubt in the validity of the model by allowing for model lack-of-fit to be tested. 

Response surface models are local Taylor expansion models which are valid only in the explored domain. It is often found that the stationary point on the response surface is remote from the explored domain and in the model may not describe any real phenomenon around the stationary point. Mathematically, a stationary point can be a maximum, a minimum, or a saddle point but it sometimes corresponds to unrealistic reponses (e.g. yield > 100%) or unattainable experimental conditions (e.g. negative concentrations of reactants). When the stationary point is outside the explored domain, the response surface is monotonous in the explored experimental domain and zx directions which correspond to small coefficients will describe rising or falling ridges. Exploring such ridges offers a means for optimizing the response even if the response surface should be oddly shaped. 

The polynomial approximation of / is a local model which is only valid in the explored experimental domain. It is not possible to extrapolate and draw any conclusions outside this domain. It is therefore important to determine a good experimental domain prior to establishing a response surface model with many parameters involved. By the methods of Steepest ascent. Chapter 10, and Sequential... 

Y block was augmented by the predicted optimum conditions from the response surface model, and the corresponding experimental yield. The model was recalculated and then used to predict the result with the bromo compound. Entry 7. Validation of the predictions by response surface modelling and experimental confirmation of optimum predicted by the response surface model is shown in Entry 7. Augmenting the X block and the Y block and recalculation of the PLS model afforded the predictions for p-methylthioacetophenone. An experimental yield of... 

Response Surface Methodology (RSM) is a well-known statistical technique (1-3) used to define the relationships of one or more process output variables (responses) to one or more process input variables (factors) when the mechanism underlying the process is either not well understood or is too complicated to allow an exact predictive model to be formulated from theory. This is a necessity in process validation, where limits must be set on the input variables of a process to assure that the product will meet predetermined specifications and quality characteristics. Response data are collected from the process under designed operating conditions, or specified settings of one or more factors, and an empirical mathematical function (model) is fitted to the data to define the relationships between process inputs and outputs. This empirical model is then used to predict the optimum ranges of the response variables and to determine the set of operating conditions which will attain that optimum. Several examples listed in Table 1 exhibit the applications of RSM to processes, factors, and responses in process validation situations. 

To demonstrate this we use the simple example that was introduced in chapter 4, that of solubility in a mixed surfactant system. The treatment is in two stages, the first being a intuitive rather than mathematical demonstration of testing for lack of fit and curvature of a response surface. Then, in section Il.B, we will carry out a more detailed, statistical analysis of the same process, showing how prediction confidence limits are calculated and the use of ANOVA in validating a model. 

Figure 9.14b showed a non-simplex domain fitting well inside an inverted f/-simplex, but two of the corners of the simplex were not only outside the domain but outside the ternary diagram. They are "imaginary" points with negative amounts of two of the components. This also can give matrices with rather better properties than if the model is defined in terms of the pure components. Here also the coefficients have no direct physical significance and it is only the prediction of the response surface within the domain that is valid. 

When (xi, X2) = (1,0), that is, when the mixture contains only component 1, Eq. (7.4) reduces to y = b =yi, where is the response observed for component 1 alone. In the same way, when (xi, X2) = (0,1) we havey = 62 = y2 -other words, the two coefficients of the additive model are equal to the response values obtained for the two pure components. If this model is valid, we will be able to predict the properties of any binary mixture without ever having to make an experiment on a binary mixture. This situation is represented geometrically in Fig. 7.3. The response surface, which is one-dimensional in this case, is simply the line joining yi to y2. The response for any mixture of composition (xi, X2) will be given by... 

Indeed, once the mathematical models are validated by classical statistical tools (e.gi, ANOVA, or analysis of the variance analysis of the lack of ht) (15). we can draw the response. surfaces representing t)ic evolution of the responses in the whole domain studied, when two factors are varying and the third one is fixed. From the different diagrams of isoresprmse curves, we can determine the influence of the different factors considered on the responses-... 

Modeling of the two paths to fit phosphate concentration data by solving the rate equations in an optimization of the rate constants reveals that depending on the initial guess, two different, but apparently valid, results may be achieved. Figure 3 is the response surface of the optimization plotted as a function of the rate constants kz and k. The response F k) is defined as... 

Statistical design of experiments (DOE) is often used in the early stages of process optimization. This is followed by a validation of the predictive model using actual plant production data. The response surface model described below captures the process performance window and shows the effect of changing composition and extruder screw speed on blend properties. A validation step can be easily implemented. 

The response surface design results are analysed by building, interpreting and validating an empirical model describing the relationship between responses and the studied factors. A second-order polynomial model is selected usually because frequently only two or three important factors are optimised, but for more factors the models would be similar. Equations (3.11) and (3.12) present the models for two x and x ) and three factors (xi, X2 and x ), respectively. 

The latest DoE was focussing on the axial and swirl stream temperature (Tswiri) the rotation speed (n) in a face-centred central composite (CCF) response surface design with those three factors (/=3) on three levels. Levels were set linearly as mentioned in section Improved Experimental Setup so that N = 2 +2/+1 = 15 experiments were required for this model. The centre point was repeated five times to ensure reproducibility and reasonable model validity. Particle size, span, particle shape, surface roughness, flowabDity and BET surface area were chosen as responses to evaluate the significant effects of the factors on these particle properties [34, 35]. 

If possible one should evaluate the significance of the regression coefficients as explained in Section 2.1.2 and eliminate from the model those considered nonsignificant. A new multiple regression is then performed with the simplified model. It is preferable also to validate the fit of the model and its prediction accuracy. The former is usually done by considering the residuals between the experimental and predicted responses, because this does not require replicate determinations as is needed for the ANOVA procedure. The validation of the prediction accuracy requires that additional experiments are carried out, which are then predicted with the model. The selection of the optimal conditions is often, but not necessarily, done with the aid of visual representation of the response surface, describing y as a function of pairs of variables. The final decision is often a multicriterion problem where multicriterion decision making techniques are applied (Section 5). 

Application of the URP method to the base-isolated building model with hysteretic responses has been shown. The validity of the URP method using second-order Taylor series approximatimi and RSM (response surface method) is demonstrated through the comparison with the other method (Monte Carlo Simulation (MCS)). 

The percolation model of adsorption response outlined in this section is based on assumption of existence of a broad spread between heights of inter-crystalline energy barriers in polycrystals. This assumption is valid for numerous polycrystalline semiconductors [145, 146] and for oxides of various metals in particular. The latter are characterized by practically stoichiometric content of surface-adjacent layers. It will be shown in the next chapter that these are these oxides that are characterized by chemisorption-caused response in their electrophysical parameters mainly generated by adsorption charging of adsorbent surface [32, 52, 155]. The availability of broad spread in heights of inter-crystalline barriers in above polycrystallites was experimentally proved by various techniques. These are direct measurements of the drop of potentials on probe contacts during mapping microcrystal pattern [145] and the studies of the value of exponential factor of ohmic electric conductivity of the material which was L/l times lower than the expected one in case of identical... 

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