Parameter estimation multiple regression

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. 

From this conclusion follows, that a factorial design can be used to fit a response surface model to account for main effects and interaction effects. In the concluding section of this chapter is discussed how the properties of the model matrix X influence the quality of the estimated parameters in multiple regression. It is shown that factorial design have optimum qualities. 

The following expressions can be used to estimate the temperature and enthalpy of steam. The expressions are based upon multiple regression analysis. The equation for temperature is accurate to within 1.5 % at 1,000 psia. The expression for latent heat is accurate to within + 3 % at 1,000 psia. Input data required to use these equations is the steam pressure in psia. The parameters in the equations are defined as t for temperature in F, for latent heat in Btu/lb, and P for pressure in psia. 

Experimental polymer rheology data obtained in a capillary rheometer at different temperatures is used to determine the unknown coefficients in Equations 11 - 12. Multiple linear regression is used for parameter estimation. The values of these coefficients for three different polymers is shown in Table I. The polymer rheology is shown in Figures 2 - 4. 

Techniques for parameter estimation vary considerably. If consistent values for model parameters cannot be obtained, the investigators may decide that the model is itself unreliable and should be changed. Thus, model choice and parameter estimation are interactive. A number of workers have discussed generalised procedures [16—18]. Yeh [19] developed numerical algorithms and showed that multiple linear regression could be used successfully if the reaction scheme consisted of steps such as those shown in eqn. (41) or... 

The higher-level parameters in the multiple regression model enable quantification of how chemicals influence each other in relation to the measured response. Suppose that fil 2 (the estimated function parameter for the first-level interaction term between chemical 1 and chemical 2) in a regression model... 

To undertake the parameters estimation of the rate constants, deactivation coefficient and coking thermal factors, a combination linear and non-linear multiple parameters regression techniques were applied. The form of deactivation coefficient can be expressed as ... 

Even though many different covariates may be collected in an experiment, it may not be desirable to enter all these in a multiple regression model. First, not all covariates may be statistically significant—they have no predictive power. Second, a model with too many covariates produces models that have variances, e.g., standard errors, residual errors, etc., that are larger than simpler models. On the other hand, too few covariates lead to models with biased parameter estimates, mean square error, and predictive capabilities. As previously stated, model selection should follow Occam s razor, which basically states the simpler model is always chosen over more complex models. ... 

K. C. Yeh, Kinetic parameter estimation by numerical algorithms and multiple liner regression Theoretical, J. Pharm. Sci. 66,1688-1691 (1977). 

Holloway (12) has derived a set of parameters for estimating the NMR chemical shifts of organotin derivatives from a multiple regression analysis on a large body of data. Such a set of parameters could prove valuable in the assigning of chemical shifts in complex tin compounds. 

Full details for estimation of the model are given in Schwartz et al. (1994, p. 214). Application of this model toward multiple-parameter estimation is illustrated by Schwartz et al. (1994) a similar model and estimation procedure has been applied to longitudinal regression (Jacob et al., in press). These models al-... 

Although Eq. [29] has the appearance of a multiple regression, remember that the parameter estimates were not calculated by OLS. Instead they were found by a biased regression method. Consequently, these parameters, which are referred to as pseudo-p s, will not in general equal the OLS values because they have been shrunken, (some more than others). However, as more components are added into the latent variable model (Eq. [27]), i.e., as p increases toward k, these pseudo-p s approach the values obtained by OLS. In the limit p = k, Eq. [29] will be identical to the OLS model, a result that will be illustrated later when we apply the methods to a real dataset. 

The partial derivative is obtained from parameters found by regressing vapor-liquid equilibrium (VLE) data for the solvent mixture. The estimated solubilities are not very sensitive to these parameters. There are multiple ways to obtain the Ay values. One is to use data from solute solubilities in the pure solvents and regress the parameters A°2 and A separately. Then,... 

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