Experimental design matrix

Measuring the multicomponent diffusion matrix experimental design and... 

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. 

In the development of a SE-HPLC method the variables that may be manipulated and optimized are the column (matrix type, particle and pore size, and physical dimension), buffer system (type and ionic strength), pH, and solubility additives (e.g., organic solvents, detergents). Once a column and mobile phase system have been selected the system parameters of protein load (amount of material and volume) and flow rate should also be optimized. A beneficial approach to the development of a SE-HPLC method is to optimize the multiple variables by the use of statistical experimental design. Also, information about the physical and chemical properties such as pH or ionic strength, solubility, and especially conditions that promote aggregation can be applied to the development of a SE-HPLC assay. Typical problems encountered during the development of a SE-HPLC assay are protein insolubility and column stationary phase... 

An experimental design Is sometimes taken to be a "test matrix" that Indicates the conditions associated with each test measurement. However, for statistical purposes the test matrix must reflect both the "experimental design" and the "data analysis" characteristics that are required to Identify separately the magnitudes of the different... 

The set of selected wavelengths (i.e. the experimental design) affects the variance-covariance matrix, and thus the precision of the results. For example, the set 22, 24 and 26 (Table 41.5) gives a less precise result than the set 22, 32 and 24 (Table 41.7). The best set of wavelengths can be derived in the same way as for multiple linear regression, i.e. the determinant of the dispersion matrix (h h) which contains the absorptivities, should be maximized. 

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... 

For inversion of the matrix XTX it is necessary that a sufficient number of spectra for different concentration steps have been measured. The concentration vectors must vary independently from each other. For this reason, experimental design (see, e.g., Deming and Morgan [1993]) can be... 

Sections on matrix algebra, analytic geometry, experimental design, instrument and system calibration, noise, derivatives and their use in data analysis, linearity and nonlinearity are described. Collaborative laboratory studies, using ANOVA, testing for systematic error, ranking tests for collaborative studies, and efficient comparison of two analytical methods are included. Discussion on topics such as the limitations in analytical accuracy and brief introductions to the statistics of spectral searches and the chemometrics of imaging spectroscopy are included. 

If we are dealing with a simple experimental system consisting of only two components, then one component is considered the unknown while the other is designated the matrix. The unknown component and the matrix are designated by the subscripts 1 and 2, respectively. Equation (12) can be rewritten as... 

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... 

Let us now locate a series of second experiments from x 2 = -5 to x 2 - +5. The X matrix for this changing experimental design may be represented as... 

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... 

Equation 7.1 is one of the most important relationships in the area of experimental design. As we have seen in this chapter, the precision of estimated parameter values is contained in the variance-covariance matrix V the smaller the elements of V, the more precise will be the parameter estimates. As we shall see in Chapter 11, the precision of estimating the response surface is also directly related to V the smaller the elements of V, the less fuzzy will be our view of the estimated surface. 

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. 

The lower left panel in Figure 13.2 shows the central composite design in the two factors X, and X2. The factor domain extends from -5 to +5 in each factor dimension. The coordinate axes in this panel are rotated 45° to correspond to the orientation of the axes in the panel above. Each black dot represents a distinctly different factor combination, or design point. The pattern of dots shows a central composite design centered at (Xj = 0, Xj = 0). The factorial points are located 2 units from the center. The star points are located 4 units from the center. The three concentric circles indicate that the center point has been replicated a total of four times. The experimental design matrix is... 

Figure 13.3 shows a similar set of four panels for a slightly different central composite design. The lower left panel shows the placement of experiments in factor space (i.e., it shows the experimental design). The upper left panel shows the normalized uncertainty as a function of factors x, and x. The upper right panel shows the normalized information as a function of factors x, and Xj. The lower right panel plots normalized information as a function of factor x, for X2 = -5, -4, -3, -2, -1, and 0. The experimental design matrix is... 

Figure 13.5 shows still another central composite design. The experimental design matrix is... 

The rotatable central composite design in Figure 13.7 is related to the rotatable central composite design in Figure 13.3 through expansion by a factor of V2 the square points expand from 2 to 2 2 from the center the star points expand from 2 2 to 4 from the center. The experimental design matrix is... 

In Figure 13.9, instead of carrying out four replicate experiments at the center point (as in Figure 13.2), the four replicates are carried out such that one experiment is moved to each of the existing four factorial points. 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. 

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