Design matrix

Restoring of SD of parameters of stress field is based on the effect of acoustoelasticity. Its fundamental problem is determination of relationship between US wave parameters and components of stresses. To use in practice acoustoelasticity for SDS diagnosing, it is designed matrix theory [Bobrenco, 1991]. For the description of the elastic waves spreading in the medium it uses matrices of velocity v of US waves spreading, absolute A and relative... 

Country of origin and state of ripeness of 16 olive oils. The last 4 columns contain the same information in the form of a coded design matrix... 

At this point we introduce the formal notation, which is commonly used in literature, and which is further used throughout this chapter. In the new notation we replace the parameter vector b in the calibration example by a vector x, which is called the state vector. In the multicomponent kinetic system the state vector x contains the concentrations of the compounds in the reaction mixture at a given time. Thus x is the vector which is estimated by the filter. The response of the measurement device, e.g., the absorbance at a given wavelength, is denoted by z. The absorbtivities at a given wavelength which relate the measured absorbance to the concentrations of the compounds in the mixture, or the design matrix in the calibration experiment (x in eq. (41.3)) are denoted by h. ... 

Table 5.9. Design matrix for three factors at two levels (+ and — stand for +1 and —1) ...

The coefficients a,- are estimated from the results of experiments carried out according to a design matrix such as Table 5.9 which shows a 23 plan matrix. The significance of the several factors are tested by comparing the coefficients with the experimental error, to be exact, by testing whether the confidence intervals Aai include 0 or not. The experimental error can be estimated by repeated measurements of each experiment or - as it is done frequently in a more effective way - by replications at the centre of the plan (so-called zero replications ), see Fig. 5.2. 

The following rows of the design matrix is generated by shifting the first row cyclically one place (N — 2 times). 

Table 5.10. Plackett-Burman design matrix lor N 8 experiments and consequently m = 7 factors (including dummy variables) at two levels... 

With Eq. (6.126) and a GAUSsian activation function the output of the hidden neurons (the RBF design matrix) becomes... 

F design matrix for linear regression (mxnp or nsxnp)... 

Finally, we turn to an entirely different question involving confidence limits. Suppose we were to carry out the experiments indicated by the design matrix of Equation 11.15 a second time. We would probably not obtain the same set of responses we did the first time (Equation 11.16), but instead would have a different Y matrix. This would lead to a different set of parameter estimates, B, and a predicted response surface that in general would not be the same as that shown in Figure 11.4. A third repetition of the experiments would lead to a third predicted response surface, and so on. The question, then, is what limits can we construct about these response surfaces so that in a given percentage of cases, those limits will include the entire true response 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... 

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... 

Although it is true that the first three columns of plus and minus signs in Table 14.3 are equivalent to the abbreviated coded experimental design matrix D, the signs in Table 14.3 are used for a slightly different purpose than they were Table 14.2. In fact, as we will see, the eight columns of signs in Table 14.3 are equivalent to the matrix of parameter coefficients, X. 

The experimental design matrix for the above nine panels (experiments) is... 

The X matrix that corresponds to Equation 14.50 is derived from the design matrix given in Equation 14.49 by adding an initial column of one s. Then, using abbreviated sign notation for the levels. 

Consider the situation of a researcher who believes that the rate of an enzyme catalyzed reaction is affected not only by factors such as temperature, substrate concentration, and pH (see Section 11.1), but also by the concentration of sodium ion ([Na ]) in solution with the enzyme. To investigate this hypothesis, the researcher designs a set of experiments in which all factors are kept constant but one the concentration of sodium ion is varied from 0 to 10 millimolar (mA/) according to the design matrix... 

A better description of the design matrix would involve not only the concentration of sodium ion (x mM), but also the time at which each experiment was carried out (xj, min). If we begin our measurement of time with the first experiment, then the design matrix for the previous set of experiments would be... 

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