Responses estimated, matrix

Using matrix least squares techniques (see Section 5.2), the chosen linear model may be fit to the data to obtain a set of parameter estimates, B, from which predicted values of response, y, may be obtained. It is convenient to define a matrix of estimated responses, F. 

In a sense, calculating the mean replicate response removes the effect of purely experimental uncertainty from the data. It is not unreasonable, then, to expect that the deviation of these mean replicate responses from the estimated responses is due to a lack of fit of the model to the data. The matrix of lack-of-fit deviations, L, is obtained by subtracting f from J... 

Using this response matrix and the known concentration of only one of the components (c), the regression coefficients in Equation 5-19 can be estimated as... 

In matrix notation, the model is written by augmenting the instrument response vector, x, with a column of ones, producing the response matrix as shown in the middle of Figure 5.2 (y = Xb + e). Least-squares estimates of the model parameters bQ and bl are computed by... 

In the mid-1980s a new technique called rank annihilation factor analysis (RAFA and GRAM) was developed which performed this process non-iteratively [42, 43]. Now an exact solution can be estimated directly on the sample matrix if the pure component response matrix of the analyte of interest is known. The third technique uses a range of concentrations yielding several pure component data matrices. This approach, called trilinear decomposition (TLD), stabilizes the prediction of the unknown sample due to the range of known response matrices... 

The second step in PCR is the estimation of the regression vector b. Either a subset of the scores, U, or the random-error reduced response matrix, R, can be related to the dependent variable, c. However, both options are equivalent. The orthonormal character of the scores and loadings make... 

Ma et al. [1997] have discussed an in vitro human placenta model for drug testing. This was a two-compartment perfusion system using human trophoblast cells attached to a chemically modified polyethylene terephthalate fibrous matrix as a cell culture scaffold. This system is a CCA in the same sense as the two-compartment system used to estimate response to dioxin. 

Preprocessing of multivariate and multiway data sets prior to regression and discriminant analysis follow the general principles outlined earher with few exceptions. In general, the response matrix (i.e., the data that are to be predicted) should be mean centered because this serves an additional purpose in regression and classification models. By centering both the dependent and independent variables, any possible differences in offsets are ranoved. Row normalization can be implemented if the priority is to establish a relationship between variables, rather than estimate the magnitude of the response, or to stabilize the impact of differently concentrated samples on models, as previously described. For example, if the calibration model is intended to predict a concentration from data that follow the Beer-Lambert law (e.g., fluorescence), then it is crucial not to normalize as this would cause the loss of concentration information. If, on the other hand, the model is intended to classify samples, then normahzation may help the model focus on patterns rather than on concentration-induced variations. 

The aim of all the foregoing methods of factor analysis is to decompose a data-set into physically meaningful factors, for instance pure spectra from a HPLC-DAD data-set. After those factors have been obtained, quantitation should be possible by calculating the contribution of each factor in the rows of the data matrix. By ITTFA (see Section 34.2.6) for example, one estimates the elution profiles of each individual compound. However, for quantitation the peak areas have to be correlated to the concentration by a calibration step. This is particularly important when using a diode array detector because the response factors (absorptivity) may considerably vary with the compound considered. Some methods of factor analysis require the presence of a pure variable for each factor. In that case quantitation becomes straightforward and does not need a multivariate approach because full selectivity is available. 

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. ... 

This choice of Qi yields maximum likelihood estimates of the parameters if the error terms in each response variable and for each experiment (eu, i=l,...N j=l,...,w) are all identically and independently distributed (i.i.d) normally with zero mean and variance, o . Namely, (e,) = 0 and COV(s,) = a I where I is the mxm identity matrix. 

Thus, the error in the solution vector is expected to be large for an ill-conditioned problem and small for a well-conditioned one. In parameter estimation, vector b is comprised of a linear combination of the response variables (measurements) which contain the error terms. Matrix A does not depend explicitly on the response variables, it depends only on the parameter sensitivity coefficients which depend only on the independent variables (assumed to be known precisely) and on the estimated parameter vector k which incorporates the uncertainty in the data. As a result, we expect most of the uncertainty in Equation 8.29 to be present in Ab. 

Step 2. Given the current estimate of the parameters, k , compute the parameter sensitivity matrix, G the response variables f(x k[Pg.161]

Step 6. Based on the additional measurement of the response variables, estimate the parameter vector and its covariance matrix. 

Validate routine methods, i.e., define the conditions under which the assay results are meaningful.115 To do that, one must select samples that are truly representative of the product stream. This may be a difficult task when the process is still under development and the product stream variable. The linearity of detector response should be defined over a range much broader than that expected to be encountered. Interference from the sample matrix and bias from analyte loss in preparation or separation often can be inferred from studies of linearity. Explicit detection or quantitation limits should be established. The precision (run-to-run repeatability) and accuracy (comparison with known standards) can be estimated with standards. Sample stability should be explored and storage conditions defined. 

It can be shown that if the uncertainties associated with the measurements of the response are approximately normally distributed (see Equation 3.8), then parameter estimates obtained from these measurements are also normally distributed. The standard deviation of the estimate of a parameter will be called the standard uncertainty, s, of the parameter estimate (it is usually called the standard error ) and can be calculated from the matrix if an estimate of is available. 

Both the parameter estimate bo and the standard uncertainty of its estimate depend on the Y matrix of experimental responses (see Equations 5.36, 5.28, and 5.43) if one set of experiments yields responses that agree closely, the standard uncertainty of bg will be small if a different set of experiments happens to produce responses that are dissimilar, the standard uncertainty of bg will be large. Thus, not only is the estimation of the parameter Pq itself subject to uncertainty, but the estimation of its standard uncertainty is also subject to uncertainty. 

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. 

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 ... 

It is easy to see that the estimate of (the MEAN in the classical treatment) is obtained by multiplying each value of response in the single column of the Y matrix by a +1 from the top row of the X matrix in Equation 14.7, and then multiplying the sum of products by one-eighth (or, equivalently, dividing the sum by eight). This division by eight is the source of the divisor listed for the MEAN in Table 14.3. 

Assume the gloss retention responses associated with Equation 14.49 are (in order) 98, 84, 70, 106, 92, 90, 114, 112, and 98. Using matrix least squares and the model of Equation 14.50, what is the estimated effect of the additive (h,) Using matrix... 

Fundamental Parameters (FP) are universal standardless, factory built-in calibration programs that describe the physics of the detector s response to pure elements, correction factors for overlapping peaks, and a number of other parameters to estimate element concentration while theoretically correcting for matrix discrepancies (e.g., Figure 1987). FP should be used for accurately measuring samples of unknown chemical composition in which concentrations of light and heavy elements may vary from ppm to high percent levels. 

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