Full-quadratic model

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... 

The three empirical models could be used as response surfaces, and from the simplest to the most complex, these were the additive (first-order) model, the interactive model, and the full-quadratic (second-order) model. The additive and interactive models could be considered as special cases of the full-quadratic model. 

The full quadratic model in two factors was, Y =6q-f6j X -f X +b X +b X, which has two added second order, or curvature terms added for each of the two factors. This... 

The following full quadratic model containing six coefficients was used to describe the responses, where b are the regression coefficients given by the model ... 

We take first the full quadratic model and calculate the lines of equal response. As there are 3 variables, the factor space is in 3 dimensions, and to represent this on a 2-dimensional paper surface or computer screen, we need to take slices with one of the variables held constant. Here the polysorbate concentration, X, was chosen as the fixed variable. Taking x, = -1, 0, +1 (real values 3.7, 4.0, 4.3% polysorbate), lines of equal turbidity are plotted in the X2, X plane (propylene glycol, sucrose invert medium), as shown in figure 5.7. 

We need a better model, and the next step, naturally, is to include proper quadratic terms, in addition to the cross (interaction) terms already in the model. To have a sufficient number of degrees of freedom, we must include levels other than +1 in the training set runs. Using all 52 runs in the table to fit a full quadratic model, we arrive at the equation... 

Linear and quadratic models were fitted to the degradation results of all herbicides. For tebuthiuron, the linear model does not show significant lack of fit but does not have any significant coefficients either. The full quadratic model also shows no significant lack of fit but the quadratic term on the ferrioxalate concentration is statistically significant at the 95% confidence level. These calculations are left to the interested reader. Here, we shall discuss only the results obtained for the other two herbicides. 

Exercise 7.6. Do a -test to check if the coefficients of the full quadratic model for the anal3d ical signals of the membrane study are significant. 

Fit the full second-order (quadratic) model to the data. 

The responses of nisin and lactic acid were correlated by nonlinear regression using the following full quadratic polynomial model ... 

The most evident design would appear to be a three-level factorial design. An example of a three-level factorial design is shown in Fig. 6.1.. A full three-level factorial design, 3, can be used to obtain quadratic models. However, unless / is small (/ = 2) the design requires a number of experiments (= V) that is not often feasible. An example of the use of a 3" design can be found in Ref. 51]. The two factors, the pH and the percentage acetonitrile were examined to evaluate their influence on the retention and resolution of three isoxazolyl penicillin antibiotics. The centre point experiment was triplicated to evaluate experimental error. 

As explained in Section 6.2 simple empirical models such as those of Eq. (6.1) and Eq. (6.2) are usually applied. They can be easily generalized to more than two variables. Usually not all possible terms are included. For instance, when including three variables one could include a ternary interaction (i.e. a term in. vi.vi.vy) in Eq. (6.1) or terms with different exponents in Eq. (6.2). such as. vi.v , but in practice this is very unusual. The models are nearly always restricted to the terms in the individual variables and binary interactions for the linear models of Eq. (6.1), and additionally include quadratic terms for individual variables for the quadratic models of Eq. (6.2). To obtain the actual model, the coefficients must be computed. In the case of the full factorial design, this can be done by using Eq. (6.5) and dividing by 2 (see Section 6.4.1). In many other applications such as those of Section 6.4.3 there are more experiments than coefficients in the model. For instance, for a three-variable central composite design, the model of Eq. (6.2)... 

The model predictive control used includes all features of Quadratic Dynamic Matrix Control [19], furthermore it is able to take into account soft output constraints as a non linear optimization. The programs are written in C++ with Fortran libraries. The manipulated inputs (shown in cm Vs) calculated by predictive control are imposed to the full nonlinear model of the SMB. The control simulations were made to study the tracking of both purities and the influence of disturbances of feed flow rate or feed composition. Only partial results are shown. 

Figure 6.51 Cell-surviving fractions Sf(D) as a function of radiation dose D in Gy (top panel (i)). Bottom panel (ii), as the Fe-plot (Full effect), shows the so-called reactivity R(D) given by product of the reciprocal dose and the negative natural logarithm of Sf(D), as the ordinate versus D as the abscissa. Any departure of experimental data from a straight line indicates inadequacy of the LQ model for the Fe-plot. Experiment (symbols) [60] the mean clonogenic surviving fractions Sf(D) (panel (i)) and R(D) = —(1/D)ln(SF) (panel (ii)) for the asynchronous V79 Chinese hamster cells irradiated hypoxically by 250 kVp X-ray with a concurrent 30 min exposure to the sulfhydryl-binding agent, N-ethylmaleimide, of low-concentration 0.75 //M. Theories (present computations) solid curve - PLQ (Pad linear-quadratic) model and dotted curve - LQ model (the straight-line a + pD on the bottom panel).

CCD is one of the most common designs generally used in response surface modeling, which allows for the determination of both linear and quadratic models. Full uniformly routable CCDs present the following characteristics ... 

Although the design was constructed according to Taguchi s method, it is in fact a full factorial 2 3 design, which may be analysed in the usual way. The authors of the original article showed the necessity of introducing quadratic terms in and Xj. The postulated model therefore contains the 6 terms of the quadratic model, the... 

The coefficients of the statistically significant terms of the fitted models for the five responses are given in Table 7A.8. The special cubic model is the one that best fits all properties except cohesivity, for which the quadratic model is sufficient. Of all fitted models, only that for firmness shows some lack of fit, but the number of level combinations in the design is not sufficient for fitting a full cubic model. The coefficient values clearly indicate that starch (xi) is the most important component, but its effect depends on the proportions of the other two ingredients, sugar and powdered milk. 

In the experiments and studies carried out, a full quadratic empirical model including all main effects, second-order effects, and two- and three-way interactions were accounted via multiple linear regression analysis to determine the properties. Terms that were not significant in the model (<95 % confidence) were eliminated one at a time. The general empirical model for all the properties evaluated is given in Eq. (5.4). ANOVA present simple empirical formulation of the properties to be... 

Statistical design of experiment (DOE) is an efficient procedure for finding the optimum molar ratio for copolymers having the best property profile. Based on the concepts of response-surface (RS) methodology, developed by Box and Wilson [11], there are four models or polynominals (Table III) useful in our study. For three components, in general, if there are seven to nine experimental data points, the linear, quadratic and special cubic will be applicable for use in predictions. If there are ten or more data points, the full cubic model will also be applicable. At the start of the effort, one prepares a fair number of copolymers with different AA IA NVP ratios and tests for a property one wishes to optimize, with the data fit to the statistical models. Based on the models, new copolymers, with different ratios, are prepared and tested for the desired property improvement. This type procedure significantly lowers the number of copolymers that needs to be prepared and evaluated, in order to identify the ratio needed to give the best mechanical property. 

The optimisation of the parameters affecting Cd(II) adsorption by the adsorbent was performed with a 2 full factorial design. A Doehlert matrix was built to obtain an optimum set of experimental conditions from a quadratic model. 

It is obvious that to be able to estimate the quadratic coefficients, p, in equation (14), it is necessary to have at least three distinct levels or settings for the variables. This suggests that a suitable design for estimating the coefficients of the second-order model would be a single replicate of a three-level full factorial design in p variables. This design... 

If there are only p=2, or p=3 variables then a full factorial design is often feasible. However, the number of runs required becomes prohibitively large as the number of variables increases. For example, with p=5 variables, the second-order model requires the estimation of 21 coefficients the mean, five main effects, five pure quadratic terms, and ten two-factor interactions. The three-level full factorial design would require... 

In the remainder of this section it is desired to obtain the relative, constant-pressure heat capacity of the liquid at x=j and the concentration fluctuation factor for all compositions. Since the latter equation is complicated, it is not written out in full here. This has been done in Eqs. (37)-(45) of the paper by Liao et al. (1982) for the special case that 14 = 34 = 0 and / l3 is the only nonzero cubic interaction term, i.e., the version of the model applied here to the Ga-Sb and In-Sb binaries. Bhatia and Hargrove (1974) have given equations for the composition fluctuation factor at zero wave number for the special cases of complete association or dissociation and only quadratic interaction coefficients. 

The full factorial central composite design includes factorial points, star points, and center points. The corresponding model is the complete quadratic surface between the response and the factors, as given by Eq. 1 ... 

FIGURE 8.8 Examples of simplex lattices for (a) linear, (b) quadratic, (c) full cubic, and (d) special cubic models. 

The value of the hyperparameter jt may be chosen by considering the prior expected number of active effects. Illustrative calculations are now given for a full second-order model with / factors, and for subsets of active effects that include linear and quadratic main effects and linear x linear interactions. Thus, the full model contains / linear effects, / quadratic effects, and ( ) linear x linear interaction effects. Prior probabilities on the subsets being active have the form of (22) and (23) above. A straightforward extension of the calculations of Bingham and Chipman (2002) yields an expected number of active effects as... 

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