Single Response Models

Let us first consider models that have only one measured variable (w=l). We shall consider two cases. One where we know precisely the value of the experimental error variance and the other when we have an estimate of it. Namely, there is quantifiable uncertainty in our estimate of the experimental error variance. 

In this case we assume that we know precisely the value of the standard experimental error in the measurements (of). Using Equation 11.2 we obtain an estimate of the experimental error variance under the assumption that the model is adequate. Therefore, to test whether the model is adequate we simply need to test the hypothesis 

Since Og is known exactly (i.e., there is no uncertainty in its value, it is a given number) the above hypothesis test is done through a y2-test. Namely, 

Let us assume that Og is not known exactly, however, we have performed n repeated measurements of the response variable. From this small sample of multiple measurements we can determine the sample mean and sample variance. If 

Sg is the sample estimate of Og, estimated from the n repeated measurements it is given by 

Minimum Volume Design. The volume of the hyperellipsoid is given by 

Spherical Shape Design. The next experiment is designed at those settings which maximize the smallest eigenvalue of X X, which leads to the maximum contraction of the largest principal axis of the ellipsoid [Hosten, 1974], When the models are nonlinear, Xis replaced by J. 

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The above procedure will be followed in the following three examples with two, three and four parameter linear single response models... 

Obviously, it is very important that the next experiment has maximum discriminating power. Let us illustrate this point with a very simple example where simple common sense arguments can lead us to a satisfactory design. Assume that we have the following two rival single-response models, each with two parameters and one independent variable ... 

Box and Hill (1967) proposed a criterion that incorporates the uncertainties associated with model predictions. For two rival single-response models the proposed divergence expression takes the form,... 

The treatment below is initially restricted to single-response models in which only one dependent variable is considered. At the end of this section some remarks on the treatment of multircsponse models are given. 

The methods of Chapter 6 are not appropriate for multiresponse investigations unless the responses have known relative precisions and independent, unbiased normal distributions of error. These restrictions come from the error model in Eq. (6.1-2). Single-response models were treated under these assumptions by Gauss (1809, 1823) and less completely by Legendre (1805), co-discoverer of the method of least squares. Aitken (1935) generalized weighted least squares to multiple responses with a specified error covariance matrix his method was extended to nonlinear parameter estimation by Bard and Lapidus (1968) and Bard (1974). However, least squares is not suitable for multiresponse problems unless information is given about the error covariance matrix we may consider such applications at another time. 

For the single-response model, using a quadratic expansion for S(0), the likelihood function (8.101) is data-translated by the least-squares estimate 0ls- Thus the noninformative prior is uniform in this same parameter 0, p 0) c, at least in the region of appreciable nonzero likelihood. Again, a uniform prior in 0 is justified on the grounds of being noninformative. The Bayesian most probable estimate 0m therefore agrees with the maximum likehhood and least squares estimates. 

Search methods consist of the optimization of an appropriate objective function, 0(0), which is usually the likelihood or posterior distribution functions or, for a single-response model with a normal distribution of errors, the sum of squares of the residuals. The latter case is implied in the following discussion. Various gradient and direct search techniques for single-response objective functions are discussed elsewhere (see, e.g., Bard, 1974, Chap. 5 Draper and Smith, 1981, Chap. 10). 

The original formulations of MPC (i.e., DMC and IDCOM) were based on empirical hnear models expressed in either step-response or impulse-response form. For simphcity, we will consider only a singleinput, single-output (SISO) model. However, the SISO model can be easily generalized to the MIMO models that are used in industrial applications. The step response model relating a single controlled variable y and a single manipiilated variable u can be expressed as... 

Analysis of most (perhaps 65%) pharmacokinetic data from clinical trials starts and stops with noncompartmental analysis (NCA). NCA usually includes calculating the area under the curve (AUC) of concentration versus time, or under the first-moment curve (AUMC, from a graph of concentration multiplied by time versus time). Calculation of AUC and AUMC facilitates simple calculations for some standard pharmacokinetic parameters and collapses measurements made at several sampling times into a single number representing exposure. The approach makes few assumptions, has few parameters, and allows fairly rigorous statistical description of exposure and how it is affected by dose. An exposure response model may be created. With respect to descriptive dimensions these dose-exposure and exposure-response models... 

The simple linear regression model which has a single response variable, a single independent variable and two unknown parameters. 

For the single response linear regression model (w=l), Equations (3.17a) and (3.17b) reduce to... 

Problems that can be described by a multiple linear regression model (i.e., they have a single response variable, 1) can be readily solved by available software. We will demonstrate such problems can be solved by using Microsoft Excel and SigmaPlot . 

The direct optimization of a single response formulation modelled by either a normal or pseudocomponent equation is accomplished by the incorporation of the component constraints in the Complex algorithm. Multiresponse optimization to achieve a "balanced" set of property values is possible by the combination of response desirability factors and the Complex algorithm. Examples from the literature are analyzed to demonstrate the utility of these techniques. 

In principle, any type of process model can be used to predict future values of the controlled outputs. For example, one can use a physical model based on first principles (e.g., mass and energy balances), a linear model (e.g., transfer function, step response model, or state space-model), or a nonlinear model (e.g., neural nets). Because most industrial applications of MPC have relied on linear dynamic models, later on we derive the MPC equations for a single-input/single-output (SISO) model. The SISO model, however, can be easily generalized to the MIMO models that are used in industrial applications (Lee et al., 1994). One model that can be used in MPC is called the step response model, which relates a single controlled variable y with a single manipulated variable u (based on previous changes in u) as follows ... 

The spring is elastically storing energy. With time this energy is dissipated by flow within the dashpot. An experiment performed using the application of rapid stress in which the stress is monitored with time is called a stress relaxation experiment. For a single Maxwell model we require only two of the three model parameters to describe the decay of stress with time. These three parameters are the elastic modulus G, the viscosity r and the relaxation time rm. The exponential decay described in Equation (4.16) represents a linear response. As the strain is increased past a critical value this simple decay is lost. 

This result, that the low frequency limit of the in phase component of the viscosity equates to the viscosity of the dashpot, means that for a single Maxwell model it is possible to replace rj by rj(0). Thus far we have concentrated on the description of experimental responses to the application of a strain. Similar constructions can be developed for the application of a stress. For example the application of an oscillating stress to a sample gives rise to an oscillating strain. We can define a complex compliance J which is the ratio of the strain to the stress. We will explore the relationship between different experiments and the resulting models in Section 4.6. 

A stress relaxation experiment can be performed on a wide range of materials. If we perform such a test on a real material a number of deviations are normally observed from the behaviour of a single Maxwell model. Some of these deviations are associated with the application of the strain itself. For example it is very difficult to apply an instantaneous strain to a sample. This influences the measured response at short experimental times. It is often difficult to apply a strain small enough to provide a linear response. A Maxwell model is only applicable to linear responses. Even if you were to imagine an experiment where a strain is... 

This is the measure used in a test where concentration or dose is plotted on the x axis and the percentage of maximum response is plotted on the y axis. It is a laboratory result of a test performed under a single set of circumstances or on a single animal model. 

The multivariate element of chemometrics indicates that more than one response variable of the analyzer is used to build a model. This is often done out of necessity, because no single response variable from the analyzer has sufficient selectivity to monitor a specific property without experiencing interferences from other properties. 

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