Canonical Analysis of the Response surface

Having obtained an adequate mathematical second-order model of a research subject for k 3, we can obtain a geometric interpretation in a two- or three-dimensional space. To obtain this interpretation, it is necessary to transform the second-order model into a typical-canonical form. Canonic transformation of a regression model is terminated in the form  

For k 3, it is possible, after canonic transformation, to determine to which type of geometric surface the given equation corresponds. It is known from mathematical analysis that there exist 17 second-order surfaces of standard form. By a canonic regression model, we can make this sorting by extreme types  

When the response surface has an extreme, then all coefficients of a canonic equation have the same signs and the center of the figure is close to the center of experiment. A saddle-type surface has a canonic equation where all coefficients have different signs. In a crest-type surface some canonic equation coefficients are insignificant and the center of the figure is far away from the center of experiment. To obtain a surface approximated by a second-order model for two factors, it is possible to get four kinds of contour curves-graphs of constant values  

Canonic transformation of a regression corresponds to transfer of coordinate beginning into a new point S and to replacement of the old coordinate axes (X, 

The next phase is characterized by rotating coordinate axes with reference to the old ones. 

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Example. Canonical analysis of the response surface model of the enamine synthesis... 

The synthesis of 2-trimethylsilyoxy-l,3-butadiene by treatment of methyl vinyl ketone with chlorotrimethylsilane, lithium bromide and triethylamine in tetrahydrofuran was discussed in section 12.5.6. It was discussed how the stoichiometry of the reaction was determined by canonical analysis of the response surface model, and how this analysis made it possible to establish experimental conditions which afforded a quantitative conversion. However, before the response surface model could be established it was necessary to find a reaction system worth optimizing. 

A worked-out example of the above technique was given in Chapter 12 (pp. 276-280) in the context of canonical analysis of a response surface model. 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

Perform a canonical analysis on the fitted equation y = 5.13 + O.lblXi, - 0.373x + 0.517x, - 1.33x i - 0.758xi,X2,.. What are the coordinates of the stationary point What are the characteristics of the response surface in the region of the stationary point (see Table 12.3) ... 

Canonical analysis achieves this geometric interpretation of the response surface by transforming the estimated polynomial model into a simpler form. The origin of the factor space is first translated to the stationary point of the estimated response surface, the point at which the partial derivatives of the response with respect to all of the factors are simultaneously equal to zero (see Section 10.5). The new factor... 

In more complicated cases, canonical analysis of the different models can be useful to explore the experimental conditions, especially when ridge systems occur. Another technique to optimize several responses simultaneously is to weigh them together into one single criterion, a Desirability function, which can be used as a criterion for the optimization. This technique is not recommended as a general method for optimization of many responses. It can, however, be of value as a tool for evaluation of simultaneous mapping by response surface models. Desirability functions are discussed below. 

A quantitative model which describes the variation of the response in the optimum experimental domain can be accomplished by a quadratic response surface model. From such models a more precise location of the optimum conditions can be achieved. Canonical analysis of the model may reveal underlying interdependencies of the experimental variables, and this may give clues to an understanding of the basic mechanism of the reaction. 

Optimization. The influence of the reaction parameters on the yield of the reaction were studied using response surface methodology (RSM). A central composite design was used and both canonical analysis of the second-order equation and discussion of the isoresponse curves were used for the interpretation of the results. 

The technique allows immediate interpretation of the regression equation by including the linear and interaction (cross-product) terms in the constant term (To or stationary point), thus simplifying the subsequent evaluation of the canonical form of the regression equation. The first report of canonical analysis in the statistical literature was by Box and Wilson [37] for determining optimal conditions in chemical reactions. Canonical analysis, or canonical reduction, was described as an efficient method to explore an empirical response surface to suggest areas for further experimentation. In canonical analysis or canonical reduction, second-order regression equations... 

Response-surface methodology has been used extensively for determining areas of process operation providing maximum profit. For example, the succinct representation of the rate surface of Eq. (114) indicates that increasing values of X3 will increase the rate r. If some response other than reaction rate is considered to be more indicative of process performance (such as cost, yield, or selectivity), the canonical analysis would be performed on this response to indicate areas of improved process performance. This information... 

This is the case in canonical analysis of response surface model, where the coefficient matrix is symmetric. 

A second-order model with square terms can describe a variety of differently shaped response surfaces. It is rather difficult to comprehend how the surface is shaped by mere inspection of the algebraic expression of the model. A canonical analysis constitutes a mathematical transformation of the original model into a form which only contains quadratic terms ... 

Orthogonal matrices are useful and may be used to accomplish variable transformations which lead to simplification of many different type of problems. Orthogonal matrices were used in the canonical transformation of response surface modek. Such matrices are also used in Factor Analysis. 

Transformation to the canonical form is usually carried out by computer. Equation 6.1 may be used directly to optimize a single response (3). It is a purely numercal method, relatively difficult to master, and it is now little used by the experimenter. It is also possible, although more difficult, to apply the canonical analysis method to multiple responses. The technique has been mentioned at this point, not as an optimization method to be undertaken by the non-specialist, but to demonstrate the basic forms of the second order response surface. All the examples cited in this chapter show one or other of these different shapes. 

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