Experimental design model matrix

Kelkar and McCarthy (1995) proposed another method to use the feedforward experiments to develop a kinetic model in a CSTR. An initial experimental design is augmented in a stepwise manner with additional experiments until a satisfactory model is developed. For augmenting data, experiments are selected in a way to increase the determinant of the correlation matrix. The method is demonstrated on kinetic model development for the aldol condensation of acetone over a mixed oxide catalyst. 

If matrix A is ill-conditioned at the optimum (i.e., at k=k ), there is not much we can do. We are faced with a truly ill-conditioned problem and the estimated parameters will have highly questionable values with unacceptably large estimated variances. Probably, the most productive thing to do is to reexamine the structure and dependencies of the mathematical model and try to reformulate a better posed problem. Sequential experimental design techniques can also aid us in... 

Because variables in models are often highly correlated, when experimental data are collected, the xrx matrix in Equation 2.9 can be badly conditioned (see Appendix A), and thus the estimates of the values of the coefficients in a model can have considerable associated uncertainty. The method of factorial experimental design forces the data to be orthogonal and avoids this problem. This method allows you to determine the relative importance of each input variable and thus to develop a parsimonious model, one that includes only the most important variables and effects. Factorial experiments also represent efficient experimentation. You systematically plan and conduct experiments in which all of the variables are changed simultaneously rather than one at a time, thus reducing the number of experiments needed. 

D of the X matrix be nonzero. This latter requirement can be seen from Equations 5.14 and 5.15. Elements a and c of the X matrix associated with the present model are both equal to unity (see Equations 5.10 and 5.7) thus, with this model, the condition for a nonzero determinant (see Equation 5.12) is that element b (x,) not equal element d (, 2). When the experimental design consists of two experiments carried out at different levels of the factor jc, (x, x 2 see Figure 5.1), the condition is satisfied. 

Because the determinant is equal to zero, the (X X) matrix cannot be inverted, and a unique solution does not exist. An interpretation of the zero determinant is that the slope P, and the response intercept Po are both undefined (see Equations 5.14 and 5.15). This interpretation is consistent with the experimental design used and the model attempted the best straight line through the two points would have infinite slope (a vertical line) and the response intercept would not exist (see Figure 5.8). 

In this chapter, we will examine the variance-covariance matrix to see how the location of experiments in factor space (i.e., the experimental design) affects the individual variances and covariances of the parameter estimates. Throughout this section we will be dealing with the specific two-parameter first-order model y, = Pq + + li only the resulting principles are entirely general, however, and can be... 

Consideration of the effect of experimental design on the elements of the variance-covariance matrix leads naturally to the area of optimal design [Box, Hunter, and Hunter (1978), Evans (1979), and Wolters and Kateman (1990)]. Let us suppose that our purpose in carrying out two experiments is to obtain good estimates of the intercept and slope for the model yj, = Po + Pi i, + r,. We might want to know what levels of the factor x , we should use to obtain the most precise estimates of po and... 

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

Verify that the (X X) matrix obtained for the model of Equation 12.24 and the experimental design of Figure 12.9 cannot be inverted. 

Figure 13.14 shows a star design that can be used to fit a two-factor FSOP model. The experimental design matrix is... 

The striking feature of this design is the set of six spikes in both the normalized uncertainty and normalized information surfaces. These spikes are an extreme expression of the basic idea that experiments provide information. Even if the experimental design is not a good match for the model even if the iX X) matrix is ill conditioned even if the model doesn t fit the data very well, there is still high-quality information at the points where experiments have been carried out. 

From the technical viewpoint, the matrix inversion (C C) in Equation 12.36 can be very unstable if any two of the analyte concentrations in the calibration standards happen to be highly correlated to one another. This translates to the need for careful experimental design in the preparation of calibration standards for CLS modeling, which is particularly challenging because multiple constituents must be considered. In addi-... 

It first introduces the reader to the fundamentals of experimental design. Systems theory, response surface concepts, and basic statistics serve as a basis for the further development of matrix least squares and hypothesis testing. The effects of different experimental designs and different models on the variance-covariance matrix and on the analysis of variance (ANOVA) are extensively discussed. Applications and advanced topics such as confidence bands, rotatability, and confounding complete the text. Numerous worked examples are presented. 

Response Surfaces. 3. Basic Statistics. 4. One Experiment. 5. Two Experiments. 6. Hypothesis Testing. 7. The Variance-Covariance Matrix. 8. Three Experiments. 9. Analysis of Variance (ANOVA) for Linear Models. 10. A Ten-Experiment Example. 11. Approximating a Region of a Multifactor Response Surface. 12. Additional Multifactor Concepts and Experimental Designs. Append- ices Matrix Algebra. Critical Values of t. Critical Values of F, a = 0.05. Index. 

At this point, it is helpful to introduce some notation that will be used to further describe experimental designs and response-surface modeling. As was described earlier, all possible operating conditions are represented as combinations of the values of the input variables. Each particular combination is a point in the operating region of a process and is called a treatment. These sets of points can be denoted in matrix form... 

Also, we use SL to denote a set of L experimental points in the same factor space, called candidate points. The set of candidate points will be used as a source of points that might possibly be included in the experimental design, XN. The information matrix of the /V-point design, XN, will be denoted as above by Mw = FT where Mw is the information matrix for some model (Equation 8.59). By the following formula, we denote the variance of the prediction at point x-. 

When only one factor is involved in the experiment, the predictive ability is often visualised by confidence bands. The size of these confidence bands depends on the magnitude of the experimental error. The shape , however, depends on the experimental design, and can be obtained from the design matrix (Section 2.2.3) and is influenced by the arrangement of experiments, replication procedure and mathematical model. The concept of leverage is used as a measure of such confidence. The mathematical definition is given by... 

The true model parameters (ft, ftj(...) are partial derivatives of the response function / and cannot be measured directly. It is, however, possible to otain estimates, ft, bV], bVl, of these parameters by multiple regression methods in which the polynomial model is fitted to known experimental results obtained by varying the settings of xr. These variations will then define an experimental design and are conveniently displayed as a design matrix, D, in which the rows describe the settings in the individual experiments and the columns describe the variations of the experimental variables over the series of experiments. 

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