Analysis canonical

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. 

To find the coordinates of the stationary point, we first differentiate the full second-order polynomial model with respect to each of the factors and set each derivative equal to zero. For two-factor models we obtain 

The coordinates of the stationary point (s, and 5,2) re those values of x, and Xj that simultaneously satisfy both of these partial derivatives. Equation 12.66 may be rewritten as 

Let us also define a it x 1 matrix of first-order parameter estimates, f  

Finally, we define a k x k matrix of second-order parameter estimates, S  

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Thus, we see that CCA forms a canonical analysis, namely a decomposition of each data set into a set of mutually orthogonal components. A similar type of decomposition is at the heart of many types of multivariate analysis, e.g. PCA and PLS. Under the assumption of multivariate normality for both populations the canonical correlations can be tested for significance [6]. Retaining only the significant canonical correlations may allow for a considerable dimension reduction. 

R. Gittins, Canonical Analysis. A Review with Applications in Ecology. Springer-Verlag, Berlin, 1985. 

Canonical analysis, or canonical reduction, is a technique used to reduce a second-order regression equation, such as Eq. (9), to an equation consisting of a constant and squared terms, as follows ... 

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... 

A reported application of canonical analysis involved a novel combination of the canonical form of the regression equation with a computer-aided grid search technique to optimize controlled drug release from a pellet system prepared by extrusion and spheronization [28,29]. Formulation factors were used as independent variables, and in vitro dissolution was the main response, or dependent variable. Both a minimum and a maximum drug release rate was predicted and verified by preparation and testing of the predicted formulations. Excellent agreement between the predicted values and the actual values was evident for the four-component pellet system in this study. 

Fig. 14 Two-dimensional representation of the rigid rotation and translation involved in canonical analysis. 

R. E. O Connor, N. R. Bohidar, and J. B. Schwartz, Optimization by Canonical Analysis Applied to controlled Release Pellets, Abstracts of the 1st National AAPA Meeting. Washington, DC. Nov. 2-6, 1986. 

The parameter estimates obtained by a linear least-squares analysis of the original coded data (Cl) are shown in Table XV. After a canonical analysis, this equation becomes... 

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... 

Anticipating a later section on canonical analysis of second-order polynomial models, we will show that the first-order term can be made to equal zero if we code the model using the stationary point as the center of the symmetrical design. For this new system of coding, c, = 10 2/3 and (see Section 8.5). 

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... 

Canonical analysis of Equation 11.63 gives y2 = 8.008 — 0.6285xf — 0.08280xf. Compare this with Equation 11.80 which suggests that yx = 8.009 — 0.1571x — 0.02069x2, keeping in mind that in translation of axes, dXi = dXi = 1 whereas in Equation 11.63, d = dX2 = 2. Comment. 

FIGURE 4.8 Canonical analysis of Emmental cheeses from different countries. (From Shintu and Caldarelli, 2006.)... 

FIGURE 4.16 Canonical analysis score plot performed considering 66 dried meat samples coming from different countries (CH, Switzerland BR, Brazil CA, Canada US, USA AU, Australia). (From Shintu ef ai, 2007.)... 

In the case of an extreme experiment, we are faced with determining optimum coordinates from the obtained mathematical model. In that case, canonical analysis or methods of nonlinear programming are mostly used. The obtained optimum coordinates on a research, lab level are the starting point for a switch from lab to pilot-plant or full-scale levels3 . The procedure is in principle repeated in a larger sys-... 

Sometimes canonical correlation or canonical analysis is referred to as a central technique with factor and correspondence analysis considered in one branch (having no causal concepts) and multivariate regression and discriminant analysis in the other branch (based on causal concepts). 

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